3.7.17 \(\int x (d+e x^2)^3 (a+b \text {ArcSin}(c x)) \, dx\) [617]

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

[Out]

-1/1024*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)/c^8/e+1/8*(e*x^2+d)^4*(
a+b*arcsin(c*x))/e+5/3072*b*(2*c^2*d+e)*(40*c^4*d^2+40*c^2*d*e+21*e^2)*x*(-c^2*x^2+1)^(1/2)/c^7+1/1536*b*(104*
c^4*d^2+104*c^2*d*e+35*e^2)*x*(e*x^2+d)*(-c^2*x^2+1)^(1/2)/c^5+7/384*b*(2*c^2*d+e)*x*(e*x^2+d)^2*(-c^2*x^2+1)^
(1/2)/c^3+1/64*b*x*(e*x^2+d)^3*(-c^2*x^2+1)^(1/2)/c

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Rubi [A]
time = 0.20, antiderivative size = 258, normalized size of antiderivative = 1.00, 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 = {4813, 427, 542, 396, 222} \begin {gather*} \frac {\left (d+e x^2\right )^4 (a+b \text {ArcSin}(c x))}{8 e}-\frac {b \text {ArcSin}(c x) \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 )}{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}+\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 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} \end {gather*}

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 222

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 396

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 427

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 542

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 4813

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]

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*}

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Mathematica [A]
time = 0.14, size = 232, normalized size = 0.90 \begin {gather*} \frac {c x \left (384 a c^7 x \left (4 d^3+6 d^2 e x^2+4 d e^2 x^4+e^3 x^6\right )+b \sqrt {1-c^2 x^2} \left (105 e^3+10 c^2 e^2 \left (48 d+7 e x^2\right )+8 c^4 e \left (108 d^2+40 d e x^2+7 e^2 x^4\right )+16 c^6 \left (48 d^3+36 d^2 e x^2+16 d e^2 x^4+3 e^3 x^6\right )\right )\right )+3 b \left (-256 c^6 d^3-288 c^4 d^2 e-160 c^2 d e^2-35 e^3+128 c^8 \left (4 d^3 x^2+6 d^2 e x^4+4 d e^2 x^6+e^3 x^8\right )\right ) \text {ArcSin}(c x)}{3072 c^8} \end {gather*}

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 = 0.13, size = 376, normalized size = 1.46

method	result	size
derivativedivides	\(\frac {\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{4} a}{8 c^{6} e}+\frac {b \left (\frac {\arcsin \left (c x \right ) c^{8} d^{4}}{8 e}+\frac {\arcsin \left (c x \right ) c^{8} d^{3} x^{2}}{2}+\frac {3 e \arcsin \left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e^{2} \arcsin \left (c x \right ) c^{8} d \,x^{6}}{2}+\frac {e^{3} \arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {c^{8} d^{4} \arcsin \left (c x \right )+4 c^{6} d^{3} e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+6 c^{4} d^{2} e^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+4 d \,c^{2} e^{3} \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )+e^{4} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )}{8 e}\right )}{c^{6}}}{c^{2}}\)	\(376\)
default	\(\frac {\frac {\left (c^{2} e \,x^{2}+c^{2} d \right )^{4} a}{8 c^{6} e}+\frac {b \left (\frac {\arcsin \left (c x \right ) c^{8} d^{4}}{8 e}+\frac {\arcsin \left (c x \right ) c^{8} d^{3} x^{2}}{2}+\frac {3 e \arcsin \left (c x \right ) c^{8} d^{2} x^{4}}{4}+\frac {e^{2} \arcsin \left (c x \right ) c^{8} d \,x^{6}}{2}+\frac {e^{3} \arcsin \left (c x \right ) c^{8} x^{8}}{8}-\frac {c^{8} d^{4} \arcsin \left (c x \right )+4 c^{6} d^{3} e \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+6 c^{4} d^{2} e^{2} \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )+4 d \,c^{2} e^{3} \left (-\frac {c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{6}-\frac {5 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{24}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{16}+\frac {5 \arcsin \left (c x \right )}{16}\right )+e^{4} \left (-\frac {c^{7} x^{7} \sqrt {-c^{2} x^{2}+1}}{8}-\frac {7 c^{5} x^{5} \sqrt {-c^{2} x^{2}+1}}{48}-\frac {35 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{192}-\frac {35 c x \sqrt {-c^{2} x^{2}+1}}{128}+\frac {35 \arcsin \left (c x \right )}{128}\right )}{8 e}\right )}{c^{6}}}{c^{2}}\)	\(376\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(1/8*(c^2*e*x^2+c^2*d)^4*a/c^6/e+b/c^6*(1/8/e*arcsin(c*x)*c^8*d^4+1/2*arcsin(c*x)*c^8*d^3*x^2+3/4*e*arcs
in(c*x)*c^8*d^2*x^4+1/2*e^2*arcsin(c*x)*c^8*d*x^6+1/8*e^3*arcsin(c*x)*c^8*x^8-1/8/e*(c^8*d^4*arcsin(c*x)+4*c^6
*d^3*e*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+6*c^4*d^2*e^2*(-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))+4*d*c^2*e^3*(-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))+e^4*(-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)))))

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Maxima [A]
time = 0.50, size = 342, normalized size = 1.33 \begin {gather*} \frac {1}{8} \, a x^{8} e^{3} + \frac {1}{2} \, a d x^{6} e^{2} + \frac {3}{4} \, a d^{2} x^{4} e + \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 (c x\right )}{c^{3}}\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 (c x\right )}{c^{5}}\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 (c x\right )}{c^{7}}\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 (c x\right )}{c^{9}}\right )} c\right )} b e^{3} \end {gather*}

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*x^8*e^3 + 1/2*a*d*x^6*e^2 + 3/4*a*d^2*x^4*e + 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*x)/c^3))*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*x)/c^5)*c)*b*d^2*e + 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 + 1/3072*(384*x^8*ar
csin(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 + 1
05*sqrt(-c^2*x^2 + 1)*x/c^8 - 105*arcsin(c*x)/c^9)*c)*b*e^3

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Fricas [A]
time = 1.80, size = 267, normalized size = 1.03 \begin {gather*} \frac {384 \, a c^{8} x^{8} e^{3} + 1536 \, a c^{8} d x^{6} e^{2} + 2304 \, a c^{8} d^{2} x^{4} e + 1536 \, a c^{8} d^{3} x^{2} + 3 \, {\left (512 \, b c^{8} d^{3} x^{2} - 256 \, b c^{6} d^{3} + {\left (128 \, b c^{8} x^{8} - 35 \, b\right )} e^{3} + 32 \, {\left (16 \, b c^{8} d x^{6} - 5 \, b c^{2} d\right )} e^{2} + 96 \, {\left (8 \, b c^{8} d^{2} x^{4} - 3 \, b c^{4} d^{2}\right )} e\right )} \arcsin \left (c x\right ) + {\left (768 \, b c^{7} d^{3} x + {\left (48 \, b c^{7} x^{7} + 56 \, b c^{5} x^{5} + 70 \, b c^{3} x^{3} + 105 \, b c x\right )} e^{3} + 32 \, {\left (8 \, b c^{7} d x^{5} + 10 \, b c^{5} d x^{3} + 15 \, b c^{3} d x\right )} e^{2} + 288 \, {\left (2 \, b c^{7} d^{2} x^{3} + 3 \, b c^{5} d^{2} x\right )} e\right )} \sqrt {-c^{2} x^{2} + 1}}{3072 \, c^{8}} \end {gather*}

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*x^8*e^3 + 1536*a*c^8*d*x^6*e^2 + 2304*a*c^8*d^2*x^4*e + 1536*a*c^8*d^3*x^2 + 3*(512*b*c^8*d^
3*x^2 - 256*b*c^6*d^3 + (128*b*c^8*x^8 - 35*b)*e^3 + 32*(16*b*c^8*d*x^6 - 5*b*c^2*d)*e^2 + 96*(8*b*c^8*d^2*x^4
- 3*b*c^4*d^2)*e)*arcsin(c*x) + (768*b*c^7*d^3*x + (48*b*c^7*x^7 + 56*b*c^5*x^5 + 70*b*c^3*x^3 + 105*b*c*x)*e
^3 + 32*(8*b*c^7*d*x^5 + 10*b*c^5*d*x^3 + 15*b*c^3*d*x)*e^2 + 288*(2*b*c^7*d^2*x^3 + 3*b*c^5*d^2*x)*e)*sqrt(-c
^2*x^2 + 1))/c^8

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Sympy [A]
time = 1.20, size = 483, normalized size = 1.87 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (238) = 476\).
time = 0.42, size = 597, normalized size = 2.31 \begin {gather*} \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 {\sqrt {-c^{2} x^{2} + 1} b d^{3} x}{4 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b d^{3} \arcsin \left (c x\right )}{2 \, c^{2}} - \frac {3 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2} e x}{16 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d^{3}}{2 \, c^{2}} + \frac {b d^{3} \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d^{2} e \arcsin \left (c x\right )}{4 \, c^{4}} + \frac {15 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} e x}{32 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d e^{2} x}{12 \, c^{5}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b d^{2} e \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b d e^{2} \arcsin \left (c x\right )}{2 \, c^{6}} - \frac {13 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e^{2} x}{48 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b e^{3} x}{64 \, c^{7}} + \frac {15 \, b d^{2} e \arcsin \left (c x\right )}{32 \, c^{4}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d e^{2} \arcsin \left (c x\right )}{2 \, c^{6}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b e^{3} \arcsin \left (c x\right )}{8 \, c^{8}} + \frac {11 \, \sqrt {-c^{2} x^{2} + 1} b d e^{2} x}{32 \, c^{5}} + \frac {25 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e^{3} x}{384 \, c^{7}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )} b d e^{2} \arcsin \left (c x\right )}{2 \, c^{6}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b e^{3} \arcsin \left (c x\right )}{2 \, c^{8}} - \frac {163 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{3} x}{1536 \, c^{7}} + \frac {11 \, b d e^{2} \arcsin \left (c x\right )}{32 \, c^{6}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b e^{3} \arcsin \left (c x\right )}{4 \, c^{8}} + \frac {93 \, \sqrt {-c^{2} x^{2} + 1} b e^{3} x}{1024 \, c^{7}} + \frac {{\left (c^{2} x^{2} - 1\right )} b e^{3} \arcsin \left (c x\right )}{2 \, c^{8}} + \frac {93 \, b e^{3} \arcsin \left (c x\right )}{1024 \, c^{8}} \end {gather*}

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]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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