Optimal. Leaf size=46 \[ \frac {2 B \sqrt {a+b x^3}}{3 b^2}-\frac {2 (A b-a B)}{3 b^2 \sqrt {a+b x^3}} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {444, 43} \begin {gather*} \frac {2 B \sqrt {a+b x^3}}{3 b^2}-\frac {2 (A b-a B)}{3 b^2 \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 444
Rubi steps
\begin {align*} \int \frac {x^2 \left (A+B x^3\right )}{\left (a+b x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {A+B x}{(a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {A b-a B}{b (a+b x)^{3/2}}+\frac {B}{b \sqrt {a+b x}}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 (A b-a B)}{3 b^2 \sqrt {a+b x^3}}+\frac {2 B \sqrt {a+b x^3}}{3 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 33, normalized size = 0.72 \begin {gather*} \frac {2 \left (2 a B-A b+b B x^3\right )}{3 b^2 \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.04, size = 33, normalized size = 0.72 \begin {gather*} \frac {2 \left (2 a B-A b+b B x^3\right )}{3 b^2 \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.55, size = 41, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (B b x^{3} + 2 \, B a - A b\right )} \sqrt {b x^{3} + a}}{3 \, {\left (b^{3} x^{3} + a b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 38, normalized size = 0.83 \begin {gather*} \frac {2 \, \sqrt {b x^{3} + a} B}{3 \, b^{2}} + \frac {2 \, {\left (B a - A b\right )}}{3 \, \sqrt {b x^{3} + a} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 30, normalized size = 0.65 \begin {gather*} -\frac {2 \left (-B b \,x^{3}+A b -2 B a \right )}{3 \sqrt {b \,x^{3}+a}\, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.56, size = 47, normalized size = 1.02 \begin {gather*} \frac {2}{3} \, B {\left (\frac {\sqrt {b x^{3} + a}}{b^{2}} + \frac {a}{\sqrt {b x^{3} + a} b^{2}}\right )} - \frac {2 \, A}{3 \, \sqrt {b x^{3} + a} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.61, size = 33, normalized size = 0.72 \begin {gather*} \frac {2\,B\,a-2\,A\,b+2\,B\,\left (b\,x^3+a\right )}{3\,b^2\,\sqrt {b\,x^3+a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.08, size = 75, normalized size = 1.63 \begin {gather*} \begin {cases} - \frac {2 A}{3 b \sqrt {a + b x^{3}}} + \frac {4 B a}{3 b^{2} \sqrt {a + b x^{3}}} + \frac {2 B x^{3}}{3 b \sqrt {a + b x^{3}}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{3}}{3} + \frac {B x^{6}}{6}}{a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________