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

Optimal. Leaf size=512 \[ d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e x^4 \left (a+b \sin ^{-1}(c x)\right ) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 (3 d h+e g) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {b e^2 x^4 \sqrt {1-c^2 x^2} (3 d h+e g)}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+12 e^2 (3 d h+e g)\right )}{225 c^3}+\frac {b e x^3 \sqrt {1-c^2 x^2} \left (9 c^2 \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^2 h\right )}{144 c^3}-\frac {b \sin ^{-1}(c x) \left (24 c^4 d^2 (d g+3 e f)+9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^3 h\right )}{96 c^6}+\frac {b \sqrt {1-c^2 x^2} \left (75 x \left (24 c^4 d^2 (d g+3 e f)+9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^3 h\right )+32 \left (225 c^4 d^3 f+50 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+24 e^2 (3 d h+e g)\right )\right )}{7200 c^5} \]

[Out]

-1/96*b*(24*c^4*d^2*(d*g+3*e*f)+5*e^3*h+9*c^2*e*(3*d^2*h+3*d*e*g+e^2*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*e*(3*d^
2*h+3*d*e*g+e^2*f)*x^4*(a+b*arcsin(c*x))+1/5*e^2*(3*d*h+e*g)*x^5*(a+b*arcsin(c*x))+1/6*e^3*h*x^6*(a+b*arcsin(c
*x))+1/225*b*(12*e^2*(3*d*h+e*g)+25*c^2*d*(d^2*h+3*d*e*g+3*e^2*f))*x^2*(-c^2*x^2+1)^(1/2)/c^3+1/144*b*e*(5*e^2
*h+9*c^2*(3*d^2*h+3*d*e*g+e^2*f))*x^3*(-c^2*x^2+1)^(1/2)/c^3+1/25*b*e^2*(3*d*h+e*g)*x^4*(-c^2*x^2+1)^(1/2)/c+1
/36*b*e^3*h*x^5*(-c^2*x^2+1)^(1/2)/c+1/7200*b*(7200*c^4*d^3*f+768*e^2*(3*d*h+e*g)+1600*c^2*d*(d^2*h+3*d*e*g+3*
e^2*f)+75*(24*c^4*d^2*(d*g+3*e*f)+5*e^3*h+9*c^2*e*(3*d^2*h+3*d*e*g+e^2*f))*x)*(-c^2*x^2+1)^(1/2)/c^5

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Rubi [A]  time = 2.51, antiderivative size = 509, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4749, 12, 1809, 780, 216} \[ \frac {1}{4} e x^4 \left (a+b \sin ^{-1}(c x)\right ) \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} d^2 x^2 (d g+3 e f) \left (a+b \sin ^{-1}(c x)\right )+d^3 f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 x^5 (3 d h+e g) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {b e x^3 \sqrt {1-c^2 x^2} \left (e^2 \left (\frac {5 h}{c^2}+9 f\right )+27 d^2 h+27 d e g\right )}{144 c}+\frac {b x^2 \sqrt {1-c^2 x^2} \left (25 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+12 e^2 (3 d h+e g)\right )}{225 c^3}+\frac {b \sqrt {1-c^2 x^2} \left (75 x \left (9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+24 c^4 d^2 (d g+3 e f)+5 e^3 h\right )+32 \left (50 c^2 d \left (d^2 h+3 d e g+3 e^2 f\right )+225 c^4 d^3 f+24 e^2 (3 d h+e g)\right )\right )}{7200 c^5}-\frac {b \sin ^{-1}(c x) \left (9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+24 c^4 d^2 (d g+3 e f)+5 e^3 h\right )}{96 c^6}+\frac {b e^2 x^4 \sqrt {1-c^2 x^2} (3 d h+e g)}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(12*e^2*(e*g + 3*d*h) + 25*c^2*d*(3*e^2*f + 3*d*e*g + d^2*h))*x^2*Sqrt[1 - c^2*x^2])/(225*c^3) + (b*e*(27*d
*e*g + 27*d^2*h + e^2*(9*f + (5*h)/c^2))*x^3*Sqrt[1 - c^2*x^2])/(144*c) + (b*e^2*(e*g + 3*d*h)*x^4*Sqrt[1 - c^
2*x^2])/(25*c) + (b*e^3*h*x^5*Sqrt[1 - c^2*x^2])/(36*c) + (b*(32*(225*c^4*d^3*f + 24*e^2*(e*g + 3*d*h) + 50*c^
2*d*(3*e^2*f + 3*d*e*g + d^2*h)) + 75*(24*c^4*d^2*(3*e*f + d*g) + 5*e^3*h + 9*c^2*e*(e^2*f + 3*d*e*g + 3*d^2*h
))*x)*Sqrt[1 - c^2*x^2])/(7200*c^5) - (b*(24*c^4*d^2*(3*e*f + d*g) + 5*e^3*h + 9*c^2*e*(e^2*f + 3*d*e*g + 3*d^
2*h))*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*(e^2*f + 3*d*e*g + 3*d^2*h)*x^4*(a + b*ArcSin[c*x
]))/4 + (e^2*(e*g + 3*d*h)*x^5*(a + b*ArcSin[c*x]))/5 + (e^3*h*x^6*(a + b*ArcSin[c*x]))/6

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\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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x \left (10 d^3 (6 f+x (3 g+2 h x))+15 d^2 e x (6 f+x (4 g+3 h x))+3 d e^2 x^2 (20 f+3 x (5 g+4 h x))+e^3 x^3 (15 f+2 x (6 g+5 h x))\right )}{60 \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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{60} (b c) \int \frac {x \left (10 d^3 (6 f+x (3 g+2 h x))+15 d^2 e x (6 f+x (4 g+3 h x))+3 d e^2 x^2 (20 f+3 x (5 g+4 h x))+e^3 x^3 (15 f+2 x (6 g+5 h x))\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-360 c^2 d^3 f-180 c^2 d^2 (3 e f+d g) x-120 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right ) x^2-10 e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3-72 c^2 e^2 (e g+3 d h) x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{360 c}\\ &=\frac {b e^2 (e g+3 d h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \int \frac {x \left (1800 c^4 d^3 f+900 c^4 d^2 (3 e f+d g) x+24 c^2 \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2+50 c^2 e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3\right )}{\sqrt {1-c^2 x^2}} \, dx}{1800 c^3}\\ &=\frac {b e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^2 (e g+3 d h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-7200 c^6 d^3 f-150 c^2 \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x-96 c^4 \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{7200 c^5}\\ &=\frac {b \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^2 (e g+3 d h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \int \frac {x \left (96 c^4 \left (225 c^4 d^3 f+24 e^2 (e g+3 d h)+50 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right )+450 c^4 \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x\right )}{\sqrt {1-c^2 x^2}} \, dx}{21600 c^7}\\ &=\frac {b \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^2 (e g+3 d h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b \left (32 \left (225 c^4 d^3 f+24 e^2 (e g+3 d h)+50 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right )+75 \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{7200 c^5}+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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{96 c^5}\\ &=\frac {b \left (12 e^2 (e g+3 d h)+25 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right ) x^2 \sqrt {1-c^2 x^2}}{225 c^3}+\frac {b e \left (5 e^2 h+9 c^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x^3 \sqrt {1-c^2 x^2}}{144 c^3}+\frac {b e^2 (e g+3 d h) x^4 \sqrt {1-c^2 x^2}}{25 c}+\frac {b e^3 h x^5 \sqrt {1-c^2 x^2}}{36 c}+\frac {b \left (32 \left (225 c^4 d^3 f+24 e^2 (e g+3 d h)+50 c^2 d \left (3 e^2 f+3 d e g+d^2 h\right )\right )+75 \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 h\right )\right ) x\right ) \sqrt {1-c^2 x^2}}{7200 c^5}-\frac {b \left (24 c^4 d^2 (3 e f+d g)+5 e^3 h+9 c^2 e \left (e^2 f+3 d e g+3 d^2 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} e \left (e^2 f+3 d e g+3 d^2 h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} e^2 (e g+3 d h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} e^3 h x^6 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.56, size = 463, normalized size = 0.90 \[ a d^3 f x+\frac {1}{4} a e x^4 \left (3 d^2 h+3 d e g+e^2 f\right )+\frac {1}{3} a d x^3 \left (d^2 h+3 d e g+3 e^2 f\right )+\frac {1}{2} a d^2 x^2 (d g+3 e f)+\frac {1}{5} a e^2 x^5 (3 d h+e g)+\frac {1}{6} a e^3 h x^6-\frac {b \sin ^{-1}(c x) \left (24 c^4 d^2 (d g+3 e f)+9 c^2 e \left (3 d^2 h+3 d e g+e^2 f\right )+5 e^3 h\right )}{96 c^6}+\frac {b \sqrt {1-c^2 x^2} \left (2 c^4 \left (100 d^3 (36 f+x (9 g+4 h x))+75 d^2 e x (36 f+x (16 g+9 h x))+3 d e^2 x^2 (400 f+9 x (25 g+16 h x))+e^3 x^3 (225 f+4 x (36 g+25 h x))\right )+c^2 \left (1600 d^3 h+75 d^2 e (64 g+27 h x)+3 d e^2 \left (1600 f+675 g x+384 h x^2\right )+e^3 x \left (675 f+384 g x+250 h x^2\right )\right )+3 e^2 (768 d h+256 e g+125 e h x)\right )}{7200 c^5}+\frac {1}{60} b x \sin ^{-1}(c x) \left (10 d^3 (6 f+x (3 g+2 h x))+15 d^2 e x (6 f+x (4 g+3 h x))+3 d e^2 x^2 (20 f+3 x (5 g+4 h x))+e^3 x^3 (15 f+2 x (6 g+5 h x))\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(f + g*x + h*x^2)*(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*(e^2*f + 3*d*e*g + 3*
d^2*h)*x^4)/4 + (a*e^2*(e*g + 3*d*h)*x^5)/5 + (a*e^3*h*x^6)/6 + (b*Sqrt[1 - c^2*x^2]*(3*e^2*(256*e*g + 768*d*h
+ 125*e*h*x) + c^2*(1600*d^3*h + 75*d^2*e*(64*g + 27*h*x) + e^3*x*(675*f + 384*g*x + 250*h*x^2) + 3*d*e^2*(16
00*f + 675*g*x + 384*h*x^2)) + 2*c^4*(100*d^3*(36*f + x*(9*g + 4*h*x)) + 75*d^2*e*x*(36*f + x*(16*g + 9*h*x))
+ 3*d*e^2*x^2*(400*f + 9*x*(25*g + 16*h*x)) + e^3*x^3*(225*f + 4*x*(36*g + 25*h*x)))))/(7200*c^5) - (b*(24*c^4
*d^2*(3*e*f + d*g) + 5*e^3*h + 9*c^2*e*(e^2*f + 3*d*e*g + 3*d^2*h))*ArcSin[c*x])/(96*c^6) + (b*x*(10*d^3*(6*f
+ x*(3*g + 2*h*x)) + 15*d^2*e*x*(6*f + x*(4*g + 3*h*x)) + 3*d*e^2*x^2*(20*f + 3*x*(5*g + 4*h*x)) + e^3*x^3*(15
*f + 2*x*(6*g + 5*h*x)))*ArcSin[c*x])/60

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fricas [A]  time = 0.74, size = 676, normalized size = 1.32 \[ \frac {1200 \, a c^{6} e^{3} h x^{6} + 7200 \, a c^{6} d^{3} f x + 1440 \, {\left (a c^{6} e^{3} g + 3 \, a c^{6} d e^{2} h\right )} x^{5} + 1800 \, {\left (a c^{6} e^{3} f + 3 \, a c^{6} d e^{2} g + 3 \, a c^{6} d^{2} e h\right )} x^{4} + 2400 \, {\left (3 \, a c^{6} d e^{2} f + 3 \, a c^{6} d^{2} e g + a c^{6} d^{3} h\right )} x^{3} + 3600 \, {\left (3 \, a c^{6} d^{2} e f + a c^{6} d^{3} g\right )} x^{2} + 15 \, {\left (80 \, b c^{6} e^{3} h x^{6} + 480 \, b c^{6} d^{3} f x + 96 \, {\left (b c^{6} e^{3} g + 3 \, b c^{6} d e^{2} h\right )} x^{5} + 120 \, {\left (b c^{6} e^{3} f + 3 \, b c^{6} d e^{2} g + 3 \, b c^{6} d^{2} e h\right )} x^{4} + 160 \, {\left (3 \, b c^{6} d e^{2} f + 3 \, b c^{6} d^{2} e g + b c^{6} d^{3} h\right )} x^{3} + 240 \, {\left (3 \, b c^{6} d^{2} e f + b c^{6} d^{3} g\right )} x^{2} - 45 \, {\left (8 \, b c^{4} d^{2} e + b c^{2} e^{3}\right )} f - 15 \, {\left (8 \, b c^{4} d^{3} + 9 \, b c^{2} d e^{2}\right )} g - 5 \, {\left (27 \, b c^{2} d^{2} e + 5 \, b e^{3}\right )} h\right )} \arcsin \left (c x\right ) + {\left (200 \, b c^{5} e^{3} h x^{5} + 288 \, {\left (b c^{5} e^{3} g + 3 \, b c^{5} d e^{2} h\right )} x^{4} + 50 \, {\left (9 \, b c^{5} e^{3} f + 27 \, b c^{5} d e^{2} g + {\left (27 \, b c^{5} d^{2} e + 5 \, b c^{3} e^{3}\right )} h\right )} x^{3} + 32 \, {\left (75 \, b c^{5} d e^{2} f + 3 \, {\left (25 \, b c^{5} d^{2} e + 4 \, b c^{3} e^{3}\right )} g + {\left (25 \, b c^{5} d^{3} + 36 \, b c^{3} d e^{2}\right )} h\right )} x^{2} + 2400 \, {\left (3 \, b c^{5} d^{3} + 2 \, b c^{3} d e^{2}\right )} f + 192 \, {\left (25 \, b c^{3} d^{2} e + 4 \, b c e^{3}\right )} g + 64 \, {\left (25 \, b c^{3} d^{3} + 36 \, b c d e^{2}\right )} h + 75 \, {\left (9 \, {\left (8 \, b c^{5} d^{2} e + b c^{3} e^{3}\right )} f + 3 \, {\left (8 \, b c^{5} d^{3} + 9 \, b c^{3} d e^{2}\right )} g + {\left (27 \, b c^{3} d^{2} e + 5 \, b c e^{3}\right )} h\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{7200 \, c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7200*(1200*a*c^6*e^3*h*x^6 + 7200*a*c^6*d^3*f*x + 1440*(a*c^6*e^3*g + 3*a*c^6*d*e^2*h)*x^5 + 1800*(a*c^6*e^3
*f + 3*a*c^6*d*e^2*g + 3*a*c^6*d^2*e*h)*x^4 + 2400*(3*a*c^6*d*e^2*f + 3*a*c^6*d^2*e*g + a*c^6*d^3*h)*x^3 + 360
0*(3*a*c^6*d^2*e*f + a*c^6*d^3*g)*x^2 + 15*(80*b*c^6*e^3*h*x^6 + 480*b*c^6*d^3*f*x + 96*(b*c^6*e^3*g + 3*b*c^6
*d*e^2*h)*x^5 + 120*(b*c^6*e^3*f + 3*b*c^6*d*e^2*g + 3*b*c^6*d^2*e*h)*x^4 + 160*(3*b*c^6*d*e^2*f + 3*b*c^6*d^2
*e*g + b*c^6*d^3*h)*x^3 + 240*(3*b*c^6*d^2*e*f + b*c^6*d^3*g)*x^2 - 45*(8*b*c^4*d^2*e + b*c^2*e^3)*f - 15*(8*b
*c^4*d^3 + 9*b*c^2*d*e^2)*g - 5*(27*b*c^2*d^2*e + 5*b*e^3)*h)*arcsin(c*x) + (200*b*c^5*e^3*h*x^5 + 288*(b*c^5*
e^3*g + 3*b*c^5*d*e^2*h)*x^4 + 50*(9*b*c^5*e^3*f + 27*b*c^5*d*e^2*g + (27*b*c^5*d^2*e + 5*b*c^3*e^3)*h)*x^3 +
32*(75*b*c^5*d*e^2*f + 3*(25*b*c^5*d^2*e + 4*b*c^3*e^3)*g + (25*b*c^5*d^3 + 36*b*c^3*d*e^2)*h)*x^2 + 2400*(3*b
*c^5*d^3 + 2*b*c^3*d*e^2)*f + 192*(25*b*c^3*d^2*e + 4*b*c*e^3)*g + 64*(25*b*c^3*d^3 + 36*b*c*d*e^2)*h + 75*(9*
(8*b*c^5*d^2*e + b*c^3*e^3)*f + 3*(8*b*c^5*d^3 + 9*b*c^3*d*e^2)*g + (27*b*c^3*d^2*e + 5*b*c*e^3)*h)*x)*sqrt(-c
^2*x^2 + 1))/c^6

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giac [B]  time = 0.50, size = 1313, normalized size = 2.56 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*arcsin(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 + sqrt(-c^2*x^2 + 1)*b*d^3*f/c - 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 + 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 - 1/9*(-c^2*x^2 + 1)^(3/2)*b*d^3*h/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 + 15/32*sqrt(-c^2*x^2 + 1)*b*d^2*h*x*e/c^3 + 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 + 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 + sqrt(-c^2*x^2 + 1
)*b*d^2*g*e/c^3 + 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 +
3/2*(c^2*x^2 - 1)*b*d*g*arcsin(c*x)*e^2/c^4 + 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 + 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 + 15/32*b*d*g*arcsi
n(c*x)*e^2/c^4 + 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 + 5/32*b*f*arcsin(c*x)*e^3/c^4 + 1/2*(c^2*x^2 - 1)^2*b*h*arcsin(c*
x)*e^3/c^6 - 2/15*(-c^2*x^2 + 1)^(3/2)*b*g*e^3/c^5 + 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 + 1/5*sqrt(-c^2*x^2 + 1)*b*g*e^3/c^5 + 11/96*
b*h*arcsin(c*x)*e^3/c^6

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maple [A]  time = 0.02, size = 705, normalized size = 1.38 \[ \frac {\frac {a \left (\frac {e^{3} h \,c^{6} x^{6}}{6}+\frac {\left (3 d c \,e^{2} h +e^{3} c g \right ) c^{5} x^{5}}{5}+\frac {\left (3 c^{2} d^{2} e h +3 d \,c^{2} e^{2} g +e^{3} f \,c^{2}\right ) c^{4} x^{4}}{4}+\frac {\left (c^{3} d^{3} h +3 c^{3} d^{2} e g +3 d \,c^{3} e^{2} f \right ) c^{3} x^{3}}{3}+\frac {\left (c^{4} d^{3} g +3 c^{4} d^{2} e f \right ) c^{2} x^{2}}{2}+c^{6} d^{3} f x \right )}{c^{5}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{3} h \,c^{6} x^{6}}{6}+\frac {3 \arcsin \left (c x \right ) c^{6} x^{5} d \,e^{2} h}{5}+\frac {\arcsin \left (c x \right ) c^{6} x^{5} e^{3} g}{5}+\frac {3 \arcsin \left (c x \right ) c^{6} x^{4} d^{2} e h}{4}+\frac {3 \arcsin \left (c x \right ) c^{6} x^{4} d \,e^{2} g}{4}+\frac {\arcsin \left (c x \right ) c^{6} x^{4} e^{3} f}{4}+\frac {\arcsin \left (c x \right ) c^{6} x^{3} d^{3} h}{3}+\arcsin \left (c x \right ) c^{6} x^{3} d^{2} e g +\arcsin \left (c x \right ) c^{6} x^{3} d \,e^{2} f +\frac {\arcsin \left (c x \right ) c^{6} x^{2} d^{3} g}{2}+\frac {3 \arcsin \left (c x \right ) c^{6} x^{2} d^{2} e f}{2}+\arcsin \left (c x \right ) c^{6} d^{3} f x -\frac {e^{3} h \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 )}{6}-\frac {\left (36 d c \,e^{2} h +12 e^{3} c 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 )}{60}-\frac {\left (45 c^{2} d^{2} e h +45 d \,c^{2} e^{2} g +15 e^{3} f \,c^{2}\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 )}{60}-\frac {\left (20 c^{3} d^{3} h +60 c^{3} d^{2} e g +60 d \,c^{3} 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 )}{60}-\frac {\left (30 c^{4} d^{3} g +90 c^{4} d^{2} e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{60}+c^{5} d^{3} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{5}}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a/c^5*(1/6*e^3*h*c^6*x^6+1/5*(3*c*d*e^2*h+c*e^3*g)*c^5*x^5+1/4*(3*c^2*d^2*e*h+3*c^2*d*e^2*g+c^2*e^3*f)*c^
4*x^4+1/3*(c^3*d^3*h+3*c^3*d^2*e*g+3*c^3*d*e^2*f)*c^3*x^3+1/2*(c^4*d^3*g+3*c^4*d^2*e*f)*c^2*x^2+c^6*d^3*f*x)+b
/c^5*(1/6*arcsin(c*x)*e^3*h*c^6*x^6+3/5*arcsin(c*x)*c^6*x^5*d*e^2*h+1/5*arcsin(c*x)*c^6*x^5*e^3*g+3/4*arcsin(c
*x)*c^6*x^4*d^2*e*h+3/4*arcsin(c*x)*c^6*x^4*d*e^2*g+1/4*arcsin(c*x)*c^6*x^4*e^3*f+1/3*arcsin(c*x)*c^6*x^3*d^3*
h+arcsin(c*x)*c^6*x^3*d^2*e*g+arcsin(c*x)*c^6*x^3*d*e^2*f+1/2*arcsin(c*x)*c^6*x^2*d^3*g+3/2*arcsin(c*x)*c^6*x^
2*d^2*e*f+arcsin(c*x)*c^6*d^3*f*x-1/6*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))-1/60*(36*c*d*e^2*h+12*c*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))-1/60*(45*c^2*d^2*e*h+45*c^2*d*e^2*g+15*c^2*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))-1/60*(20*c^3*d^3*h+60*c^3*d^2*e*g+60*
c^3*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))-1/60*(30*c^4*d^3*g+90*c^4*d^2*e*f)*(-1/2
*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+c^5*d^3*f*(-c^2*x^2+1)^(1/2)))

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maxima [A]  time = 0.45, size = 859, normalized size = 1.68 \[ \frac {1}{6} \, a e^{3} h x^{6} + \frac {1}{5} \, a e^{3} g x^{5} + \frac {3}{5} \, a d e^{2} h x^{5} + \frac {1}{4} \, a e^{3} f x^{4} + \frac {3}{4} \, a d e^{2} g x^{4} + \frac {3}{4} \, a d^{2} e h x^{4} + a d e^{2} f x^{3} + a d^{2} e g x^{3} + \frac {1}{3} \, a d^{3} h x^{3} + \frac {3}{2} \, a d^{2} e f x^{2} + \frac {1}{2} \, a d^{3} g x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{2} e f + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d e^{2} f + \frac {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b e^{3} f + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d^{3} g + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{2} e g + \frac {3}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d e^{2} g + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b e^{3} g + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b d^{3} h + \frac {3}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b d^{2} e h + \frac {1}{25} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d e^{2} h + \frac {1}{288} \, {\left (48 \, x^{6} \arcsin \left (c x\right ) + {\left (\frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac {10 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \arcsin \left (c x\right )}{c^{7}}\right )} c\right )} b e^{3} h + a d^{3} f x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d^{3} f}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*e^3*h*x^6 + 1/5*a*e^3*g*x^5 + 3/5*a*d*e^2*h*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 + 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*arcsin(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*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*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*sqr
t(-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
+ a*d^3*f*x + (c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*d^3*f/c

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

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),x)

[Out]

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

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sympy [A]  time = 7.17, size = 1263, normalized size = 2.47 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**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 + 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 + 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*e**3*f*x**4/4 + a*e**3*g*x**5/5 +
a*e**3*h*x**6/6 + b*d**3*f*x*asin(c*x) + b*d**3*g*x**2*asin(c*x)/2 + b*d**3*h*x**3*asin(c*x)/3 + 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 + 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*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*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) + 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) + 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*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*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) + 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) + 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)/(32*c**3) + 4*b*d*e**2*h*x**2*sqrt(-c
**2*x**2 + 1)/(25*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) - 9*b*d**2*e*h*asin(c*x)/(32*c**4) - 9*b*d*e**2*g*a
sin(c*x)/(32*c**4) - 3*b*e**3*f*asin(c*x)/(32*c**4) + 8*b*d*e**2*h*sqrt(-c**2*x**2 + 1)/(25*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) - 5*b*e**3*h*asin(c*x)/(96*c**6)
, Ne(c, 0)), (a*(d**3*f*x + d**3*g*x**2/2 + d**3*h*x**3/3 + 3*d**2*e*f*x**2/2 + d**2*e*g*x**3 + 3*d**2*e*h*x**
4/4 + d*e**2*f*x**3 + 3*d*e**2*g*x**4/4 + 3*d*e**2*h*x**5/5 + e**3*f*x**4/4 + e**3*g*x**5/5 + e**3*h*x**6/6),
True))

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