3.2037 \(\int \frac {x^5}{(a+\frac {b}{x^3})^{3/2}} \, dx\)

Optimal. Leaf size=95 \[ \frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{4 a^{7/2}}-\frac {5 b^2}{4 a^3 \sqrt {a+\frac {b}{x^3}}}-\frac {5 b x^3}{12 a^2 \sqrt {a+\frac {b}{x^3}}}+\frac {x^6}{6 a \sqrt {a+\frac {b}{x^3}}} \]

[Out]

5/4*b^2*arctanh((a+b/x^3)^(1/2)/a^(1/2))/a^(7/2)-5/4*b^2/a^3/(a+b/x^3)^(1/2)-5/12*b*x^3/a^2/(a+b/x^3)^(1/2)+1/
6*x^6/a/(a+b/x^3)^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{4 a^{7/2}}+\frac {5 x^6 \sqrt {a+\frac {b}{x^3}}}{6 a^2}-\frac {5 b x^3 \sqrt {a+\frac {b}{x^3}}}{4 a^3}-\frac {2 x^6}{3 a \sqrt {a+\frac {b}{x^3}}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b/x^3)^(3/2),x]

[Out]

(-5*b*Sqrt[a + b/x^3]*x^3)/(4*a^3) - (2*x^6)/(3*a*Sqrt[a + b/x^3]) + (5*Sqrt[a + b/x^3]*x^6)/(6*a^2) + (5*b^2*
ArcTanh[Sqrt[a + b/x^3]/Sqrt[a]])/(4*a^(7/2))

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^5}{\left (a+\frac {b}{x^3}\right )^{3/2}} \, dx &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^3 (a+b x)^{3/2}} \, dx,x,\frac {1}{x^3}\right )\right )\\ &=-\frac {2 x^6}{3 a \sqrt {a+\frac {b}{x^3}}}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,\frac {1}{x^3}\right )}{3 a}\\ &=-\frac {2 x^6}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {5 \sqrt {a+\frac {b}{x^3}} x^6}{6 a^2}+\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,\frac {1}{x^3}\right )}{4 a^2}\\ &=-\frac {5 b \sqrt {a+\frac {b}{x^3}} x^3}{4 a^3}-\frac {2 x^6}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {5 \sqrt {a+\frac {b}{x^3}} x^6}{6 a^2}-\frac {\left (5 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^3}\right )}{8 a^3}\\ &=-\frac {5 b \sqrt {a+\frac {b}{x^3}} x^3}{4 a^3}-\frac {2 x^6}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {5 \sqrt {a+\frac {b}{x^3}} x^6}{6 a^2}-\frac {(5 b) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^3}}\right )}{4 a^3}\\ &=-\frac {5 b \sqrt {a+\frac {b}{x^3}} x^3}{4 a^3}-\frac {2 x^6}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {5 \sqrt {a+\frac {b}{x^3}} x^6}{6 a^2}+\frac {5 b^2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{4 a^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 96, normalized size = 1.01 \[ \frac {\sqrt {a} x^{3/2} \left (2 a^2 x^6-5 a b x^3-15 b^2\right )+15 b^{5/2} \sqrt {\frac {a x^3}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x^{3/2}}{\sqrt {b}}\right )}{12 a^{7/2} x^{3/2} \sqrt {a+\frac {b}{x^3}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b/x^3)^(3/2),x]

[Out]

(Sqrt[a]*x^(3/2)*(-15*b^2 - 5*a*b*x^3 + 2*a^2*x^6) + 15*b^(5/2)*Sqrt[1 + (a*x^3)/b]*ArcSinh[(Sqrt[a]*x^(3/2))/
Sqrt[b]])/(12*a^(7/2)*Sqrt[a + b/x^3]*x^(3/2))

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fricas [A]  time = 1.18, size = 243, normalized size = 2.56 \[ \left [\frac {15 \, {\left (a b^{2} x^{3} + b^{3}\right )} \sqrt {a} \log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \, {\left (2 \, a x^{6} + b x^{3}\right )} \sqrt {a} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right ) + 4 \, {\left (2 \, a^{3} x^{9} - 5 \, a^{2} b x^{6} - 15 \, a b^{2} x^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{48 \, {\left (a^{5} x^{3} + a^{4} b\right )}}, -\frac {15 \, {\left (a b^{2} x^{3} + b^{3}\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} x^{3} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right ) - 2 \, {\left (2 \, a^{3} x^{9} - 5 \, a^{2} b x^{6} - 15 \, a b^{2} x^{3}\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{24 \, {\left (a^{5} x^{3} + a^{4} b\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b/x^3)^(3/2),x, algorithm="fricas")

[Out]

[1/48*(15*(a*b^2*x^3 + b^3)*sqrt(a)*log(-8*a^2*x^6 - 8*a*b*x^3 - b^2 - 4*(2*a*x^6 + b*x^3)*sqrt(a)*sqrt((a*x^3
+ b)/x^3)) + 4*(2*a^3*x^9 - 5*a^2*b*x^6 - 15*a*b^2*x^3)*sqrt((a*x^3 + b)/x^3))/(a^5*x^3 + a^4*b), -1/24*(15*(
a*b^2*x^3 + b^3)*sqrt(-a)*arctan(2*sqrt(-a)*x^3*sqrt((a*x^3 + b)/x^3)/(2*a*x^3 + b)) - 2*(2*a^3*x^9 - 5*a^2*b*
x^6 - 15*a*b^2*x^3)*sqrt((a*x^3 + b)/x^3))/(a^5*x^3 + a^4*b)]

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giac [A]  time = 0.25, size = 117, normalized size = 1.23 \[ -\frac {1}{12} \, b^{2} {\left (\frac {15 \, \arctan \left (\frac {\sqrt {\frac {a x^{3} + b}{x^{3}}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {8}{a^{3} \sqrt {\frac {a x^{3} + b}{x^{3}}}} - \frac {9 \, a \sqrt {\frac {a x^{3} + b}{x^{3}}} - \frac {7 \, {\left (a x^{3} + b\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{x^{3}}}{{\left (a - \frac {a x^{3} + b}{x^{3}}\right )}^{2} a^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b/x^3)^(3/2),x, algorithm="giac")

[Out]

-1/12*b^2*(15*arctan(sqrt((a*x^3 + b)/x^3)/sqrt(-a))/(sqrt(-a)*a^3) + 8/(a^3*sqrt((a*x^3 + b)/x^3)) - (9*a*sqr
t((a*x^3 + b)/x^3) - 7*(a*x^3 + b)*sqrt((a*x^3 + b)/x^3)/x^3)/((a - (a*x^3 + b)/x^3)^2*a^3))

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maple [C]  time = 0.05, size = 3910, normalized size = 41.16 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a+b/x^3)^(3/2),x)

[Out]

1/12/((a*x^3+b)/x^3)^(3/2)/x^5*(a*x^3+b)/a^5*(-8*I*((-a*x+(-a^2*b)^(1/3))*(2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^
2*b)^(1/3))*(-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/a^2*x)^(1/2)*3^(1/2)*x^2*a^2*b^2-7*I*(a*x^4+b*x)^
(1/2)*((-a*x+(-a^2*b)^(1/3))*(2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))*(-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-
(-a^2*b)^(1/3))/a^2*x)^(1/2)*3^(1/2)*((a*x^3+b)*x)^(1/2)*x*a^2*b+90*I*(-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^
2*b)^(1/3))*a*x)^(1/2)*((2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(
1/2)*((-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticPi((
-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^
(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-a^2*b)^(2/3)*3^(1/2)*((a*x^3+b)*x)^(1/2)*b^2+180*I*(-(I*3^(1/2)
-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2)*((2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))/(1+I*3^(1/
2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b
)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2),((I*3^(1/2)+3)*(I*3^(
1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-a^2*b)^(1/3)*3^(1/2)*((a*x^3+b)*x)^(1/2)*x*a*b^2+90*I*(-(I*3^(1/
2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2)*((2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))/(1+I*3^(
1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2
*b)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2),(I*3^(1/2)-1)/(I*3
^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*3^(1/2)*((a*x^3+b)*x)^(1/2)*x^2*a^2
*b^2-90*I*(-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2)*((2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^
2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/(I*3^
(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2)
,((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-a^2*b)^(2/3)*3^(1/2)*((a*x^3+b)*x)^(1/2)*b
^2+90*(-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2)*((2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)
^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/(I*3^(1/2
)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2),((I
*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*((a*x^3+b)*x)^(1/2)*x^2*a^2*b^2+2*I*(a*x^4+b*x)^
(1/2)*((-a*x+(-a^2*b)^(1/3))*(2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))*(-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-
(-a^2*b)^(1/3))/a^2*x)^(1/2)*3^(1/2)*((a*x^3+b)*x)^(1/2)*x^4*a^3-90*(-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*
b)^(1/3))*a*x)^(1/2)*((2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/
2)*((-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticPi((-(
I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1
/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*((a*x^3+b)*x)^(1/2)*x^2*a^2*b^2-6*(a*x^4+b*x)^(1/2)*((-a*x+(-a^2*b)
^(1/3))*(2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))*(-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/a^2*x
)^(1/2)*((a*x^3+b)*x)^(1/2)*x^4*a^3-180*(-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2)*((2*a*x
+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((-2*a*x+I*3^(1/2)*(-a^2*
b)^(1/3)-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-
a*x+(-a^2*b)^(1/3))*a*x)^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-a^2*b)^(1/3)
*((a*x^3+b)*x)^(1/2)*x*a*b^2+180*(-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2)*((2*a*x+I*3^(1
/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3
)-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-
a^2*b)^(1/3))*a*x)^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))
^(1/2))*(-a^2*b)^(1/3)*((a*x^3+b)*x)^(1/2)*x*a*b^2-180*I*(-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a
*x)^(1/2)*((2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((-2*a*x
+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3
)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I
*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-a^2*b)^(1/3)*3^(1/2)*((a*x^3+b)*x)^(1/2)*x*a*b^2+90*(-(I*3^(1/2)-3)/(I*3^(1/
2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2)*((2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-
a^2*b)^(1/3)))^(1/2)*((-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1
/2)*EllipticF((-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(1+I
*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-a^2*b)^(2/3)*((a*x^3+b)*x)^(1/2)*b^2-90*(-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+
(-a^2*b)^(1/3))*a*x)^(1/2)*((2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(-a*x+(-a^2*b)^(1/3)
))^(1/2)*((-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*Elliptic
Pi((-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(
I*3^(1/2)-1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-a^2*b)^(2/3)*((a*x^3+b)*x)^(1/2)*b^2-90*I*(-(I*3^(1/2)-3)/(
I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2)*((2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))/(1+I*3^(1/2))/(
-a*x+(-a^2*b)^(1/3)))^(1/2)*((-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/
3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3))*a*x)^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-
1)/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*3^(1/2)*((a*x^3+b)*x)^(1/2)*x^2*a^2*b^2+21*(a*x^4+b*x)^(1/2)*((-a*x+(-a
^2*b)^(1/3))*(2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))*(-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/
a^2*x)^(1/2)*((a*x^3+b)*x)^(1/2)*x*a^2*b+24*x^2*b^2*a^2*((-a*x+(-a^2*b)^(1/3))*(2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)
+(-a^2*b)^(1/3))*(-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/a^2*x)^(1/2))/(I*3^(1/2)-3)/((-a*x+(-a^2*b)^
(1/3))*(2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)+(-a^2*b)^(1/3))*(-2*a*x+I*3^(1/2)*(-a^2*b)^(1/3)-(-a^2*b)^(1/3))/a^2*x)
^(1/2)

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maxima [A]  time = 1.98, size = 122, normalized size = 1.28 \[ -\frac {15 \, {\left (a + \frac {b}{x^{3}}\right )}^{2} b^{2} - 25 \, {\left (a + \frac {b}{x^{3}}\right )} a b^{2} + 8 \, a^{2} b^{2}}{12 \, {\left ({\left (a + \frac {b}{x^{3}}\right )}^{\frac {5}{2}} a^{3} - 2 \, {\left (a + \frac {b}{x^{3}}\right )}^{\frac {3}{2}} a^{4} + \sqrt {a + \frac {b}{x^{3}}} a^{5}\right )}} - \frac {5 \, b^{2} \log \left (\frac {\sqrt {a + \frac {b}{x^{3}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{3}}} + \sqrt {a}}\right )}{8 \, a^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b/x^3)^(3/2),x, algorithm="maxima")

[Out]

-1/12*(15*(a + b/x^3)^2*b^2 - 25*(a + b/x^3)*a*b^2 + 8*a^2*b^2)/((a + b/x^3)^(5/2)*a^3 - 2*(a + b/x^3)^(3/2)*a
^4 + sqrt(a + b/x^3)*a^5) - 5/8*b^2*log((sqrt(a + b/x^3) - sqrt(a))/(sqrt(a + b/x^3) + sqrt(a)))/a^(7/2)

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mupad [B]  time = 1.93, size = 96, normalized size = 1.01 \[ \frac {x^6\,\sqrt {a+\frac {b}{x^3}}}{6\,a^2}-\frac {2\,b^2}{3\,a^3\,\sqrt {a+\frac {b}{x^3}}}+\frac {5\,b^2\,\ln \left (x^6\,\left (\sqrt {a+\frac {b}{x^3}}-\sqrt {a}\right )\,{\left (\sqrt {a+\frac {b}{x^3}}+\sqrt {a}\right )}^3\right )}{8\,a^{7/2}}-\frac {7\,b\,x^3\,\sqrt {a+\frac {b}{x^3}}}{12\,a^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a + b/x^3)^(3/2),x)

[Out]

(x^6*(a + b/x^3)^(1/2))/(6*a^2) - (2*b^2)/(3*a^3*(a + b/x^3)^(1/2)) + (5*b^2*log(x^6*((a + b/x^3)^(1/2) - a^(1
/2))*((a + b/x^3)^(1/2) + a^(1/2))^3))/(8*a^(7/2)) - (7*b*x^3*(a + b/x^3)^(1/2))/(12*a^3)

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sympy [A]  time = 6.16, size = 110, normalized size = 1.16 \[ \frac {x^{\frac {15}{2}}}{6 a \sqrt {b} \sqrt {\frac {a x^{3}}{b} + 1}} - \frac {5 \sqrt {b} x^{\frac {9}{2}}}{12 a^{2} \sqrt {\frac {a x^{3}}{b} + 1}} - \frac {5 b^{\frac {3}{2}} x^{\frac {3}{2}}}{4 a^{3} \sqrt {\frac {a x^{3}}{b} + 1}} + \frac {5 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{\frac {3}{2}}}{\sqrt {b}} \right )}}{4 a^{\frac {7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(a+b/x**3)**(3/2),x)

[Out]

x**(15/2)/(6*a*sqrt(b)*sqrt(a*x**3/b + 1)) - 5*sqrt(b)*x**(9/2)/(12*a**2*sqrt(a*x**3/b + 1)) - 5*b**(3/2)*x**(
3/2)/(4*a**3*sqrt(a*x**3/b + 1)) + 5*b**2*asinh(sqrt(a)*x**(3/2)/sqrt(b))/(4*a**(7/2))

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