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

Optimal. Leaf size=258 \[ \frac{\left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e}+\frac{b x \sqrt{1-c^2 x^2} \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{1536 c^5}+\frac{5 b x \sqrt{1-c^2 x^2} \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )}{3072 c^7}-\frac{b \left (288 c^4 d^2 e^2+256 c^6 d^3 e+128 c^8 d^4+160 c^2 d e^3+35 e^4\right ) \sin ^{-1}(c x)}{1024 c^8 e}+\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^3}{64 c}+\frac{7 b x \sqrt{1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{384 c^3} \]

[Out]

(5*b*(2*c^2*d + e)*(40*c^4*d^2 + 40*c^2*d*e + 21*e^2)*x*Sqrt[1 - c^2*x^2])/(3072*c^7) + (b*(104*c^4*d^2 + 104*
c^2*d*e + 35*e^2)*x*Sqrt[1 - c^2*x^2]*(d + e*x^2))/(1536*c^5) + (7*b*(2*c^2*d + e)*x*Sqrt[1 - c^2*x^2]*(d + e*
x^2)^2)/(384*c^3) + (b*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^3)/(64*c) - (b*(128*c^8*d^4 + 256*c^6*d^3*e + 288*c^4*d
^2*e^2 + 160*c^2*d*e^3 + 35*e^4)*ArcSin[c*x])/(1024*c^8*e) + ((d + e*x^2)^4*(a + b*ArcSin[c*x]))/(8*e)

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Rubi [A]  time = 0.267678, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {4729, 416, 528, 388, 216} \[ \frac{\left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e}+\frac{b x \sqrt{1-c^2 x^2} \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) \left (d+e x^2\right )}{1536 c^5}+\frac{5 b x \sqrt{1-c^2 x^2} \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right )}{3072 c^7}-\frac{b \left (288 c^4 d^2 e^2+256 c^6 d^3 e+128 c^8 d^4+160 c^2 d e^3+35 e^4\right ) \sin ^{-1}(c x)}{1024 c^8 e}+\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^3}{64 c}+\frac{7 b x \sqrt{1-c^2 x^2} \left (2 c^2 d+e\right ) \left (d+e x^2\right )^2}{384 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*b*(2*c^2*d + e)*(40*c^4*d^2 + 40*c^2*d*e + 21*e^2)*x*Sqrt[1 - c^2*x^2])/(3072*c^7) + (b*(104*c^4*d^2 + 104*
c^2*d*e + 35*e^2)*x*Sqrt[1 - c^2*x^2]*(d + e*x^2))/(1536*c^5) + (7*b*(2*c^2*d + e)*x*Sqrt[1 - c^2*x^2]*(d + e*
x^2)^2)/(384*c^3) + (b*x*Sqrt[1 - c^2*x^2]*(d + e*x^2)^3)/(64*c) - (b*(128*c^8*d^4 + 256*c^6*d^3*e + 288*c^4*d
^2*e^2 + 160*c^2*d*e^3 + 35*e^4)*ArcSin[c*x])/(1024*c^8*e) + ((d + e*x^2)^4*(a + b*ArcSin[c*x]))/(8*e)

Rule 4729

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^(p + 1
)*(a + b*ArcSin[c*x]))/(2*e*(p + 1)), x] - Dist[(b*c)/(2*e*(p + 1)), Int[(d + e*x^2)^(p + 1)/Sqrt[1 - c^2*x^2]
, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -1]

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

Rule 528

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

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 x \left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e}-\frac{(b c) \int \frac{\left (d+e x^2\right )^4}{\sqrt{1-c^2 x^2}} \, dx}{8 e}\\ &=\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^3}{64 c}+\frac{\left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e}+\frac{b \int \frac{\left (d+e x^2\right )^2 \left (-d \left (8 c^2 d+e\right )-7 e \left (2 c^2 d+e\right ) x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{64 c e}\\ &=\frac{7 b \left (2 c^2 d+e\right ) x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^2}{384 c^3}+\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^3}{64 c}+\frac{\left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e}-\frac{b \int \frac{\left (d+e x^2\right ) \left (d \left (48 c^4 d^2+20 c^2 d e+7 e^2\right )+e \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x^2\right )}{\sqrt{1-c^2 x^2}} \, dx}{384 c^3 e}\\ &=\frac{b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \sqrt{1-c^2 x^2} \left (d+e x^2\right )}{1536 c^5}+\frac{7 b \left (2 c^2 d+e\right ) x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^2}{384 c^3}+\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^3}{64 c}+\frac{\left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e}+\frac{b \int \frac{-d \left (192 c^6 d^3+184 c^4 d^2 e+132 c^2 d e^2+35 e^3\right )-5 e \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x^2}{\sqrt{1-c^2 x^2}} \, dx}{1536 c^5 e}\\ &=\frac{5 b \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x \sqrt{1-c^2 x^2}}{3072 c^7}+\frac{b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \sqrt{1-c^2 x^2} \left (d+e x^2\right )}{1536 c^5}+\frac{7 b \left (2 c^2 d+e\right ) x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^2}{384 c^3}+\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^3}{64 c}+\frac{\left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e}-\frac{\left (b \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right )\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{1024 c^7 e}\\ &=\frac{5 b \left (2 c^2 d+e\right ) \left (40 c^4 d^2+40 c^2 d e+21 e^2\right ) x \sqrt{1-c^2 x^2}}{3072 c^7}+\frac{b \left (104 c^4 d^2+104 c^2 d e+35 e^2\right ) x \sqrt{1-c^2 x^2} \left (d+e x^2\right )}{1536 c^5}+\frac{7 b \left (2 c^2 d+e\right ) x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^2}{384 c^3}+\frac{b x \sqrt{1-c^2 x^2} \left (d+e x^2\right )^3}{64 c}-\frac{b \left (128 c^8 d^4+256 c^6 d^3 e+288 c^4 d^2 e^2+160 c^2 d e^3+35 e^4\right ) \sin ^{-1}(c x)}{1024 c^8 e}+\frac{\left (d+e x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 e}\\ \end{align*}

Mathematica [A]  time = 0.199539, size = 232, normalized size = 0.9 \[ \frac{c x \left (384 a c^7 x \left (6 d^2 e x^2+4 d^3+4 d e^2 x^4+e^3 x^6\right )+b \sqrt{1-c^2 x^2} \left (16 c^6 \left (36 d^2 e x^2+48 d^3+16 d e^2 x^4+3 e^3 x^6\right )+8 c^4 e \left (108 d^2+40 d e x^2+7 e^2 x^4\right )+10 c^2 e^2 \left (48 d+7 e x^2\right )+105 e^3\right )\right )+3 b \sin ^{-1}(c x) \left (128 c^8 \left (6 d^2 e x^4+4 d^3 x^2+4 d e^2 x^6+e^3 x^8\right )-288 c^4 d^2 e-256 c^6 d^3-160 c^2 d e^2-35 e^3\right )}{3072 c^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(384*a*c^7*x*(4*d^3 + 6*d^2*e*x^2 + 4*d*e^2*x^4 + e^3*x^6) + b*Sqrt[1 - c^2*x^2]*(105*e^3 + 10*c^2*e^2*(4
8*d + 7*e*x^2) + 8*c^4*e*(108*d^2 + 40*d*e*x^2 + 7*e^2*x^4) + 16*c^6*(48*d^3 + 36*d^2*e*x^2 + 16*d*e^2*x^4 + 3
*e^3*x^6))) + 3*b*(-256*c^6*d^3 - 288*c^4*d^2*e - 160*c^2*d*e^2 - 35*e^3 + 128*c^8*(4*d^3*x^2 + 6*d^2*e*x^4 +
4*d*e^2*x^6 + e^3*x^8))*ArcSin[c*x])/(3072*c^8)

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Maple [A]  time = 0.005, size = 369, normalized size = 1.4 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{a}{{c}^{6}} \left ({\frac{{e}^{3}{c}^{8}{x}^{8}}{8}}+{\frac{{c}^{8}d{e}^{2}{x}^{6}}{2}}+{\frac{3\,{c}^{8}{d}^{2}e{x}^{4}}{4}}+{\frac{{x}^{2}{c}^{8}{d}^{3}}{2}} \right ) }+{\frac{b}{{c}^{6}} \left ({\frac{\arcsin \left ( cx \right ){e}^{3}{c}^{8}{x}^{8}}{8}}+{\frac{\arcsin \left ( cx \right ){c}^{8}d{e}^{2}{x}^{6}}{2}}+{\frac{3\,\arcsin \left ( cx \right ){c}^{8}{d}^{2}e{x}^{4}}{4}}+{\frac{\arcsin \left ( cx \right ){d}^{3}{c}^{8}{x}^{2}}{2}}-{\frac{{e}^{3}}{8} \left ( -{\frac{{c}^{7}{x}^{7}}{8}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{7\,{c}^{5}{x}^{5}}{48}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{35\,{c}^{3}{x}^{3}}{192}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{35\,cx}{128}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{35\,\arcsin \left ( cx \right ) }{128}} \right ) }-{\frac{{c}^{2}d{e}^{2}}{2} \left ( -{\frac{{c}^{5}{x}^{5}}{6}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{5\,{c}^{3}{x}^{3}}{24}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{5\,cx}{16}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{5\,\arcsin \left ( cx \right ) }{16}} \right ) }-{\frac{3\,{c}^{4}{d}^{2}e}{4} \left ( -{\frac{{c}^{3}{x}^{3}}{4}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{3\,cx}{8}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{3\,\arcsin \left ( cx \right ) }{8}} \right ) }-{\frac{{d}^{3}{c}^{6}}{2} \left ( -{\frac{cx}{2}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{\arcsin \left ( cx \right ) }{2}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(a/c^6*(1/8*e^3*c^8*x^8+1/2*c^8*d*e^2*x^6+3/4*c^8*d^2*e*x^4+1/2*x^2*c^8*d^3)+b/c^6*(1/8*arcsin(c*x)*e^3*
c^8*x^8+1/2*arcsin(c*x)*c^8*d*e^2*x^6+3/4*arcsin(c*x)*c^8*d^2*e*x^4+1/2*arcsin(c*x)*d^3*c^8*x^2-1/8*e^3*(-1/8*
c^7*x^7*(-c^2*x^2+1)^(1/2)-7/48*c^5*x^5*(-c^2*x^2+1)^(1/2)-35/192*c^3*x^3*(-c^2*x^2+1)^(1/2)-35/128*c*x*(-c^2*
x^2+1)^(1/2)+35/128*arcsin(c*x))-1/2*c^2*d*e^2*(-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))-3/4*c^4*d^2*e*(-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))-1/2*d^3*c^6*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))))

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Maxima [A]  time = 1.48887, size = 529, normalized size = 2.05 \begin{align*} \frac{1}{8} \, a e^{3} x^{8} + \frac{1}{2} \, a d e^{2} x^{6} + \frac{3}{4} \, a d^{2} e x^{4} + \frac{1}{2} \, a d^{3} x^{2} + \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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} b d^{3} + \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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} b d^{2} e + \frac{1}{96} \,{\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 (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} b d e^{2} + \frac{1}{3072} \,{\left (384 \, x^{8} \arcsin \left (c x\right ) +{\left (\frac{48 \, \sqrt{-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac{56 \, \sqrt{-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac{70 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac{105 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{8}} - \frac{105 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{8}}\right )} c\right )} b e^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a*e^3*x^8 + 1/2*a*d*e^2*x^6 + 3/4*a*d^2*e*x^4 + 1/2*a*d^3*x^2 + 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^3 + 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^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*d^2*e + 1/96*(48*x^6*a
rcsin(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*d*e^2 + 1/3072*(384*x^8*arcsin(c*x) + (48*sqrt(-c^2*x^2 + 1)*x^7
/c^2 + 56*sqrt(-c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(-c^2*x^2 + 1)*x^3/c^6 + 105*sqrt(-c^2*x^2 + 1)*x/c^8 - 105*arcs
in(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^8))*c)*b*e^3

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Fricas [A]  time = 2.15815, size = 636, normalized size = 2.47 \begin{align*} \frac{384 \, a c^{8} e^{3} x^{8} + 1536 \, a c^{8} d e^{2} x^{6} + 2304 \, a c^{8} d^{2} e x^{4} + 1536 \, a c^{8} d^{3} x^{2} + 3 \,{\left (128 \, b c^{8} e^{3} x^{8} + 512 \, b c^{8} d e^{2} x^{6} + 768 \, b c^{8} d^{2} e x^{4} + 512 \, b c^{8} d^{3} x^{2} - 256 \, b c^{6} d^{3} - 288 \, b c^{4} d^{2} e - 160 \, b c^{2} d e^{2} - 35 \, b e^{3}\right )} \arcsin \left (c x\right ) +{\left (48 \, b c^{7} e^{3} x^{7} + 8 \,{\left (32 \, b c^{7} d e^{2} + 7 \, b c^{5} e^{3}\right )} x^{5} + 2 \,{\left (288 \, b c^{7} d^{2} e + 160 \, b c^{5} d e^{2} + 35 \, b c^{3} e^{3}\right )} x^{3} + 3 \,{\left (256 \, b c^{7} d^{3} + 288 \, b c^{5} d^{2} e + 160 \, b c^{3} d e^{2} + 35 \, b c e^{3}\right )} x\right )} \sqrt{-c^{2} x^{2} + 1}}{3072 \, c^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3072*(384*a*c^8*e^3*x^8 + 1536*a*c^8*d*e^2*x^6 + 2304*a*c^8*d^2*e*x^4 + 1536*a*c^8*d^3*x^2 + 3*(128*b*c^8*e^
3*x^8 + 512*b*c^8*d*e^2*x^6 + 768*b*c^8*d^2*e*x^4 + 512*b*c^8*d^3*x^2 - 256*b*c^6*d^3 - 288*b*c^4*d^2*e - 160*
b*c^2*d*e^2 - 35*b*e^3)*arcsin(c*x) + (48*b*c^7*e^3*x^7 + 8*(32*b*c^7*d*e^2 + 7*b*c^5*e^3)*x^5 + 2*(288*b*c^7*
d^2*e + 160*b*c^5*d*e^2 + 35*b*c^3*e^3)*x^3 + 3*(256*b*c^7*d^3 + 288*b*c^5*d^2*e + 160*b*c^3*d*e^2 + 35*b*c*e^
3)*x)*sqrt(-c^2*x^2 + 1))/c^8

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Sympy [A]  time = 16.9054, size = 483, normalized size = 1.87 \begin{align*} \begin{cases} \frac{a d^{3} x^{2}}{2} + \frac{3 a d^{2} e x^{4}}{4} + \frac{a d e^{2} x^{6}}{2} + \frac{a e^{3} x^{8}}{8} + \frac{b d^{3} x^{2} \operatorname{asin}{\left (c x \right )}}{2} + \frac{3 b d^{2} e x^{4} \operatorname{asin}{\left (c x \right )}}{4} + \frac{b d e^{2} x^{6} \operatorname{asin}{\left (c x \right )}}{2} + \frac{b e^{3} x^{8} \operatorname{asin}{\left (c x \right )}}{8} + \frac{b d^{3} x \sqrt{- c^{2} x^{2} + 1}}{4 c} + \frac{3 b d^{2} e x^{3} \sqrt{- c^{2} x^{2} + 1}}{16 c} + \frac{b d e^{2} x^{5} \sqrt{- c^{2} x^{2} + 1}}{12 c} + \frac{b e^{3} x^{7} \sqrt{- c^{2} x^{2} + 1}}{64 c} - \frac{b d^{3} \operatorname{asin}{\left (c x \right )}}{4 c^{2}} + \frac{9 b d^{2} e x \sqrt{- c^{2} x^{2} + 1}}{32 c^{3}} + \frac{5 b d e^{2} x^{3} \sqrt{- c^{2} x^{2} + 1}}{48 c^{3}} + \frac{7 b e^{3} x^{5} \sqrt{- c^{2} x^{2} + 1}}{384 c^{3}} - \frac{9 b d^{2} e \operatorname{asin}{\left (c x \right )}}{32 c^{4}} + \frac{5 b d e^{2} x \sqrt{- c^{2} x^{2} + 1}}{32 c^{5}} + \frac{35 b e^{3} x^{3} \sqrt{- c^{2} x^{2} + 1}}{1536 c^{5}} - \frac{5 b d e^{2} \operatorname{asin}{\left (c x \right )}}{32 c^{6}} + \frac{35 b e^{3} x \sqrt{- c^{2} x^{2} + 1}}{1024 c^{7}} - \frac{35 b e^{3} \operatorname{asin}{\left (c x \right )}}{1024 c^{8}} & \text{for}\: c \neq 0 \\a \left (\frac{d^{3} x^{2}}{2} + \frac{3 d^{2} e x^{4}}{4} + \frac{d e^{2} x^{6}}{2} + \frac{e^{3} x^{8}}{8}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x**2+d)**3*(a+b*asin(c*x)),x)

[Out]

Piecewise((a*d**3*x**2/2 + 3*a*d**2*e*x**4/4 + a*d*e**2*x**6/2 + a*e**3*x**8/8 + b*d**3*x**2*asin(c*x)/2 + 3*b
*d**2*e*x**4*asin(c*x)/4 + b*d*e**2*x**6*asin(c*x)/2 + b*e**3*x**8*asin(c*x)/8 + b*d**3*x*sqrt(-c**2*x**2 + 1)
/(4*c) + 3*b*d**2*e*x**3*sqrt(-c**2*x**2 + 1)/(16*c) + b*d*e**2*x**5*sqrt(-c**2*x**2 + 1)/(12*c) + b*e**3*x**7
*sqrt(-c**2*x**2 + 1)/(64*c) - b*d**3*asin(c*x)/(4*c**2) + 9*b*d**2*e*x*sqrt(-c**2*x**2 + 1)/(32*c**3) + 5*b*d
*e**2*x**3*sqrt(-c**2*x**2 + 1)/(48*c**3) + 7*b*e**3*x**5*sqrt(-c**2*x**2 + 1)/(384*c**3) - 9*b*d**2*e*asin(c*
x)/(32*c**4) + 5*b*d*e**2*x*sqrt(-c**2*x**2 + 1)/(32*c**5) + 35*b*e**3*x**3*sqrt(-c**2*x**2 + 1)/(1536*c**5) -
5*b*d*e**2*asin(c*x)/(32*c**6) + 35*b*e**3*x*sqrt(-c**2*x**2 + 1)/(1024*c**7) - 35*b*e**3*asin(c*x)/(1024*c**
8), Ne(c, 0)), (a*(d**3*x**2/2 + 3*d**2*e*x**4/4 + d*e**2*x**6/2 + e**3*x**8/8), True))

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Giac [B]  time = 1.23898, size = 987, normalized size = 3.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(-c^2*x^2 + 1)*b*d^3*x/c + 1/2*(c^2*x^2 - 1)*b*d^3*arcsin(c*x)/c^2 - 3/16*(-c^2*x^2 + 1)^(3/2)*b*d^2*x
*e/c^3 + 1/2*(c^2*x^2 - 1)*a*d^3/c^2 + 1/4*b*d^3*arcsin(c*x)/c^2 + 3/4*(c^2*x^2 - 1)^2*b*d^2*arcsin(c*x)*e/c^4
+ 15/32*sqrt(-c^2*x^2 + 1)*b*d^2*x*e/c^3 + 3/4*(c^2*x^2 - 1)^2*a*d^2*e/c^4 + 3/2*(c^2*x^2 - 1)*b*d^2*arcsin(c
*x)*e/c^4 + 1/12*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d*x*e^2/c^5 + 1/2*(c^2*x^2 - 1)^3*b*d*arcsin(c*x)*e^2/c^
6 + 3/2*(c^2*x^2 - 1)*a*d^2*e/c^4 + 15/32*b*d^2*arcsin(c*x)*e/c^4 - 13/48*(-c^2*x^2 + 1)^(3/2)*b*d*x*e^2/c^5 +
1/2*(c^2*x^2 - 1)^3*a*d*e^2/c^6 + 3/2*(c^2*x^2 - 1)^2*b*d*arcsin(c*x)*e^2/c^6 + 1/64*(c^2*x^2 - 1)^3*sqrt(-c^
2*x^2 + 1)*b*x*e^3/c^7 + 11/32*sqrt(-c^2*x^2 + 1)*b*d*x*e^2/c^5 + 1/8*(c^2*x^2 - 1)^4*b*arcsin(c*x)*e^3/c^8 +
3/2*(c^2*x^2 - 1)^2*a*d*e^2/c^6 + 3/2*(c^2*x^2 - 1)*b*d*arcsin(c*x)*e^2/c^6 + 25/384*(c^2*x^2 - 1)^2*sqrt(-c^2
*x^2 + 1)*b*x*e^3/c^7 + 1/8*(c^2*x^2 - 1)^4*a*e^3/c^8 + 1/2*(c^2*x^2 - 1)^3*b*arcsin(c*x)*e^3/c^8 + 3/2*(c^2*x
^2 - 1)*a*d*e^2/c^6 + 11/32*b*d*arcsin(c*x)*e^2/c^6 - 163/1536*(-c^2*x^2 + 1)^(3/2)*b*x*e^3/c^7 + 1/2*(c^2*x^2
- 1)^3*a*e^3/c^8 + 3/4*(c^2*x^2 - 1)^2*b*arcsin(c*x)*e^3/c^8 + 93/1024*sqrt(-c^2*x^2 + 1)*b*x*e^3/c^7 + 3/4*(
c^2*x^2 - 1)^2*a*e^3/c^8 + 1/2*(c^2*x^2 - 1)*b*arcsin(c*x)*e^3/c^8 + 1/2*(c^2*x^2 - 1)*a*e^3/c^8 + 93/1024*b*a
rcsin(c*x)*e^3/c^8