3.2011 \(\int (a+\frac {b}{x^3})^{3/2} x^2 \, dx\)

Optimal. Leaf size=58 \[ \frac {1}{3} x^3 \left (a+\frac {b}{x^3}\right )^{3/2}-b \sqrt {a+\frac {b}{x^3}}+\sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right ) \]

[Out]

1/3*(a+b/x^3)^(3/2)*x^3+b*arctanh((a+b/x^3)^(1/2)/a^(1/2))*a^(1/2)-b*(a+b/x^3)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 47, 50, 63, 208} \[ \frac {1}{3} x^3 \left (a+\frac {b}{x^3}\right )^{3/2}-b \sqrt {a+\frac {b}{x^3}}+\sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^3)^(3/2)*x^2,x]

[Out]

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

Rule 47

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 \left (a+\frac {b}{x^3}\right )^{3/2} x^2 \, dx &=-\left (\frac {1}{3} \operatorname {Subst}\left (\int \frac {(a+b x)^{3/2}}{x^2} \, dx,x,\frac {1}{x^3}\right )\right )\\ &=\frac {1}{3} \left (a+\frac {b}{x^3}\right )^{3/2} x^3-\frac {1}{2} b \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\frac {1}{x^3}\right )\\ &=-b \sqrt {a+\frac {b}{x^3}}+\frac {1}{3} \left (a+\frac {b}{x^3}\right )^{3/2} x^3-\frac {1}{2} (a b) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^3}\right )\\ &=-b \sqrt {a+\frac {b}{x^3}}+\frac {1}{3} \left (a+\frac {b}{x^3}\right )^{3/2} x^3-a \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^3}}\right )\\ &=-b \sqrt {a+\frac {b}{x^3}}+\frac {1}{3} \left (a+\frac {b}{x^3}\right )^{3/2} x^3+\sqrt {a} b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 49, normalized size = 0.84 \[ -\frac {2 b \sqrt {a+\frac {b}{x^3}} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {a x^3}{b}\right )}{3 \sqrt {\frac {a x^3}{b}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^3)^(3/2)*x^2,x]

[Out]

(-2*b*Sqrt[a + b/x^3]*Hypergeometric2F1[-3/2, -1/2, 1/2, -((a*x^3)/b)])/(3*Sqrt[1 + (a*x^3)/b])

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fricas [A]  time = 1.17, size = 151, normalized size = 2.60 \[ \left [\frac {1}{4} \, \sqrt {a} b \log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \, {\left (2 \, a x^{6} + b x^{3}\right )} \sqrt {a} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right ) + \frac {1}{3} \, {\left (a x^{3} - 2 \, b\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}, -\frac {1}{2} \, \sqrt {-a} b \arctan \left (\frac {2 \, \sqrt {-a} x^{3} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right ) + \frac {1}{3} \, {\left (a x^{3} - 2 \, b\right )} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*sqrt(a)*b*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)) + 1/3*(a*
x^3 - 2*b)*sqrt((a*x^3 + b)/x^3), -1/2*sqrt(-a)*b*arctan(2*sqrt(-a)*x^3*sqrt((a*x^3 + b)/x^3)/(2*a*x^3 + b)) +
1/3*(a*x^3 - 2*b)*sqrt((a*x^3 + b)/x^3)]

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giac [A]  time = 0.35, size = 55, normalized size = 0.95 \[ \frac {1}{3} \, \sqrt {a x^{4} + b x} a x - \frac {1}{3} \, {\left (\frac {3 \, a \arctan \left (\frac {\sqrt {a + \frac {b}{x^{3}}}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 2 \, \sqrt {a + \frac {b}{x^{3}}}\right )} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(a*x^4 + b*x)*a*x - 1/3*(3*a*arctan(sqrt(a + b/x^3)/sqrt(-a))/sqrt(-a) + 2*sqrt(a + b/x^3))*b

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maple [C]  time = 0.02, size = 3559, normalized size = 61.36 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*((a*x^3+b)/x^3)^(3/2)*x^3/a*(-18*I*(-a^2*b)^(2/3)*3^(1/2)*(-(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))*x^2*b-I*3^(1/2)*(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)*x^3*a^2-36*I*(-
a^2*b)^(1/3)*3^(1/2)*(-(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))*x^3*a*b-18*I*3^(1/2)*(-(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))*x^4*a^2*b+36*I*(-a^2*b)^(1/3)*3^(1/2)
*(-(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))*x^3*a*b+18*I*3^(1/2)*(-(
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))*x^4*a^2*b-18*(-(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))*x^4*a^2*b+18*(-(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))*
x^4*a^2*b+36*(-a^2*b)^(1/3)*(-(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))*x^3*a*b-36*(-a^2*b)^(1/3)*
(-(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))*x^3*a*b-18*(-a^2*b)^(2/3)
*(-(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))*x^2*b+18*(-a^2*b)^(2/3)*(-(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)*E
llipticPi((-(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))*x^2*b+18*I*(-a^2*b)^(2/3)*3^(1/2)*(-(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))*x^2*b+3*(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)*x^3*a^2+2*I*3^(1/2)*(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*b-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*b)/(a*x^3+b)/((a*x^3+b)*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 = 1.96, size = 66, normalized size = 1.14 \[ \frac {1}{3} \, \sqrt {a + \frac {b}{x^{3}}} a x^{3} - \frac {1}{2} \, \sqrt {a} b \log \left (\frac {\sqrt {a + \frac {b}{x^{3}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{3}}} + \sqrt {a}}\right ) - \frac {2}{3} \, \sqrt {a + \frac {b}{x^{3}}} b \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(a + b/x^3)*a*x^3 - 1/2*sqrt(a)*b*log((sqrt(a + b/x^3) - sqrt(a))/(sqrt(a + b/x^3) + sqrt(a))) - 2/3*s
qrt(a + b/x^3)*b

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mupad [B]  time = 1.79, size = 69, normalized size = 1.19 \[ \frac {a\,x^3\,\sqrt {a+\frac {b}{x^3}}}{3}-\frac {2\,b\,\sqrt {a+\frac {b}{x^3}}}{3}+\frac {\sqrt {a}\,b\,\ln \left (x^6\,\left (\sqrt {a+\frac {b}{x^3}}-\sqrt {a}\right )\,{\left (\sqrt {a+\frac {b}{x^3}}+\sqrt {a}\right )}^3\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b/x^3)^(3/2),x)

[Out]

(a*x^3*(a + b/x^3)^(1/2))/3 - (2*b*(a + b/x^3)^(1/2))/3 + (a^(1/2)*b*log(x^6*((a + b/x^3)^(1/2) - a^(1/2))*((a
+ b/x^3)^(1/2) + a^(1/2))^3))/2

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sympy [B]  time = 3.00, size = 100, normalized size = 1.72 \[ \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a} x^{\frac {3}{2}}}{\sqrt {b}} \right )} + \frac {a^{2} x^{\frac {9}{2}}}{3 \sqrt {b} \sqrt {\frac {a x^{3}}{b} + 1}} - \frac {a \sqrt {b} x^{\frac {3}{2}}}{3 \sqrt {\frac {a x^{3}}{b} + 1}} - \frac {2 b^{\frac {3}{2}}}{3 x^{\frac {3}{2}} \sqrt {\frac {a x^{3}}{b} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a)*b*asinh(sqrt(a)*x**(3/2)/sqrt(b)) + a**2*x**(9/2)/(3*sqrt(b)*sqrt(a*x**3/b + 1)) - a*sqrt(b)*x**(3/2)/
(3*sqrt(a*x**3/b + 1)) - 2*b**(3/2)/(3*x**(3/2)*sqrt(a*x**3/b + 1))

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