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

Optimal. Leaf size=484 \[ \frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+2 d e g+e^2 f\right )+\frac{1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 i+2 d e h+e^2 g\right )+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d x^2 (d g+2 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e x^5 (2 d i+e h) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^2 i x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{b x^2 \sqrt{1-c^2 x^2} \left (25 c^2 \left (d^2 h+2 d e g+e^2 f\right )+12 e (2 d i+e h)\right )}{225 c^3}+\frac{b \sqrt{1-c^2 x^2} \left (75 x \left (9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+24 c^4 d (d g+2 e f)+5 e^2 i\right )+32 \left (50 c^2 \left (d^2 h+2 d e g+e^2 f\right )+225 c^4 d^2 f+24 e (2 d i+e h)\right )\right )}{7200 c^5}-\frac{b \sin ^{-1}(c x) \left (9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+24 c^4 d (d g+2 e f)+5 e^2 i\right )}{96 c^6}+\frac{b x^3 \sqrt{1-c^2 x^2} \left (9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+5 e^2 i\right )}{144 c^3}+\frac{b e x^4 \sqrt{1-c^2 x^2} (2 d i+e h)}{25 c}+\frac{b e^2 i x^5 \sqrt{1-c^2 x^2}}{36 c} \]

[Out]

(b*(25*c^2*(e^2*f + 2*d*e*g + d^2*h) + 12*e*(e*h + 2*d*i))*x^2*Sqrt[1 - c^2*x^2])/(225*c^3) + (b*(5*e^2*i + 9*
c^2*(e^2*g + 2*d*e*h + d^2*i))*x^3*Sqrt[1 - c^2*x^2])/(144*c^3) + (b*e*(e*h + 2*d*i)*x^4*Sqrt[1 - c^2*x^2])/(2
5*c) + (b*e^2*i*x^5*Sqrt[1 - c^2*x^2])/(36*c) + (b*(32*(225*c^4*d^2*f + 50*c^2*(e^2*f + 2*d*e*g + d^2*h) + 24*
e*(e*h + 2*d*i)) + 75*(24*c^4*d*(2*e*f + d*g) + 5*e^2*i + 9*c^2*(e^2*g + 2*d*e*h + d^2*i))*x)*Sqrt[1 - c^2*x^2
])/(7200*c^5) - (b*(24*c^4*d*(2*e*f + d*g) + 5*e^2*i + 9*c^2*(e^2*g + 2*d*e*h + d^2*i))*ArcSin[c*x])/(96*c^6)
+ d^2*f*x*(a + b*ArcSin[c*x]) + (d*(2*e*f + d*g)*x^2*(a + b*ArcSin[c*x]))/2 + ((e^2*f + 2*d*e*g + d^2*h)*x^3*(
a + b*ArcSin[c*x]))/3 + ((e^2*g + 2*d*e*h + d^2*i)*x^4*(a + b*ArcSin[c*x]))/4 + (e*(e*h + 2*d*i)*x^5*(a + b*Ar
cSin[c*x]))/5 + (e^2*i*x^6*(a + b*ArcSin[c*x]))/6

________________________________________________________________________________________

Rubi [A]  time = 2.54995, antiderivative size = 482, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+2 d e g+e^2 f\right )+\frac{1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 i+2 d e h+e^2 g\right )+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d x^2 (d g+2 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e x^5 (2 d i+e h) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^2 i x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{b x^2 \sqrt{1-c^2 x^2} \left (25 c^2 \left (d^2 h+2 d e g+e^2 f\right )+12 e (2 d i+e h)\right )}{225 c^3}+\frac{b \sqrt{1-c^2 x^2} \left (75 x \left (9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+24 c^4 d (d g+2 e f)+5 e^2 i\right )+32 \left (50 c^2 \left (d^2 h+2 d e g+e^2 f\right )+225 c^4 d^2 f+24 e (2 d i+e h)\right )\right )}{7200 c^5}-\frac{b \sin ^{-1}(c x) \left (9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+24 c^4 d (d g+2 e f)+5 e^2 i\right )}{96 c^6}+\frac{b x^3 \sqrt{1-c^2 x^2} \left (e^2 \left (\frac{5 i}{c^2}+9 g\right )+9 d^2 i+18 d e h\right )}{144 c}+\frac{b e x^4 \sqrt{1-c^2 x^2} (2 d i+e h)}{25 c}+\frac{b e^2 i x^5 \sqrt{1-c^2 x^2}}{36 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(25*c^2*(e^2*f + 2*d*e*g + d^2*h) + 12*e*(e*h + 2*d*i))*x^2*Sqrt[1 - c^2*x^2])/(225*c^3) + (b*(18*d*e*h + 9
*d^2*i + e^2*(9*g + (5*i)/c^2))*x^3*Sqrt[1 - c^2*x^2])/(144*c) + (b*e*(e*h + 2*d*i)*x^4*Sqrt[1 - c^2*x^2])/(25
*c) + (b*e^2*i*x^5*Sqrt[1 - c^2*x^2])/(36*c) + (b*(32*(225*c^4*d^2*f + 50*c^2*(e^2*f + 2*d*e*g + d^2*h) + 24*e
*(e*h + 2*d*i)) + 75*(24*c^4*d*(2*e*f + d*g) + 5*e^2*i + 9*c^2*(e^2*g + 2*d*e*h + d^2*i))*x)*Sqrt[1 - c^2*x^2]
)/(7200*c^5) - (b*(24*c^4*d*(2*e*f + d*g) + 5*e^2*i + 9*c^2*(e^2*g + 2*d*e*h + d^2*i))*ArcSin[c*x])/(96*c^6) +
d^2*f*x*(a + b*ArcSin[c*x]) + (d*(2*e*f + d*g)*x^2*(a + b*ArcSin[c*x]))/2 + ((e^2*f + 2*d*e*g + d^2*h)*x^3*(a
+ b*ArcSin[c*x]))/3 + ((e^2*g + 2*d*e*h + d^2*i)*x^4*(a + b*ArcSin[c*x]))/4 + (e*(e*h + 2*d*i)*x^5*(a + b*Arc
Sin[c*x]))/5 + (e^2*i*x^6*(a + b*ArcSin[c*x]))/6

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]

Rule 12

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

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 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 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]

Rubi steps

\begin{align*} \int (d+e x)^2 \left (f+g x+h x^2+107 x^3\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x \left (5 d^2 (12 f+x (6 g+x (4 h+321 x)))+2 d e x (30 f+x (20 g+3 x (5 h+428 x)))+e^2 x^2 (20 f+x (15 g+2 x (6 h+535 x)))\right )}{60 \sqrt{1-c^2 x^2}} \, dx\\ &=d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{60} (b c) \int \frac{x \left (5 d^2 (12 f+x (6 g+x (4 h+321 x)))+2 d e x (30 f+x (20 g+3 x (5 h+428 x)))+e^2 x^2 (20 f+x (15 g+2 x (6 h+535 x)))\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{107 b e^2 x^5 \sqrt{1-c^2 x^2}}{36 c}+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \int \frac{x \left (-360 c^2 d^2 f-180 c^2 d (2 e f+d g) x-120 c^2 \left (e^2 f+2 d e g+d^2 h\right ) x^2-10 \left (535 e^2+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x^3-72 c^2 e (214 d+e h) x^4\right )}{\sqrt{1-c^2 x^2}} \, dx}{360 c}\\ &=\frac{b e (214 d+e h) x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{107 b e^2 x^5 \sqrt{1-c^2 x^2}}{36 c}+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{b \int \frac{x \left (1800 c^4 d^2 f+900 c^4 d (2 e f+d g) x+24 c^2 \left (2 d e \left (1284+25 c^2 g\right )+25 c^2 d^2 h+e^2 \left (25 c^2 f+12 h\right )\right ) x^2+50 c^2 \left (535 e^2+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x^3\right )}{\sqrt{1-c^2 x^2}} \, dx}{1800 c^3}\\ &=\frac{b \left (535 e^2+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x^3 \sqrt{1-c^2 x^2}}{144 c^3}+\frac{b e (214 d+e h) x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{107 b e^2 x^5 \sqrt{1-c^2 x^2}}{36 c}+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \int \frac{x \left (-7200 c^6 d^2 f-150 c^2 \left (535 e^2+24 c^4 d (2 e f+d g)+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x-96 c^4 \left (2 d e \left (1284+25 c^2 g\right )+25 c^2 d^2 h+e^2 \left (25 c^2 f+12 h\right )\right ) x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{7200 c^5}\\ &=\frac{b \left (2 d e \left (1284+25 c^2 g\right )+25 c^2 d^2 h+e^2 \left (25 c^2 f+12 h\right )\right ) x^2 \sqrt{1-c^2 x^2}}{225 c^3}+\frac{b \left (535 e^2+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x^3 \sqrt{1-c^2 x^2}}{144 c^3}+\frac{b e (214 d+e h) x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{107 b e^2 x^5 \sqrt{1-c^2 x^2}}{36 c}+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{b \int \frac{x \left (96 c^4 \left (4 d e \left (1284+25 c^2 g\right )+2 e^2 \left (25 c^2 f+12 h\right )+25 d^2 \left (9 c^4 f+2 c^2 h\right )\right )+450 c^4 \left (535 e^2+24 c^4 d (2 e f+d g)+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x\right )}{\sqrt{1-c^2 x^2}} \, dx}{21600 c^7}\\ &=\frac{b \left (2 d e \left (1284+25 c^2 g\right )+25 c^2 d^2 h+e^2 \left (25 c^2 f+12 h\right )\right ) x^2 \sqrt{1-c^2 x^2}}{225 c^3}+\frac{b \left (535 e^2+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x^3 \sqrt{1-c^2 x^2}}{144 c^3}+\frac{b e (214 d+e h) x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{107 b e^2 x^5 \sqrt{1-c^2 x^2}}{36 c}+\frac{b \left (32 \left (4 d e \left (1284+25 c^2 g\right )+2 e^2 \left (25 c^2 f+12 h\right )+25 d^2 \left (9 c^4 f+2 c^2 h\right )\right )+75 \left (535 e^2+24 c^4 d (2 e f+d g)+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x\right ) \sqrt{1-c^2 x^2}}{7200 c^5}+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )-\frac{\left (b \left (535 e^2+24 c^4 d (2 e f+d g)+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right )\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{96 c^5}\\ &=\frac{b \left (2 d e \left (1284+25 c^2 g\right )+25 c^2 d^2 h+e^2 \left (25 c^2 f+12 h\right )\right ) x^2 \sqrt{1-c^2 x^2}}{225 c^3}+\frac{b \left (535 e^2+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x^3 \sqrt{1-c^2 x^2}}{144 c^3}+\frac{b e (214 d+e h) x^4 \sqrt{1-c^2 x^2}}{25 c}+\frac{107 b e^2 x^5 \sqrt{1-c^2 x^2}}{36 c}+\frac{b \left (32 \left (4 d e \left (1284+25 c^2 g\right )+2 e^2 \left (25 c^2 f+12 h\right )+25 d^2 \left (9 c^4 f+2 c^2 h\right )\right )+75 \left (535 e^2+24 c^4 d (2 e f+d g)+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) x\right ) \sqrt{1-c^2 x^2}}{7200 c^5}-\frac{b \left (535 e^2+24 c^4 d (2 e f+d g)+9 c^2 \left (107 d^2+e^2 g+2 d e h\right )\right ) \sin ^{-1}(c x)}{96 c^6}+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d (2 e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} \left (e^2 f+2 d e g+d^2 h\right ) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{4} \left (107 d^2+e^2 g+2 d e h\right ) x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e (214 d+e h) x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{107}{6} e^2 x^6 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.905933, size = 380, normalized size = 0.79 \[ \frac{1}{3} x^3 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 h+2 d e g+e^2 f\right )+\frac{1}{4} x^4 \left (a+b \sin ^{-1}(c x)\right ) \left (d^2 i+2 d e h+e^2 g\right )+d^2 f x \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{2} d x^2 (d g+2 e f) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e x^5 (2 d i+e h) \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} e^2 i x^6 \left (a+b \sin ^{-1}(c x)\right )+\frac{b \left (c \sqrt{1-c^2 x^2} \left (2 c^4 \left (25 d^2 (144 f+x (36 g+x (16 h+9 i x)))+2 d e x (900 f+x (400 g+9 x (25 h+16 i x)))+e^2 x^2 (400 f+x (225 g+4 x (36 h+25 i x)))\right )+c^2 \left (25 d^2 (64 h+27 i x)+2 d e \left (1600 g+675 h x+384 i x^2\right )+e^2 \left (1600 f+x \left (675 g+384 h x+250 i x^2\right )\right )\right )+3 e (512 d i+256 e h+125 e i x)\right )-75 \sin ^{-1}(c x) \left (9 c^2 \left (d^2 i+2 d e h+e^2 g\right )+24 c^4 d (d g+2 e f)+5 e^2 i\right )\right )}{7200 c^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

d^2*f*x*(a + b*ArcSin[c*x]) + (d*(2*e*f + d*g)*x^2*(a + b*ArcSin[c*x]))/2 + ((e^2*f + 2*d*e*g + d^2*h)*x^3*(a
+ b*ArcSin[c*x]))/3 + ((e^2*g + 2*d*e*h + d^2*i)*x^4*(a + b*ArcSin[c*x]))/4 + (e*(e*h + 2*d*i)*x^5*(a + b*ArcS
in[c*x]))/5 + (e^2*i*x^6*(a + b*ArcSin[c*x]))/6 + (b*(c*Sqrt[1 - c^2*x^2]*(3*e*(256*e*h + 512*d*i + 125*e*i*x)
+ c^2*(25*d^2*(64*h + 27*i*x) + 2*d*e*(1600*g + 675*h*x + 384*i*x^2) + e^2*(1600*f + x*(675*g + 384*h*x + 250
*i*x^2))) + 2*c^4*(25*d^2*(144*f + x*(36*g + x*(16*h + 9*i*x))) + 2*d*e*x*(900*f + x*(400*g + 9*x*(25*h + 16*i
*x))) + e^2*x^2*(400*f + x*(225*g + 4*x*(36*h + 25*i*x))))) - 75*(24*c^4*d*(2*e*f + d*g) + 5*e^2*i + 9*c^2*(e^
2*g + 2*d*e*h + d^2*i))*ArcSin[c*x]))/(7200*c^6)

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Maple [A]  time = 0.006, size = 674, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a/c^5*(1/6*e^2*i*c^6*x^6+1/5*(2*c*d*e*i+c*e^2*h)*c^5*x^5+1/4*(c^2*d^2*i+2*c^2*d*e*h+c^2*e^2*g)*c^4*x^4+1/
3*(c^3*d^2*h+2*c^3*d*e*g+c^3*e^2*f)*c^3*x^3+1/2*(c^4*d^2*g+2*c^4*d*e*f)*c^2*x^2+c^6*d^2*f*x)+b/c^5*(1/6*arcsin
(c*x)*e^2*i*c^6*x^6+2/5*arcsin(c*x)*c^6*x^5*d*e*i+1/5*arcsin(c*x)*c^6*x^5*e^2*h+1/4*arcsin(c*x)*c^6*x^4*d^2*i+
1/2*arcsin(c*x)*c^6*x^4*d*e*h+1/4*arcsin(c*x)*c^6*x^4*e^2*g+1/3*arcsin(c*x)*c^6*x^3*d^2*h+2/3*arcsin(c*x)*c^6*
x^3*d*e*g+1/3*arcsin(c*x)*c^6*x^3*e^2*f+1/2*arcsin(c*x)*c^6*x^2*d^2*g+arcsin(c*x)*c^6*x^2*d*e*f+arcsin(c*x)*c^
6*d^2*f*x-1/6*e^2*i*(-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*(24*c*d*e*i+12*c*e^2*h)*(-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*(15*c^2*d^2*i+30*c^2*d*e*h+15*c^2*e^2*g)*(-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^2*h+40*c^3*d*e*g+20*c^3*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^2*g+60*c^4*d*e*f)*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c
*x))+c^5*d^2*f*(-c^2*x^2+1)^(1/2)))

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Maxima [B]  time = 1.54514, size = 1231, normalized size = 2.54 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*e^2*i*x^6 + 1/5*a*e^2*h*x^5 + 2/5*a*d*e*i*x^5 + 1/4*a*e^2*g*x^4 + 1/2*a*d*e*h*x^4 + 1/4*a*d^2*i*x^4 + 1/
3*a*e^2*f*x^3 + 2/3*a*d*e*g*x^3 + 1/3*a*d^2*h*x^3 + a*d*e*f*x^2 + 1/2*a*d^2*g*x^2 + 1/2*(2*x^2*arcsin(c*x) + c
*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^2)))*b*d*e*f + 1/9*(3*x^3*arcsin(c*x) + c*(s
qrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*e^2*f + 1/4*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1
)*x/c^2 - arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^2)))*b*d^2*g + 2/9*(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*e*g + 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^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*e^2*g + 1/9*(3*x^3*arcsin(c*x) + c*(sqr
t(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*d^2*h + 1/16*(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^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*d*e*h + 1/75*(15*x^5*a
rcsin(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^2
*h + 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^2*x/sqr
t(c^2))/(sqrt(c^2)*c^4))*c)*b*d^2*i + 2/75*(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*d*e*i + 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^2*x/sqrt(c^2))/(sqrt(c^2)*c^6
))*c)*b*e^2*i + a*d^2*f*x + (c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*d^2*f/c

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Fricas [A]  time = 3.46506, size = 1426, normalized size = 2.95 \begin{align*} \frac{1200 \, a c^{6} e^{2} i x^{6} + 7200 \, a c^{6} d^{2} f x + 1440 \,{\left (a c^{6} e^{2} h + 2 \, a c^{6} d e i\right )} x^{5} + 1800 \,{\left (a c^{6} e^{2} g + 2 \, a c^{6} d e h + a c^{6} d^{2} i\right )} x^{4} + 2400 \,{\left (a c^{6} e^{2} f + 2 \, a c^{6} d e g + a c^{6} d^{2} h\right )} x^{3} + 3600 \,{\left (2 \, a c^{6} d e f + a c^{6} d^{2} g\right )} x^{2} + 15 \,{\left (80 \, b c^{6} e^{2} i x^{6} + 480 \, b c^{6} d^{2} f x - 240 \, b c^{4} d e f - 90 \, b c^{2} d e h + 96 \,{\left (b c^{6} e^{2} h + 2 \, b c^{6} d e i\right )} x^{5} + 120 \,{\left (b c^{6} e^{2} g + 2 \, b c^{6} d e h + b c^{6} d^{2} i\right )} x^{4} + 160 \,{\left (b c^{6} e^{2} f + 2 \, b c^{6} d e g + b c^{6} d^{2} h\right )} x^{3} + 240 \,{\left (2 \, b c^{6} d e f + b c^{6} d^{2} g\right )} x^{2} - 15 \,{\left (8 \, b c^{4} d^{2} + 3 \, b c^{2} e^{2}\right )} g - 5 \,{\left (9 \, b c^{2} d^{2} + 5 \, b e^{2}\right )} i\right )} \arcsin \left (c x\right ) +{\left (200 \, b c^{5} e^{2} i x^{5} + 3200 \, b c^{3} d e g + 1536 \, b c d e i + 288 \,{\left (b c^{5} e^{2} h + 2 \, b c^{5} d e i\right )} x^{4} + 50 \,{\left (9 \, b c^{5} e^{2} g + 18 \, b c^{5} d e h +{\left (9 \, b c^{5} d^{2} + 5 \, b c^{3} e^{2}\right )} i\right )} x^{3} + 32 \,{\left (25 \, b c^{5} e^{2} f + 50 \, b c^{5} d e g + 24 \, b c^{3} d e i +{\left (25 \, b c^{5} d^{2} + 12 \, b c^{3} e^{2}\right )} h\right )} x^{2} + 800 \,{\left (9 \, b c^{5} d^{2} + 2 \, b c^{3} e^{2}\right )} f + 64 \,{\left (25 \, b c^{3} d^{2} + 12 \, b c e^{2}\right )} h + 75 \,{\left (48 \, b c^{5} d e f + 18 \, b c^{3} d e h + 3 \,{\left (8 \, b c^{5} d^{2} + 3 \, b c^{3} e^{2}\right )} g +{\left (9 \, b c^{3} d^{2} + 5 \, b c e^{2}\right )} i\right )} x\right )} \sqrt{-c^{2} x^{2} + 1}}{7200 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7200*(1200*a*c^6*e^2*i*x^6 + 7200*a*c^6*d^2*f*x + 1440*(a*c^6*e^2*h + 2*a*c^6*d*e*i)*x^5 + 1800*(a*c^6*e^2*g
+ 2*a*c^6*d*e*h + a*c^6*d^2*i)*x^4 + 2400*(a*c^6*e^2*f + 2*a*c^6*d*e*g + a*c^6*d^2*h)*x^3 + 3600*(2*a*c^6*d*e
*f + a*c^6*d^2*g)*x^2 + 15*(80*b*c^6*e^2*i*x^6 + 480*b*c^6*d^2*f*x - 240*b*c^4*d*e*f - 90*b*c^2*d*e*h + 96*(b*
c^6*e^2*h + 2*b*c^6*d*e*i)*x^5 + 120*(b*c^6*e^2*g + 2*b*c^6*d*e*h + b*c^6*d^2*i)*x^4 + 160*(b*c^6*e^2*f + 2*b*
c^6*d*e*g + b*c^6*d^2*h)*x^3 + 240*(2*b*c^6*d*e*f + b*c^6*d^2*g)*x^2 - 15*(8*b*c^4*d^2 + 3*b*c^2*e^2)*g - 5*(9
*b*c^2*d^2 + 5*b*e^2)*i)*arcsin(c*x) + (200*b*c^5*e^2*i*x^5 + 3200*b*c^3*d*e*g + 1536*b*c*d*e*i + 288*(b*c^5*e
^2*h + 2*b*c^5*d*e*i)*x^4 + 50*(9*b*c^5*e^2*g + 18*b*c^5*d*e*h + (9*b*c^5*d^2 + 5*b*c^3*e^2)*i)*x^3 + 32*(25*b
*c^5*e^2*f + 50*b*c^5*d*e*g + 24*b*c^3*d*e*i + (25*b*c^5*d^2 + 12*b*c^3*e^2)*h)*x^2 + 800*(9*b*c^5*d^2 + 2*b*c
^3*e^2)*f + 64*(25*b*c^3*d^2 + 12*b*c*e^2)*h + 75*(48*b*c^5*d*e*f + 18*b*c^3*d*e*h + 3*(8*b*c^5*d^2 + 3*b*c^3*
e^2)*g + (9*b*c^3*d^2 + 5*b*c*e^2)*i)*x)*sqrt(-c^2*x^2 + 1))/c^6

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Sympy [A]  time = 10.5868, size = 1197, normalized size = 2.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.42659, size = 1917, normalized size = 3.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*a*d*i*x^5*e + 1/5*a*h*x^5*e^2 + 1/3*a*d^2*h*x^3 + 2/3*a*d*g*x^3*e + b*d^2*f*x*arcsin(c*x) + 1/3*a*f*x^3*e^
2 + a*d^2*f*x + 1/3*(c^2*x^2 - 1)*b*d^2*h*x*arcsin(c*x)/c^2 + 2/3*(c^2*x^2 - 1)*b*d*g*x*arcsin(c*x)*e/c^2 + 1/
4*sqrt(-c^2*x^2 + 1)*b*d^2*g*x/c + 1/2*sqrt(-c^2*x^2 + 1)*b*d*f*x*e/c + 1/2*(c^2*x^2 - 1)*b*d^2*g*arcsin(c*x)/
c^2 + 1/3*b*d^2*h*x*arcsin(c*x)/c^2 + 1/3*(c^2*x^2 - 1)*b*f*x*arcsin(c*x)*e^2/c^2 + (c^2*x^2 - 1)*b*d*f*arcsin
(c*x)*e/c^2 + 2/3*b*d*g*x*arcsin(c*x)*e/c^2 + 2/5*(c^2*x^2 - 1)^2*b*d*i*x*arcsin(c*x)*e/c^4 + sqrt(-c^2*x^2 +
1)*b*d^2*f/c - 1/16*(-c^2*x^2 + 1)^(3/2)*b*d^2*i*x/c^3 - 1/8*(-c^2*x^2 + 1)^(3/2)*b*d*h*x*e/c^3 + 1/2*(c^2*x^2
- 1)*a*d^2*g/c^2 + 1/4*b*d^2*g*arcsin(c*x)/c^2 + 1/4*(c^2*x^2 - 1)^2*b*d^2*i*arcsin(c*x)/c^4 + 1/3*b*f*x*arcs
in(c*x)*e^2/c^2 + 1/5*(c^2*x^2 - 1)^2*b*h*x*arcsin(c*x)*e^2/c^4 + (c^2*x^2 - 1)*a*d*f*e/c^2 + 1/2*b*d*f*arcsin
(c*x)*e/c^2 + 1/2*(c^2*x^2 - 1)^2*b*d*h*arcsin(c*x)*e/c^4 + 4/5*(c^2*x^2 - 1)*b*d*i*x*arcsin(c*x)*e/c^4 - 1/9*
(-c^2*x^2 + 1)^(3/2)*b*d^2*h/c^3 + 5/32*sqrt(-c^2*x^2 + 1)*b*d^2*i*x/c^3 - 1/16*(-c^2*x^2 + 1)^(3/2)*b*g*x*e^2
/c^3 - 2/9*(-c^2*x^2 + 1)^(3/2)*b*d*g*e/c^3 + 5/16*sqrt(-c^2*x^2 + 1)*b*d*h*x*e/c^3 + 1/4*(c^2*x^2 - 1)^2*a*d^
2*i/c^4 + 1/2*(c^2*x^2 - 1)*b*d^2*i*arcsin(c*x)/c^4 + 1/4*(c^2*x^2 - 1)^2*b*g*arcsin(c*x)*e^2/c^4 + 2/5*(c^2*x
^2 - 1)*b*h*x*arcsin(c*x)*e^2/c^4 + 1/2*(c^2*x^2 - 1)^2*a*d*h*e/c^4 + (c^2*x^2 - 1)*b*d*h*arcsin(c*x)*e/c^4 +
2/5*b*d*i*x*arcsin(c*x)*e/c^4 + 1/3*sqrt(-c^2*x^2 + 1)*b*d^2*h/c^3 - 1/9*(-c^2*x^2 + 1)^(3/2)*b*f*e^2/c^3 + 5/
32*sqrt(-c^2*x^2 + 1)*b*g*x*e^2/c^3 + 1/36*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*i*x*e^2/c^5 + 2/3*sqrt(-c^2*x^
2 + 1)*b*d*g*e/c^3 + 2/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d*i*e/c^5 + 1/2*(c^2*x^2 - 1)*a*d^2*i/c^4 + 5/3
2*b*d^2*i*arcsin(c*x)/c^4 + 1/4*(c^2*x^2 - 1)^2*a*g*e^2/c^4 + 1/2*(c^2*x^2 - 1)*b*g*arcsin(c*x)*e^2/c^4 + 1/6*
(c^2*x^2 - 1)^3*b*i*arcsin(c*x)*e^2/c^6 + 1/5*b*h*x*arcsin(c*x)*e^2/c^4 + (c^2*x^2 - 1)*a*d*h*e/c^4 + 5/16*b*d
*h*arcsin(c*x)*e/c^4 + 1/3*sqrt(-c^2*x^2 + 1)*b*f*e^2/c^3 + 1/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*h*e^2/c^
5 - 13/144*(-c^2*x^2 + 1)^(3/2)*b*i*x*e^2/c^5 - 4/15*(-c^2*x^2 + 1)^(3/2)*b*d*i*e/c^5 + 1/2*(c^2*x^2 - 1)*a*g*
e^2/c^4 + 1/6*(c^2*x^2 - 1)^3*a*i*e^2/c^6 + 5/32*b*g*arcsin(c*x)*e^2/c^4 + 1/2*(c^2*x^2 - 1)^2*b*i*arcsin(c*x)
*e^2/c^6 - 2/15*(-c^2*x^2 + 1)^(3/2)*b*h*e^2/c^5 + 11/96*sqrt(-c^2*x^2 + 1)*b*i*x*e^2/c^5 + 2/5*sqrt(-c^2*x^2
+ 1)*b*d*i*e/c^5 + 1/2*(c^2*x^2 - 1)^2*a*i*e^2/c^6 + 1/2*(c^2*x^2 - 1)*b*i*arcsin(c*x)*e^2/c^6 + 1/5*sqrt(-c^2
*x^2 + 1)*b*h*e^2/c^5 + 1/2*(c^2*x^2 - 1)*a*i*e^2/c^6 + 11/96*b*i*arcsin(c*x)*e^2/c^6