Optimal. Leaf size=46 \[ -\frac{2 (A b-a B)}{9 b^2 \left (a+b x^3\right )^{3/2}}-\frac{2 B}{3 b^2 \sqrt{a+b x^3}} \]
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Rubi [A] time = 0.0359458, antiderivative size = 46, normalized size of antiderivative = 1., 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} \[ -\frac{2 (A b-a B)}{9 b^2 \left (a+b x^3\right )^{3/2}}-\frac{2 B}{3 b^2 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{(a+b x)^{5/2}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{A b-a B}{b (a+b x)^{5/2}}+\frac{B}{b (a+b x)^{3/2}}\right ) \, dx,x,x^3\right )\\ &=-\frac{2 (A b-a B)}{9 b^2 \left (a+b x^3\right )^{3/2}}-\frac{2 B}{3 b^2 \sqrt{a+b x^3}}\\ \end{align*}
Mathematica [A] time = 0.0228209, size = 33, normalized size = 0.72 \[ -\frac{2 \left (2 a B+A b+3 b B x^3\right )}{9 b^2 \left (a+b x^3\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 30, normalized size = 0.7 \begin{align*} -{\frac{6\,bB{x}^{3}+2\,Ab+4\,Ba}{9\,{b}^{2}} \left ( b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.93893, size = 66, normalized size = 1.43 \begin{align*} -\frac{2}{9} \, B{\left (\frac{3}{\sqrt{b x^{3} + a} b^{2}} - \frac{a}{{\left (b x^{3} + a\right )}^{\frac{3}{2}} b^{2}}\right )} - \frac{2 \, A}{9 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68313, size = 111, normalized size = 2.41 \begin{align*} -\frac{2 \,{\left (3 \, B b x^{3} + 2 \, B a + A b\right )} \sqrt{b x^{3} + a}}{9 \,{\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.16649, size = 144, normalized size = 3.13 \begin{align*} \begin{cases} - \frac{2 A b}{9 a b^{2} \sqrt{a + b x^{3}} + 9 b^{3} x^{3} \sqrt{a + b x^{3}}} - \frac{4 B a}{9 a b^{2} \sqrt{a + b x^{3}} + 9 b^{3} x^{3} \sqrt{a + b x^{3}}} - \frac{6 B b x^{3}}{9 a b^{2} \sqrt{a + b x^{3}} + 9 b^{3} x^{3} \sqrt{a + b x^{3}}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{3}}{3} + \frac{B x^{6}}{6}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14252, size = 43, normalized size = 0.93 \begin{align*} -\frac{2 \,{\left (3 \,{\left (b x^{3} + a\right )} B - B a + A b\right )}}{9 \,{\left (b x^{3} + a\right )}^{\frac{3}{2}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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