3.106 \(\int (d+e x)^3 (f+g x+h x^2+i x^3) (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=684 \[ d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right ) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right ) \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e^3 i x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {b e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{36 c}+\frac {b e^3 i x^6 \sqrt {1-c^2 x^2}}{49 c}+\frac {b e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{1225 c^3}+\frac {b x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{144 c^3}-\frac {b \sin ^{-1}(c x) \left (24 c^4 d^2 (d g+3 e f)+9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{96 c^6}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (1225 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+588 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+360 e^3 i\right )}{11025 c^5}+\frac {b \sqrt {1-c^2 x^2} \left (3675 c^2 x \left (24 c^4 d^2 (d g+3 e f)+9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )+32 \left (11025 c^6 d^3 f+2450 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+1176 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+720 e^3 i\right )\right )}{352800 c^7} \]

[Out]

-1/96*b*(24*c^4*d^2*(d*g+3*e*f)+5*e^2*(3*d*i+e*h)+9*c^2*(d^3*i+3*d^2*e*h+3*d*e^2*g+e^3*f))*arcsin(c*x)/c^6+d^3
*f*x*(a+b*arcsin(c*x))+1/2*d^2*(d*g+3*e*f)*x^2*(a+b*arcsin(c*x))+1/3*d*(d^2*h+3*d*e*g+3*e^2*f)*x^3*(a+b*arcsin
(c*x))+1/4*(d^3*i+3*d^2*e*h+3*d*e^2*g+e^3*f)*x^4*(a+b*arcsin(c*x))+1/5*e*(3*d^2*i+3*d*e*h+e^2*g)*x^5*(a+b*arcs
in(c*x))+1/6*e^2*(3*d*i+e*h)*x^6*(a+b*arcsin(c*x))+1/7*e^3*i*x^7*(a+b*arcsin(c*x))+1/11025*b*(1225*c^4*d*(d^2*
h+3*d*e*g+3*e^2*f)+360*e^3*i+588*c^2*e*(3*d^2*i+3*d*e*h+e^2*g))*x^2*(-c^2*x^2+1)^(1/2)/c^5+1/144*b*(5*e^2*(3*d
*i+e*h)+9*c^2*(d^3*i+3*d^2*e*h+3*d*e^2*g+e^3*f))*x^3*(-c^2*x^2+1)^(1/2)/c^3+1/1225*b*e*(30*e^2*i+49*c^2*(3*d^2
*i+3*d*e*h+e^2*g))*x^4*(-c^2*x^2+1)^(1/2)/c^3+1/36*b*e^2*(3*d*i+e*h)*x^5*(-c^2*x^2+1)^(1/2)/c+1/49*b*e^3*i*x^6
*(-c^2*x^2+1)^(1/2)/c+1/352800*b*(352800*c^6*d^3*f+78400*c^4*d*(d^2*h+3*d*e*g+3*e^2*f)+23040*e^3*i+37632*c^2*e
*(3*d^2*i+3*d*e*h+e^2*g)+3675*c^2*(24*c^4*d^2*(d*g+3*e*f)+5*e^2*(3*d*i+e*h)+9*c^2*(d^3*i+3*d^2*e*h+3*d*e^2*g+e
^3*f))*x)*(-c^2*x^2+1)^(1/2)/c^7

________________________________________________________________________________________

Rubi [A]  time = 6.26, antiderivative size = 684, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {4749, 12, 1809, 780, 216} \[ \frac {1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right ) \left (3 d^2 e h+d^3 i+3 d e^2 g+e^3 f\right )+\frac {1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right ) \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) \left (a+b \sin ^{-1}(c x)\right )+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 x^6 (3 d i+e h) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{7} e^3 i x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {b x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (3 d^2 e h+d^3 i+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{144 c^3}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (1225 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+588 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+360 e^3 i\right )}{11025 c^5}+\frac {b \sqrt {1-c^2 x^2} \left (3675 c^2 x \left (9 c^2 \left (3 d^2 e h+d^3 i+3 d e^2 g+e^3 f\right )+24 c^4 d^2 (d g+3 e f)+5 e^2 (3 d i+e h)\right )+32 \left (2450 c^4 d \left (d^2 h+3 d e g+3 e^2 f\right )+1176 c^2 e \left (3 d^2 i+3 d e h+e^2 g\right )+11025 c^6 d^3 f+720 e^3 i\right )\right )}{352800 c^7}-\frac {b \sin ^{-1}(c x) \left (9 c^2 \left (3 d^2 e h+d^3 i+3 d e^2 g+e^3 f\right )+24 c^4 d^2 (d g+3 e f)+5 e^2 (3 d i+e h)\right )}{96 c^6}+\frac {b e x^4 \sqrt {1-c^2 x^2} \left (49 c^2 \left (3 d^2 i+3 d e h+e^2 g\right )+30 e^2 i\right )}{1225 c^3}+\frac {b e^2 x^5 \sqrt {1-c^2 x^2} (3 d i+e h)}{36 c}+\frac {b e^3 i x^6 \sqrt {1-c^2 x^2}}{49 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]),x]

[Out]

(b*(1225*c^4*d*(3*e^2*f + 3*d*e*g + d^2*h) + 360*e^3*i + 588*c^2*e*(e^2*g + 3*d*e*h + 3*d^2*i))*x^2*Sqrt[1 - c
^2*x^2])/(11025*c^5) + (b*(5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i))*x^3*Sqrt[1 - c
^2*x^2])/(144*c^3) + (b*e*(30*e^2*i + 49*c^2*(e^2*g + 3*d*e*h + 3*d^2*i))*x^4*Sqrt[1 - c^2*x^2])/(1225*c^3) +
(b*e^2*(e*h + 3*d*i)*x^5*Sqrt[1 - c^2*x^2])/(36*c) + (b*e^3*i*x^6*Sqrt[1 - c^2*x^2])/(49*c) + (b*(32*(11025*c^
6*d^3*f + 2450*c^4*d*(3*e^2*f + 3*d*e*g + d^2*h) + 720*e^3*i + 1176*c^2*e*(e^2*g + 3*d*e*h + 3*d^2*i)) + 3675*
c^2*(24*c^4*d^2*(3*e*f + d*g) + 5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i))*x)*Sqrt[1
- c^2*x^2])/(352800*c^7) - (b*(24*c^4*d^2*(3*e*f + d*g) + 5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*
d^2*e*h + d^3*i))*ArcSin[c*x])/(96*c^6) + d^3*f*x*(a + b*ArcSin[c*x]) + (d^2*(3*e*f + d*g)*x^2*(a + b*ArcSin[c
*x]))/2 + (d*(3*e^2*f + 3*d*e*g + d^2*h)*x^3*(a + b*ArcSin[c*x]))/3 + ((e^3*f + 3*d*e^2*g + 3*d^2*e*h + d^3*i)
*x^4*(a + b*ArcSin[c*x]))/4 + (e*(e^2*g + 3*d*e*h + 3*d^2*i)*x^5*(a + b*ArcSin[c*x]))/5 + (e^2*(e*h + 3*d*i)*x
^6*(a + b*ArcSin[c*x]))/6 + (e^3*i*x^7*(a + b*ArcSin[c*x]))/7

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 1809

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[(f*(c*x)^(m + q - 1)*(a + b*x^2)^(p + 1))/(b*c^(q - 1)*(m + q + 2*p + 1)), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 4749

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_), x_Symbol] :> With[{u = IntHide[ExpandExpression[Px, x], x]}, Dis
t[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b,
c}, x] && PolynomialQ[Px, x]

Rubi steps

\begin {align*} \int (d+e x)^3 \left (f+g x+h x^2+106 x^3\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x \left (70 d^3 (6 f+x (3 g+x (2 h+159 x)))+21 d e^2 x^2 (20 f+x (15 g+4 x (3 h+265 x)))+21 d^2 e x (30 f+x (20 g+3 x (5 h+424 x)))+e^3 x^3 (105 f+2 x (42 g+5 x (7 h+636 x)))\right )}{420 \sqrt {1-c^2 x^2}} \, dx\\ &=d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{420} (b c) \int \frac {x \left (70 d^3 (6 f+x (3 g+x (2 h+159 x)))+21 d e^2 x^2 (20 f+x (15 g+4 x (3 h+265 x)))+21 d^2 e x (30 f+x (20 g+3 x (5 h+424 x)))+e^3 x^3 (105 f+2 x (42 g+5 x (7 h+636 x)))\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-2940 c^2 d^3 f-1470 c^2 d^2 (3 e f+d g) x-980 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right ) x^2-735 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^3-12 e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4-490 c^2 e^2 (318 d+e h) x^5\right )}{\sqrt {1-c^2 x^2}} \, dx}{2940 c}\\ &=\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \int \frac {x \left (17640 c^4 d^3 f+8820 c^4 d^2 (3 e f+d g) x+5880 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right ) x^2+490 c^2 \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3+72 c^2 e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{17640 c^3}\\ &=\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-88200 c^6 d^3 f-44100 c^6 d^2 (3 e f+d g) x-24 c^2 \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2-2450 c^4 \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3\right )}{\sqrt {1-c^2 x^2}} \, dx}{88200 c^5}\\ &=\frac {b \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \int \frac {x \left (352800 c^8 d^3 f+7350 c^4 \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x+96 c^4 \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{352800 c^7}\\ &=\frac {b \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{11025 c^5}+\frac {b \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-96 c^4 \left (76320 e^3+11025 c^6 d^3 f+2450 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+1176 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right )-22050 c^6 \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x\right )}{\sqrt {1-c^2 x^2}} \, dx}{1058400 c^9}\\ &=\frac {b \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{11025 c^5}+\frac {b \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+\frac {b \left (32 \left (76320 e^3+11025 c^6 d^3 f+2450 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+1176 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right )+3675 c^2 \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{352800 c^7}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{96 c^5}\\ &=\frac {b \left (38160 e^3+1225 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+588 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{11025 c^5}+\frac {b \left (5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e \left (3180 e^2+49 c^2 \left (318 d^2+e^2 g+3 d e h\right )\right ) x^4 \sqrt {1-c^2 x^2}}{1225 c^3}+\frac {b e^2 (318 d+e h) x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {106 b e^3 x^6 \sqrt {1-c^2 x^2}}{49 c}+\frac {b \left (32 \left (76320 e^3+11025 c^6 d^3 f+2450 c^4 d \left (3 e^2 f+3 d e g+d^2 h\right )+1176 c^2 e \left (318 d^2+e^2 g+3 d e h\right )\right )+3675 c^2 \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{352800 c^7}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^2 (318 d+e h)+9 c^2 \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right )\right ) \sin ^{-1}(c x)}{96 c^6}+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^2 (3 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d \left (3 e^2 f+3 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} \left (106 d^3+e^3 f+3 d e^2 g+3 d^2 e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e \left (318 d^2+e^2 g+3 d e h\right ) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^2 (318 d+e h) x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {106}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.95, size = 619, normalized size = 0.90 \[ a d^3 f x+\frac {1}{3} a d x^3 \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{5} a e x^5 \left (3 d^2 i+3 d e h+e^2 g\right )+\frac {1}{2} a d^2 x^2 (d g+3 e f)+\frac {1}{4} a x^4 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+\frac {1}{6} a e^2 x^6 (3 d i+e h)+\frac {1}{7} a e^3 i x^7-\frac {b \sin ^{-1}(c x) \left (24 c^4 d^2 (d g+3 e f)+9 c^2 \left (d^3 i+3 d^2 e h+3 d e^2 g+e^3 f\right )+5 e^2 (3 d i+e h)\right )}{96 c^6}+\frac {b \sqrt {1-c^2 x^2} \left (2 c^6 \left (1225 d^3 (144 f+x (36 g+x (16 h+9 i x)))+147 d^2 e x (900 f+x (400 g+9 x (25 h+16 i x)))+147 d e^2 x^2 (400 f+x (225 g+4 x (36 h+25 i x)))+e^3 x^3 (11025 f+4 x (1764 g+25 x (49 h+36 i x)))\right )+c^4 \left (1225 d^3 (64 h+27 i x)+147 d^2 e \left (1600 g+675 h x+384 i x^2\right )+147 d e^2 \left (1600 f+x \left (675 g+384 h x+250 i x^2\right )\right )+e^3 x \left (33075 f+2 x \left (9408 g+6125 h x+4320 i x^2\right )\right )\right )+3 c^2 e \left (37632 d^2 i+147 d e (256 h+125 i x)+e^2 (12544 g+5 x (1225 h+768 i x))\right )+23040 e^3 i\right )}{352800 c^7}+\frac {1}{420} b x \sin ^{-1}(c x) \left (35 d^3 (12 f+x (6 g+x (4 h+3 i x)))+21 d^2 e x (30 f+x (20 g+3 x (5 h+4 i x)))+21 d e^2 x^2 (20 f+x (15 g+2 x (6 h+5 i x)))+e^3 x^3 (105 f+2 x (42 g+5 x (7 h+6 i x)))\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(f + g*x + h*x^2 + i*x^3)*(a + b*ArcSin[c*x]),x]

[Out]

a*d^3*f*x + (a*d^2*(3*e*f + d*g)*x^2)/2 + (a*d*(3*e^2*f + 3*d*e*g + d^2*h)*x^3)/3 + (a*(e^3*f + 3*d*e^2*g + 3*
d^2*e*h + d^3*i)*x^4)/4 + (a*e*(e^2*g + 3*d*e*h + 3*d^2*i)*x^5)/5 + (a*e^2*(e*h + 3*d*i)*x^6)/6 + (a*e^3*i*x^7
)/7 + (b*Sqrt[1 - c^2*x^2]*(23040*e^3*i + 3*c^2*e*(37632*d^2*i + 147*d*e*(256*h + 125*i*x) + e^2*(12544*g + 5*
x*(1225*h + 768*i*x))) + c^4*(1225*d^3*(64*h + 27*i*x) + 147*d^2*e*(1600*g + 675*h*x + 384*i*x^2) + 147*d*e^2*
(1600*f + x*(675*g + 384*h*x + 250*i*x^2)) + e^3*x*(33075*f + 2*x*(9408*g + 6125*h*x + 4320*i*x^2))) + 2*c^6*(
1225*d^3*(144*f + x*(36*g + x*(16*h + 9*i*x))) + 147*d^2*e*x*(900*f + x*(400*g + 9*x*(25*h + 16*i*x))) + 147*d
*e^2*x^2*(400*f + x*(225*g + 4*x*(36*h + 25*i*x))) + e^3*x^3*(11025*f + 4*x*(1764*g + 25*x*(49*h + 36*i*x)))))
)/(352800*c^7) - (b*(24*c^4*d^2*(3*e*f + d*g) + 5*e^2*(e*h + 3*d*i) + 9*c^2*(e^3*f + 3*d*e^2*g + 3*d^2*e*h + d
^3*i))*ArcSin[c*x])/(96*c^6) + (b*x*(35*d^3*(12*f + x*(6*g + x*(4*h + 3*i*x))) + 21*d^2*e*x*(30*f + x*(20*g +
3*x*(5*h + 4*i*x))) + 21*d*e^2*x^2*(20*f + x*(15*g + 2*x*(6*h + 5*i*x))) + e^3*x^3*(105*f + 2*x*(42*g + 5*x*(7
*h + 6*i*x))))*ArcSin[c*x])/420

________________________________________________________________________________________

fricas [A]  time = 0.99, size = 936, normalized size = 1.37 \[ \frac {50400 \, a c^{7} e^{3} i x^{7} + 352800 \, a c^{7} d^{3} f x + 58800 \, {\left (a c^{7} e^{3} h + 3 \, a c^{7} d e^{2} i\right )} x^{6} + 70560 \, {\left (a c^{7} e^{3} g + 3 \, a c^{7} d e^{2} h + 3 \, a c^{7} d^{2} e i\right )} x^{5} + 88200 \, {\left (a c^{7} e^{3} f + 3 \, a c^{7} d e^{2} g + 3 \, a c^{7} d^{2} e h + a c^{7} d^{3} i\right )} x^{4} + 117600 \, {\left (3 \, a c^{7} d e^{2} f + 3 \, a c^{7} d^{2} e g + a c^{7} d^{3} h\right )} x^{3} + 176400 \, {\left (3 \, a c^{7} d^{2} e f + a c^{7} d^{3} g\right )} x^{2} + 105 \, {\left (480 \, b c^{7} e^{3} i x^{7} + 3360 \, b c^{7} d^{3} f x + 560 \, {\left (b c^{7} e^{3} h + 3 \, b c^{7} d e^{2} i\right )} x^{6} + 672 \, {\left (b c^{7} e^{3} g + 3 \, b c^{7} d e^{2} h + 3 \, b c^{7} d^{2} e i\right )} x^{5} + 840 \, {\left (b c^{7} e^{3} f + 3 \, b c^{7} d e^{2} g + 3 \, b c^{7} d^{2} e h + b c^{7} d^{3} i\right )} x^{4} + 1120 \, {\left (3 \, b c^{7} d e^{2} f + 3 \, b c^{7} d^{2} e g + b c^{7} d^{3} h\right )} x^{3} + 1680 \, {\left (3 \, b c^{7} d^{2} e f + b c^{7} d^{3} g\right )} x^{2} - 315 \, {\left (8 \, b c^{5} d^{2} e + b c^{3} e^{3}\right )} f - 105 \, {\left (8 \, b c^{5} d^{3} + 9 \, b c^{3} d e^{2}\right )} g - 35 \, {\left (27 \, b c^{3} d^{2} e + 5 \, b c e^{3}\right )} h - 105 \, {\left (3 \, b c^{3} d^{3} + 5 \, b c d e^{2}\right )} i\right )} \arcsin \left (c x\right ) + {\left (7200 \, b c^{6} e^{3} i x^{6} + 9800 \, {\left (b c^{6} e^{3} h + 3 \, b c^{6} d e^{2} i\right )} x^{5} + 288 \, {\left (49 \, b c^{6} e^{3} g + 147 \, b c^{6} d e^{2} h + 3 \, {\left (49 \, b c^{6} d^{2} e + 10 \, b c^{4} e^{3}\right )} i\right )} x^{4} + 2450 \, {\left (9 \, b c^{6} e^{3} f + 27 \, b c^{6} d e^{2} g + {\left (27 \, b c^{6} d^{2} e + 5 \, b c^{4} e^{3}\right )} h + 3 \, {\left (3 \, b c^{6} d^{3} + 5 \, b c^{4} d e^{2}\right )} i\right )} x^{3} + 32 \, {\left (3675 \, b c^{6} d e^{2} f + 147 \, {\left (25 \, b c^{6} d^{2} e + 4 \, b c^{4} e^{3}\right )} g + 49 \, {\left (25 \, b c^{6} d^{3} + 36 \, b c^{4} d e^{2}\right )} h + 36 \, {\left (49 \, b c^{4} d^{2} e + 10 \, b c^{2} e^{3}\right )} i\right )} x^{2} + 117600 \, {\left (3 \, b c^{6} d^{3} + 2 \, b c^{4} d e^{2}\right )} f + 9408 \, {\left (25 \, b c^{4} d^{2} e + 4 \, b c^{2} e^{3}\right )} g + 3136 \, {\left (25 \, b c^{4} d^{3} + 36 \, b c^{2} d e^{2}\right )} h + 2304 \, {\left (49 \, b c^{2} d^{2} e + 10 \, b e^{3}\right )} i + 3675 \, {\left (9 \, {\left (8 \, b c^{6} d^{2} e + b c^{4} e^{3}\right )} f + 3 \, {\left (8 \, b c^{6} d^{3} + 9 \, b c^{4} d e^{2}\right )} g + {\left (27 \, b c^{4} d^{2} e + 5 \, b c^{2} e^{3}\right )} h + 3 \, {\left (3 \, b c^{4} d^{3} + 5 \, b c^{2} d e^{2}\right )} i\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{352800 \, c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

1/352800*(50400*a*c^7*e^3*i*x^7 + 352800*a*c^7*d^3*f*x + 58800*(a*c^7*e^3*h + 3*a*c^7*d*e^2*i)*x^6 + 70560*(a*
c^7*e^3*g + 3*a*c^7*d*e^2*h + 3*a*c^7*d^2*e*i)*x^5 + 88200*(a*c^7*e^3*f + 3*a*c^7*d*e^2*g + 3*a*c^7*d^2*e*h +
a*c^7*d^3*i)*x^4 + 117600*(3*a*c^7*d*e^2*f + 3*a*c^7*d^2*e*g + a*c^7*d^3*h)*x^3 + 176400*(3*a*c^7*d^2*e*f + a*
c^7*d^3*g)*x^2 + 105*(480*b*c^7*e^3*i*x^7 + 3360*b*c^7*d^3*f*x + 560*(b*c^7*e^3*h + 3*b*c^7*d*e^2*i)*x^6 + 672
*(b*c^7*e^3*g + 3*b*c^7*d*e^2*h + 3*b*c^7*d^2*e*i)*x^5 + 840*(b*c^7*e^3*f + 3*b*c^7*d*e^2*g + 3*b*c^7*d^2*e*h
+ b*c^7*d^3*i)*x^4 + 1120*(3*b*c^7*d*e^2*f + 3*b*c^7*d^2*e*g + b*c^7*d^3*h)*x^3 + 1680*(3*b*c^7*d^2*e*f + b*c^
7*d^3*g)*x^2 - 315*(8*b*c^5*d^2*e + b*c^3*e^3)*f - 105*(8*b*c^5*d^3 + 9*b*c^3*d*e^2)*g - 35*(27*b*c^3*d^2*e +
5*b*c*e^3)*h - 105*(3*b*c^3*d^3 + 5*b*c*d*e^2)*i)*arcsin(c*x) + (7200*b*c^6*e^3*i*x^6 + 9800*(b*c^6*e^3*h + 3*
b*c^6*d*e^2*i)*x^5 + 288*(49*b*c^6*e^3*g + 147*b*c^6*d*e^2*h + 3*(49*b*c^6*d^2*e + 10*b*c^4*e^3)*i)*x^4 + 2450
*(9*b*c^6*e^3*f + 27*b*c^6*d*e^2*g + (27*b*c^6*d^2*e + 5*b*c^4*e^3)*h + 3*(3*b*c^6*d^3 + 5*b*c^4*d*e^2)*i)*x^3
+ 32*(3675*b*c^6*d*e^2*f + 147*(25*b*c^6*d^2*e + 4*b*c^4*e^3)*g + 49*(25*b*c^6*d^3 + 36*b*c^4*d*e^2)*h + 36*(
49*b*c^4*d^2*e + 10*b*c^2*e^3)*i)*x^2 + 117600*(3*b*c^6*d^3 + 2*b*c^4*d*e^2)*f + 9408*(25*b*c^4*d^2*e + 4*b*c^
2*e^3)*g + 3136*(25*b*c^4*d^3 + 36*b*c^2*d*e^2)*h + 2304*(49*b*c^2*d^2*e + 10*b*e^3)*i + 3675*(9*(8*b*c^6*d^2*
e + b*c^4*e^3)*f + 3*(8*b*c^6*d^3 + 9*b*c^4*d*e^2)*g + (27*b*c^4*d^2*e + 5*b*c^2*e^3)*h + 3*(3*b*c^4*d^3 + 5*b
*c^2*d*e^2)*i)*x)*sqrt(-c^2*x^2 + 1))/c^7

________________________________________________________________________________________

giac [B]  time = 0.53, size = 1976, normalized size = 2.89 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

1/7*a*i*x^7*e^3 + 1/2*a*d*i*x^6*e^2 + 3/5*a*d^2*i*x^5*e + 1/4*a*d^3*i*x^4 + 1/6*a*h*x^6*e^3 + 3/5*a*d*h*x^5*e^
2 + 3/4*a*d^2*h*x^4*e + 1/3*a*d^3*h*x^3 + 1/5*a*g*x^5*e^3 + 3/4*a*d*g*x^4*e^2 + a*d^2*g*x^3*e + b*d^3*f*x*arcs
in(c*x) + 1/4*a*f*x^4*e^3 + a*d*f*x^3*e^2 + a*d^3*f*x + 1/3*(c^2*x^2 - 1)*b*d^3*h*x*arcsin(c*x)/c^2 + (c^2*x^2
- 1)*b*d^2*g*x*arcsin(c*x)*e/c^2 + 1/4*sqrt(-c^2*x^2 + 1)*b*d^3*g*x/c + 3/4*sqrt(-c^2*x^2 + 1)*b*d^2*f*x*e/c
+ 1/2*(c^2*x^2 - 1)*b*d^3*g*arcsin(c*x)/c^2 + 1/3*b*d^3*h*x*arcsin(c*x)/c^2 + (c^2*x^2 - 1)*b*d*f*x*arcsin(c*x
)*e^2/c^2 + 3/2*(c^2*x^2 - 1)*b*d^2*f*arcsin(c*x)*e/c^2 + b*d^2*g*x*arcsin(c*x)*e/c^2 + 3/5*(c^2*x^2 - 1)^2*b*
d^2*i*x*arcsin(c*x)*e/c^4 + sqrt(-c^2*x^2 + 1)*b*d^3*f/c - 1/16*(-c^2*x^2 + 1)^(3/2)*b*d^3*i*x/c^3 - 3/16*(-c^
2*x^2 + 1)^(3/2)*b*d^2*h*x*e/c^3 + 1/2*(c^2*x^2 - 1)*a*d^3*g/c^2 + 1/4*b*d^3*g*arcsin(c*x)/c^2 + 1/4*(c^2*x^2
- 1)^2*b*d^3*i*arcsin(c*x)/c^4 + b*d*f*x*arcsin(c*x)*e^2/c^2 + 3/5*(c^2*x^2 - 1)^2*b*d*h*x*arcsin(c*x)*e^2/c^4
+ 3/2*(c^2*x^2 - 1)*a*d^2*f*e/c^2 + 3/4*b*d^2*f*arcsin(c*x)*e/c^2 + 3/4*(c^2*x^2 - 1)^2*b*d^2*h*arcsin(c*x)*e
/c^4 + 6/5*(c^2*x^2 - 1)*b*d^2*i*x*arcsin(c*x)*e/c^4 - 1/9*(-c^2*x^2 + 1)^(3/2)*b*d^3*h/c^3 + 5/32*sqrt(-c^2*x
^2 + 1)*b*d^3*i*x/c^3 - 3/16*(-c^2*x^2 + 1)^(3/2)*b*d*g*x*e^2/c^3 - 1/3*(-c^2*x^2 + 1)^(3/2)*b*d^2*g*e/c^3 + 1
5/32*sqrt(-c^2*x^2 + 1)*b*d^2*h*x*e/c^3 + 1/2*(c^2*x^2 - 1)*b*d^3*i*arcsin(c*x)/c^4 + 1/5*(c^2*x^2 - 1)^2*b*g*
x*arcsin(c*x)*e^3/c^4 + 3/4*(c^2*x^2 - 1)^2*b*d*g*arcsin(c*x)*e^2/c^4 + 6/5*(c^2*x^2 - 1)*b*d*h*x*arcsin(c*x)*
e^2/c^4 + 3/2*(c^2*x^2 - 1)*b*d^2*h*arcsin(c*x)*e/c^4 + 3/5*b*d^2*i*x*arcsin(c*x)*e/c^4 + 1/3*sqrt(-c^2*x^2 +
1)*b*d^3*h/c^3 - 1/16*(-c^2*x^2 + 1)^(3/2)*b*f*x*e^3/c^3 - 1/3*(-c^2*x^2 + 1)^(3/2)*b*d*f*e^2/c^3 + 15/32*sqrt
(-c^2*x^2 + 1)*b*d*g*x*e^2/c^3 + 1/12*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d*i*x*e^2/c^5 + sqrt(-c^2*x^2 + 1)*
b*d^2*g*e/c^3 + 3/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^2*i*e/c^5 + 5/32*b*d^3*i*arcsin(c*x)/c^4 + 1/4*(c^
2*x^2 - 1)^2*b*f*arcsin(c*x)*e^3/c^4 + 2/5*(c^2*x^2 - 1)*b*g*x*arcsin(c*x)*e^3/c^4 + 1/7*(c^2*x^2 - 1)^3*b*i*x
*arcsin(c*x)*e^3/c^6 + 3/2*(c^2*x^2 - 1)*b*d*g*arcsin(c*x)*e^2/c^4 + 1/2*(c^2*x^2 - 1)^3*b*d*i*arcsin(c*x)*e^2
/c^6 + 3/5*b*d*h*x*arcsin(c*x)*e^2/c^4 + 15/32*b*d^2*h*arcsin(c*x)*e/c^4 + 5/32*sqrt(-c^2*x^2 + 1)*b*f*x*e^3/c
^3 + 1/36*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*h*x*e^3/c^5 + sqrt(-c^2*x^2 + 1)*b*d*f*e^2/c^3 + 3/25*(c^2*x^2
- 1)^2*sqrt(-c^2*x^2 + 1)*b*d*h*e^2/c^5 - 13/48*(-c^2*x^2 + 1)^(3/2)*b*d*i*x*e^2/c^5 - 2/5*(-c^2*x^2 + 1)^(3/2
)*b*d^2*i*e/c^5 + 1/2*(c^2*x^2 - 1)*b*f*arcsin(c*x)*e^3/c^4 + 1/6*(c^2*x^2 - 1)^3*b*h*arcsin(c*x)*e^3/c^6 + 1/
5*b*g*x*arcsin(c*x)*e^3/c^4 + 3/7*(c^2*x^2 - 1)^2*b*i*x*arcsin(c*x)*e^3/c^6 + 15/32*b*d*g*arcsin(c*x)*e^2/c^4
+ 3/2*(c^2*x^2 - 1)^2*b*d*i*arcsin(c*x)*e^2/c^6 + 1/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*g*e^3/c^5 - 13/144
*(-c^2*x^2 + 1)^(3/2)*b*h*x*e^3/c^5 - 2/5*(-c^2*x^2 + 1)^(3/2)*b*d*h*e^2/c^5 + 11/32*sqrt(-c^2*x^2 + 1)*b*d*i*
x*e^2/c^5 + 3/5*sqrt(-c^2*x^2 + 1)*b*d^2*i*e/c^5 + 5/32*b*f*arcsin(c*x)*e^3/c^4 + 1/2*(c^2*x^2 - 1)^2*b*h*arcs
in(c*x)*e^3/c^6 + 3/7*(c^2*x^2 - 1)*b*i*x*arcsin(c*x)*e^3/c^6 + 3/2*(c^2*x^2 - 1)*b*d*i*arcsin(c*x)*e^2/c^6 -
2/15*(-c^2*x^2 + 1)^(3/2)*b*g*e^3/c^5 + 1/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*i*e^3/c^7 + 11/96*sqrt(-c^2*
x^2 + 1)*b*h*x*e^3/c^5 + 3/5*sqrt(-c^2*x^2 + 1)*b*d*h*e^2/c^5 + 1/2*(c^2*x^2 - 1)*b*h*arcsin(c*x)*e^3/c^6 + 1/
7*b*i*x*arcsin(c*x)*e^3/c^6 + 11/32*b*d*i*arcsin(c*x)*e^2/c^6 + 1/5*sqrt(-c^2*x^2 + 1)*b*g*e^3/c^5 + 3/35*(c^2
*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*i*e^3/c^7 + 11/96*b*h*arcsin(c*x)*e^3/c^6 - 1/7*(-c^2*x^2 + 1)^(3/2)*b*i*e^3/
c^7 + 1/7*sqrt(-c^2*x^2 + 1)*b*i*e^3/c^7

________________________________________________________________________________________

maple [A]  time = 0.02, size = 932, normalized size = 1.36 \[ \frac {\frac {a \left (\frac {e^{3} i \,c^{7} x^{7}}{7}+\frac {\left (3 d c \,e^{2} i +e^{3} c h \right ) c^{6} x^{6}}{6}+\frac {\left (3 c^{2} d^{2} e i +3 d \,c^{2} e^{2} h +e^{3} c^{2} g \right ) c^{5} x^{5}}{5}+\frac {\left (c^{3} d^{3} i +3 c^{3} d^{2} e h +3 d \,c^{3} e^{2} g +e^{3} f \,c^{3}\right ) c^{4} x^{4}}{4}+\frac {\left (c^{4} d^{3} h +3 c^{4} d^{2} e g +3 d \,c^{4} e^{2} f \right ) c^{3} x^{3}}{3}+\frac {\left (c^{5} d^{3} g +3 c^{5} d^{2} e f \right ) c^{2} x^{2}}{2}+c^{7} d^{3} f x \right )}{c^{6}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{3} i \,c^{7} x^{7}}{7}+\frac {\arcsin \left (c x \right ) c^{7} x^{6} d \,e^{2} i}{2}+\frac {\arcsin \left (c x \right ) c^{7} x^{6} e^{3} h}{6}+\frac {3 \arcsin \left (c x \right ) c^{7} x^{5} d^{2} e i}{5}+\frac {3 \arcsin \left (c x \right ) c^{7} x^{5} d \,e^{2} h}{5}+\frac {\arcsin \left (c x \right ) c^{7} x^{5} e^{3} g}{5}+\frac {\arcsin \left (c x \right ) c^{7} x^{4} d^{3} i}{4}+\frac {3 \arcsin \left (c x \right ) c^{7} x^{4} d^{2} e h}{4}+\frac {3 \arcsin \left (c x \right ) c^{7} x^{4} d \,e^{2} g}{4}+\frac {\arcsin \left (c x \right ) c^{7} x^{4} e^{3} f}{4}+\frac {\arcsin \left (c x \right ) c^{7} x^{3} d^{3} h}{3}+\arcsin \left (c x \right ) c^{7} x^{3} d^{2} e g +\arcsin \left (c x \right ) c^{7} x^{3} d \,e^{2} f +\frac {\arcsin \left (c x \right ) c^{7} x^{2} d^{3} g}{2}+\frac {3 \arcsin \left (c x \right ) c^{7} x^{2} d^{2} e f}{2}+\arcsin \left (c x \right ) c^{7} d^{3} f x -\frac {e^{3} i \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}-\frac {\left (210 d c \,e^{2} i +70 e^{3} c h \right ) \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )}{420}-\frac {\left (252 c^{2} d^{2} e i +252 d \,c^{2} e^{2} h +84 e^{3} c^{2} g \right ) \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{420}-\frac {\left (105 c^{3} d^{3} i +315 c^{3} d^{2} e h +315 d \,c^{3} e^{2} g +105 e^{3} f \,c^{3}\right ) \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{420}-\frac {\left (140 c^{4} d^{3} h +420 c^{4} d^{2} e g +420 d \,c^{4} e^{2} f \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{420}-\frac {\left (210 c^{5} d^{3} g +630 c^{5} d^{2} e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{420}+c^{6} d^{3} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{6}}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x)

[Out]

1/c*(a/c^6*(1/7*e^3*i*c^7*x^7+1/6*(3*c*d*e^2*i+c*e^3*h)*c^6*x^6+1/5*(3*c^2*d^2*e*i+3*c^2*d*e^2*h+c^2*e^3*g)*c^
5*x^5+1/4*(c^3*d^3*i+3*c^3*d^2*e*h+3*c^3*d*e^2*g+c^3*e^3*f)*c^4*x^4+1/3*(c^4*d^3*h+3*c^4*d^2*e*g+3*c^4*d*e^2*f
)*c^3*x^3+1/2*(c^5*d^3*g+3*c^5*d^2*e*f)*c^2*x^2+c^7*d^3*f*x)+b/c^6*(1/7*arcsin(c*x)*e^3*i*c^7*x^7+1/2*arcsin(c
*x)*c^7*x^6*d*e^2*i+1/6*arcsin(c*x)*c^7*x^6*e^3*h+3/5*arcsin(c*x)*c^7*x^5*d^2*e*i+3/5*arcsin(c*x)*c^7*x^5*d*e^
2*h+1/5*arcsin(c*x)*c^7*x^5*e^3*g+1/4*arcsin(c*x)*c^7*x^4*d^3*i+3/4*arcsin(c*x)*c^7*x^4*d^2*e*h+3/4*arcsin(c*x
)*c^7*x^4*d*e^2*g+1/4*arcsin(c*x)*c^7*x^4*e^3*f+1/3*arcsin(c*x)*c^7*x^3*d^3*h+arcsin(c*x)*c^7*x^3*d^2*e*g+arcs
in(c*x)*c^7*x^3*d*e^2*f+1/2*arcsin(c*x)*c^7*x^2*d^3*g+3/2*arcsin(c*x)*c^7*x^2*d^2*e*f+arcsin(c*x)*c^7*d^3*f*x-
1/7*e^3*i*(-1/7*c^6*x^6*(-c^2*x^2+1)^(1/2)-6/35*c^4*x^4*(-c^2*x^2+1)^(1/2)-8/35*c^2*x^2*(-c^2*x^2+1)^(1/2)-16/
35*(-c^2*x^2+1)^(1/2))-1/420*(210*c*d*e^2*i+70*c*e^3*h)*(-1/6*c^5*x^5*(-c^2*x^2+1)^(1/2)-5/24*c^3*x^3*(-c^2*x^
2+1)^(1/2)-5/16*c*x*(-c^2*x^2+1)^(1/2)+5/16*arcsin(c*x))-1/420*(252*c^2*d^2*e*i+252*c^2*d*e^2*h+84*c^2*e^3*g)*
(-1/5*c^4*x^4*(-c^2*x^2+1)^(1/2)-4/15*c^2*x^2*(-c^2*x^2+1)^(1/2)-8/15*(-c^2*x^2+1)^(1/2))-1/420*(105*c^3*d^3*i
+315*c^3*d^2*e*h+315*c^3*d*e^2*g+105*c^3*e^3*f)*(-1/4*c^3*x^3*(-c^2*x^2+1)^(1/2)-3/8*c*x*(-c^2*x^2+1)^(1/2)+3/
8*arcsin(c*x))-1/420*(140*c^4*d^3*h+420*c^4*d^2*e*g+420*c^4*d*e^2*f)*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^
2*x^2+1)^(1/2))-1/420*(210*c^5*d^3*g+630*c^5*d^2*e*f)*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+c^6*d^3*f*
(-c^2*x^2+1)^(1/2)))

________________________________________________________________________________________

maxima [A]  time = 0.47, size = 1231, normalized size = 1.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

1/7*a*e^3*i*x^7 + 1/6*a*e^3*h*x^6 + 1/2*a*d*e^2*i*x^6 + 1/5*a*e^3*g*x^5 + 3/5*a*d*e^2*h*x^5 + 3/5*a*d^2*e*i*x^
5 + 1/4*a*e^3*f*x^4 + 3/4*a*d*e^2*g*x^4 + 3/4*a*d^2*e*h*x^4 + 1/4*a*d^3*i*x^4 + a*d*e^2*f*x^3 + a*d^2*e*g*x^3
+ 1/3*a*d^3*h*x^3 + 3/2*a*d^2*e*f*x^2 + 1/2*a*d^3*g*x^2 + 3/4*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2
- arcsin(c*x)/c^3))*b*d^2*e*f + 1/3*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)
/c^4))*b*d*e^2*f + 1/32*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*ar
csin(c*x)/c^5)*c)*b*e^3*f + 1/4*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*b*d^3*g +
1/3*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*d^2*e*g + 3/32*(8*x^4*a
rcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*b*d*e^2*g + 1/
75*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c
^6)*c)*b*e^3*g + 1/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*d^3*h +
3/32*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*
b*d^2*e*h + 1/25*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-
c^2*x^2 + 1)/c^6)*c)*b*d*e^2*h + 1/288*(48*x^6*arcsin(c*x) + (8*sqrt(-c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^2
+ 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcsin(c*x)/c^7)*c)*b*e^3*h + 1/32*(8*x^4*arcsin(c*x) + (2*sqr
t(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*b*d^3*i + 1/25*(15*x^5*arcsin(c*x
) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*b*d^2*e*i + 1/
96*(48*x^6*arcsin(c*x) + (8*sqrt(-c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 1)
*x/c^6 - 15*arcsin(c*x)/c^7)*c)*b*d*e^2*i + 1/245*(35*x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt
(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*b*e^3*i + a*d^3*f*x + (c
*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*d^3*f/c

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+e\,x\right )}^3\,\left (i\,x^3+h\,x^2+g\,x+f\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x))*(d + e*x)^3*(f + g*x + h*x^2 + i*x^3),x)

[Out]

int((a + b*asin(c*x))*(d + e*x)^3*(f + g*x + h*x^2 + i*x^3), x)

________________________________________________________________________________________

sympy [A]  time = 11.98, size = 1809, normalized size = 2.64 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(i*x**3+h*x**2+g*x+f)*(a+b*asin(c*x)),x)

[Out]

Piecewise((a*d**3*f*x + a*d**3*g*x**2/2 + a*d**3*h*x**3/3 + a*d**3*i*x**4/4 + 3*a*d**2*e*f*x**2/2 + a*d**2*e*g
*x**3 + 3*a*d**2*e*h*x**4/4 + 3*a*d**2*e*i*x**5/5 + a*d*e**2*f*x**3 + 3*a*d*e**2*g*x**4/4 + 3*a*d*e**2*h*x**5/
5 + a*d*e**2*i*x**6/2 + a*e**3*f*x**4/4 + a*e**3*g*x**5/5 + a*e**3*h*x**6/6 + a*e**3*i*x**7/7 + b*d**3*f*x*asi
n(c*x) + b*d**3*g*x**2*asin(c*x)/2 + b*d**3*h*x**3*asin(c*x)/3 + b*d**3*i*x**4*asin(c*x)/4 + 3*b*d**2*e*f*x**2
*asin(c*x)/2 + b*d**2*e*g*x**3*asin(c*x) + 3*b*d**2*e*h*x**4*asin(c*x)/4 + 3*b*d**2*e*i*x**5*asin(c*x)/5 + b*d
*e**2*f*x**3*asin(c*x) + 3*b*d*e**2*g*x**4*asin(c*x)/4 + 3*b*d*e**2*h*x**5*asin(c*x)/5 + b*d*e**2*i*x**6*asin(
c*x)/2 + b*e**3*f*x**4*asin(c*x)/4 + b*e**3*g*x**5*asin(c*x)/5 + b*e**3*h*x**6*asin(c*x)/6 + b*e**3*i*x**7*asi
n(c*x)/7 + b*d**3*f*sqrt(-c**2*x**2 + 1)/c + b*d**3*g*x*sqrt(-c**2*x**2 + 1)/(4*c) + b*d**3*h*x**2*sqrt(-c**2*
x**2 + 1)/(9*c) + b*d**3*i*x**3*sqrt(-c**2*x**2 + 1)/(16*c) + 3*b*d**2*e*f*x*sqrt(-c**2*x**2 + 1)/(4*c) + b*d*
*2*e*g*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + 3*b*d**2*e*h*x**3*sqrt(-c**2*x**2 + 1)/(16*c) + 3*b*d**2*e*i*x**4*sqr
t(-c**2*x**2 + 1)/(25*c) + b*d*e**2*f*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + 3*b*d*e**2*g*x**3*sqrt(-c**2*x**2 + 1)
/(16*c) + 3*b*d*e**2*h*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + b*d*e**2*i*x**5*sqrt(-c**2*x**2 + 1)/(12*c) + b*e**3
*f*x**3*sqrt(-c**2*x**2 + 1)/(16*c) + b*e**3*g*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + b*e**3*h*x**5*sqrt(-c**2*x**
2 + 1)/(36*c) + b*e**3*i*x**6*sqrt(-c**2*x**2 + 1)/(49*c) - b*d**3*g*asin(c*x)/(4*c**2) - 3*b*d**2*e*f*asin(c*
x)/(4*c**2) + 2*b*d**3*h*sqrt(-c**2*x**2 + 1)/(9*c**3) + 3*b*d**3*i*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 2*b*d**
2*e*g*sqrt(-c**2*x**2 + 1)/(3*c**3) + 9*b*d**2*e*h*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 4*b*d**2*e*i*x**2*sqrt(-
c**2*x**2 + 1)/(25*c**3) + 2*b*d*e**2*f*sqrt(-c**2*x**2 + 1)/(3*c**3) + 9*b*d*e**2*g*x*sqrt(-c**2*x**2 + 1)/(3
2*c**3) + 4*b*d*e**2*h*x**2*sqrt(-c**2*x**2 + 1)/(25*c**3) + 5*b*d*e**2*i*x**3*sqrt(-c**2*x**2 + 1)/(48*c**3)
+ 3*b*e**3*f*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 4*b*e**3*g*x**2*sqrt(-c**2*x**2 + 1)/(75*c**3) + 5*b*e**3*h*x*
*3*sqrt(-c**2*x**2 + 1)/(144*c**3) + 6*b*e**3*i*x**4*sqrt(-c**2*x**2 + 1)/(245*c**3) - 3*b*d**3*i*asin(c*x)/(3
2*c**4) - 9*b*d**2*e*h*asin(c*x)/(32*c**4) - 9*b*d*e**2*g*asin(c*x)/(32*c**4) - 3*b*e**3*f*asin(c*x)/(32*c**4)
+ 8*b*d**2*e*i*sqrt(-c**2*x**2 + 1)/(25*c**5) + 8*b*d*e**2*h*sqrt(-c**2*x**2 + 1)/(25*c**5) + 5*b*d*e**2*i*x*
sqrt(-c**2*x**2 + 1)/(32*c**5) + 8*b*e**3*g*sqrt(-c**2*x**2 + 1)/(75*c**5) + 5*b*e**3*h*x*sqrt(-c**2*x**2 + 1)
/(96*c**5) + 8*b*e**3*i*x**2*sqrt(-c**2*x**2 + 1)/(245*c**5) - 5*b*d*e**2*i*asin(c*x)/(32*c**6) - 5*b*e**3*h*a
sin(c*x)/(96*c**6) + 16*b*e**3*i*sqrt(-c**2*x**2 + 1)/(245*c**7), Ne(c, 0)), (a*(d**3*f*x + d**3*g*x**2/2 + d*
*3*h*x**3/3 + d**3*i*x**4/4 + 3*d**2*e*f*x**2/2 + d**2*e*g*x**3 + 3*d**2*e*h*x**4/4 + 3*d**2*e*i*x**5/5 + d*e*
*2*f*x**3 + 3*d*e**2*g*x**4/4 + 3*d*e**2*h*x**5/5 + d*e**2*i*x**6/2 + e**3*f*x**4/4 + e**3*g*x**5/5 + e**3*h*x
**6/6 + e**3*i*x**7/7), True))

________________________________________________________________________________________