3.130 \(\int \frac {x^5 (a+b \sin ^{-1}(c x))}{(d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=219 \[ -\frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^6 d^3}-\frac {2 \left (a+b \sin ^{-1}(c x)\right )}{c^6 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \sin ^{-1}(c x)}{3 c^6 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {11 b \sqrt {d-c^2 d x^2} \tanh ^{-1}(c x)}{6 c^6 d^3 \sqrt {1-c^2 x^2}}+\frac {b x \sqrt {d-c^2 d x^2}}{c^5 d^3 \sqrt {1-c^2 x^2}}-\frac {b x \sqrt {d-c^2 d x^2}}{6 c^5 d^3 \left (1-c^2 x^2\right )^{3/2}} \]

[Out]

1/3*(a+b*arcsin(c*x))/c^6/d/(-c^2*d*x^2+d)^(3/2)-2*(a+b*arcsin(c*x))/c^6/d^2/(-c^2*d*x^2+d)^(1/2)-1/6*b*x*(-c^
2*d*x^2+d)^(1/2)/c^5/d^3/(-c^2*x^2+1)^(3/2)-(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^6/d^3+b*x*(-c^2*d*x^2+d)^
(1/2)/c^5/d^3/(-c^2*x^2+1)^(1/2)+11/6*b*arctanh(c*x)*(-c^2*d*x^2+d)^(1/2)/c^6/d^3/(-c^2*x^2+1)^(1/2)

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Rubi [A]  time = 0.32, antiderivative size = 234, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4703, 4677, 8, 321, 206, 288} \[ -\frac {4 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^6 d^3}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {b x^3}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 b x \sqrt {1-c^2 x^2}}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {11 b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{6 c^6 d^2 \sqrt {d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]

[Out]

-(b*x^3)/(6*c^3*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) + (5*b*x*Sqrt[1 - c^2*x^2])/(6*c^5*d^2*Sqrt[d - c^2
*d*x^2]) + (x^4*(a + b*ArcSin[c*x]))/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (4*x^2*(a + b*ArcSin[c*x]))/(3*c^4*d^2*
Sqrt[d - c^2*d*x^2]) - (8*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c^6*d^3) + (11*b*Sqrt[1 - c^2*x^2]*ArcTa
nh[c*x])/(6*c^6*d^2*Sqrt[d - c^2*d*x^2])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 206

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

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

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

Rule 4677

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

Rule 4703

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + (-Dist[(f^2*(m - 1))/(2*e*(p +
1)), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[(b*f*n*d^IntPart[p]*(d + e*x^2
)^FracPart[p])/(2*c*(p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSi
n[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && Gt
Q[m, 1]

Rubi steps

\begin {align*} \int \frac {x^5 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 c^2 d}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {x^4}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x^3}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {d-c^2 d x^2}} \, dx}{3 c^4 d^2}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {x^2}{1-c^2 x^2} \, dx}{2 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \int \frac {x^2}{1-c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x^3}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {11 b x \sqrt {1-c^2 x^2}}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^6 d^3}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{2 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b \sqrt {1-c^2 x^2}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b \sqrt {1-c^2 x^2}\right ) \int 1 \, dx}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x^3}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {5 b x \sqrt {1-c^2 x^2}}{6 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {4 x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^6 d^3}+\frac {11 b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{6 c^6 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.34, size = 169, normalized size = 0.77 \[ \frac {\sqrt {d-c^2 d x^2} \left (\sqrt {-c^2} \left (2 a \left (3 c^4 x^4-12 c^2 x^2+8\right )+b c x \sqrt {1-c^2 x^2} \left (6 c^2 x^2-5\right )+2 b \left (3 c^4 x^4-12 c^2 x^2+8\right ) \sin ^{-1}(c x)\right )+11 i b c \left (1-c^2 x^2\right )^{3/2} F\left (\left .i \sinh ^{-1}\left (\sqrt {-c^2} x\right )\right |1\right )\right )}{6 c^4 \left (-c^2\right )^{3/2} d^3 \left (c^2 x^2-1\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(a + b*ArcSin[c*x]))/(d - c^2*d*x^2)^(5/2),x]

[Out]

(Sqrt[d - c^2*d*x^2]*(Sqrt[-c^2]*(b*c*x*Sqrt[1 - c^2*x^2]*(-5 + 6*c^2*x^2) + 2*a*(8 - 12*c^2*x^2 + 3*c^4*x^4)
+ 2*b*(8 - 12*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x]) + (11*I)*b*c*(1 - c^2*x^2)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c^2
]*x], 1]))/(6*c^4*(-c^2)^(3/2)*d^3*(-1 + c^2*x^2)^2)

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fricas [A]  time = 1.14, size = 481, normalized size = 2.20 \[ \left [\frac {11 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} - 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} \sqrt {d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) - 4 \, {\left (6 \, b c^{3} x^{3} - 5 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 8 \, {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} + {\left (3 \, b c^{4} x^{4} - 12 \, b c^{2} x^{2} + 8 \, b\right )} \arcsin \left (c x\right ) + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{24 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}}, \frac {11 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} c \sqrt {-d} x}{c^{4} d x^{4} - d}\right ) - 2 \, {\left (6 \, b c^{3} x^{3} - 5 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} - 4 \, {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} + {\left (3 \, b c^{4} x^{4} - 12 \, b c^{2} x^{2} + 8 \, b\right )} \arcsin \left (c x\right ) + 8 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{12 \, {\left (c^{10} d^{3} x^{4} - 2 \, c^{8} d^{3} x^{2} + c^{6} d^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

[1/24*(11*(b*c^4*x^4 - 2*b*c^2*x^2 + b)*sqrt(d)*log(-(c^6*d*x^6 + 5*c^4*d*x^4 - 5*c^2*d*x^2 - 4*(c^3*x^3 + c*x
)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)*sqrt(d) - d)/(c^6*x^6 - 3*c^4*x^4 + 3*c^2*x^2 - 1)) - 4*(6*b*c^3*x^3
- 5*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 8*(3*a*c^4*x^4 - 12*a*c^2*x^2 + (3*b*c^4*x^4 - 12*b*c^2*
x^2 + 8*b)*arcsin(c*x) + 8*a)*sqrt(-c^2*d*x^2 + d))/(c^10*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3), 1/12*(11*(b*c^4*
x^4 - 2*b*c^2*x^2 + b)*sqrt(-d)*arctan(2*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)*c*sqrt(-d)*x/(c^4*d*x^4 - d))
- 2*(6*b*c^3*x^3 - 5*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 4*(3*a*c^4*x^4 - 12*a*c^2*x^2 + (3*b*c^
4*x^4 - 12*b*c^2*x^2 + 8*b)*arcsin(c*x) + 8*a)*sqrt(-c^2*d*x^2 + d))/(c^10*d^3*x^4 - 2*c^8*d^3*x^2 + c^6*d^3)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [C]  time = 0.50, size = 459, normalized size = 2.10 \[ -\frac {a \,x^{4}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {4 a \,x^{2}}{c^{4} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {8 a}{3 c^{6} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{c^{5} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{2}}{c^{4} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{c^{6} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{2}}{c^{4} \left (c^{2} x^{2}-1\right )^{2} d^{3}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{6 c^{5} \left (c^{2} x^{2}-1\right )^{2} d^{3}}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{3 c^{6} \left (c^{2} x^{2}-1\right )^{2} d^{3}}-\frac {11 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{6 c^{6} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {11 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{6 c^{6} d^{3} \left (c^{2} x^{2}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x)

[Out]

-a*x^4/c^2/d/(-c^2*d*x^2+d)^(3/2)+4*a/c^4*x^2/d/(-c^2*d*x^2+d)^(3/2)-8/3*a/c^6/d/(-c^2*d*x^2+d)^(3/2)-b*(-d*(c
^2*x^2-1))^(1/2)/c^5/d^3/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-b*(-d*(c^2*x^2-1))^(1/2)/c^4/d^3/(c^2*x^2-1)*arcsin(
c*x)*x^2+b*(-d*(c^2*x^2-1))^(1/2)/c^6/d^3/(c^2*x^2-1)*arcsin(c*x)+2*b*(-d*(c^2*x^2-1))^(1/2)/c^4/(c^2*x^2-1)^2
/d^3*arcsin(c*x)*x^2-1/6*b*(-d*(c^2*x^2-1))^(1/2)/c^5/(c^2*x^2-1)^2/d^3*(-c^2*x^2+1)^(1/2)*x-5/3*b*(-d*(c^2*x^
2-1))^(1/2)/c^6/(c^2*x^2-1)^2/d^3*arcsin(c*x)-11/6*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^6/d^3/(c^2*x^
2-1)*ln(I*c*x+(-c^2*x^2+1)^(1/2)+I)+11/6*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^6/d^3/(c^2*x^2-1)*ln(I*
c*x+(-c^2*x^2+1)^(1/2)-I)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, a {\left (\frac {3 \, x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} - \frac {12 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d} + \frac {8}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{6} d}\right )} + \frac {\frac {1}{4} \, {\left ({\left (c^{8} d^{3} x^{2} - c^{6} d^{3}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \sqrt {d} {\left (\frac {2 \, x}{c^{7} d^{3} x^{2} - c^{5} d^{3}} + \frac {4 \, {\left (c^{2} x^{3} - 6 \, x\right )}}{c^{5} d^{3}} + \frac {13 \, \log \left (c x + 1\right )}{c^{6} d^{3}} - \frac {13 \, \log \left (c x - 1\right )}{c^{6} d^{3}}\right )} + 4 \, {\left (3 \, c^{4} x^{4} - 12 \, c^{2} x^{2} + 8\right )} \sqrt {d} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} b}{3 \, {\left (c^{8} d^{3} x^{2} - c^{6} d^{3}\right )} \sqrt {c x + 1} \sqrt {-c x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

-1/3*a*(3*x^4/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 12*x^2/((-c^2*d*x^2 + d)^(3/2)*c^4*d) + 8/((-c^2*d*x^2 + d)^(3/
2)*c^6*d)) + 1/3*(3*(c^8*d^3*x^2 - c^6*d^3)*sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(d)*integrate(1/3*(3*c^4*x^6 - 12
*c^2*x^4 + 8*x^2)/(c^9*d^3*x^6 - 2*c^7*d^3*x^4 + c^5*d^3*x^2 + (c^7*d^3*x^4 - 2*c^5*d^3*x^2 + c^3*d^3)*e^(log(
c*x + 1) + log(-c*x + 1))), x) + (3*c^4*x^4 - 12*c^2*x^2 + 8)*sqrt(d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1
)))*b/((c^8*d^3*x^2 - c^6*d^3)*sqrt(c*x + 1)*sqrt(-c*x + 1))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(a + b*asin(c*x)))/(d - c^2*d*x^2)^(5/2),x)

[Out]

int((x^5*(a + b*asin(c*x)))/(d - c^2*d*x^2)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*asin(c*x))/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Integral(x**5*(a + b*asin(c*x))/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)

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