Optimal. Leaf size=23 \[ \frac{2 x^3}{3 a \left (a x+b x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0076737, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {650} \[ \frac{2 x^3}{3 a \left (a x+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 650
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a x+b x^2\right )^{5/2}} \, dx &=\frac{2 x^3}{3 a \left (a x+b x^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0098519, size = 21, normalized size = 0.91 \[ \frac{2 x^3}{3 a (x (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 25, normalized size = 1.1 \begin{align*}{\frac{2\,{x}^{4} \left ( bx+a \right ) }{3\,a} \left ( b{x}^{2}+ax \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.13721, size = 100, normalized size = 4.35 \begin{align*} -\frac{x^{2}}{{\left (b x^{2} + a x\right )}^{\frac{3}{2}} b} - \frac{a x}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}} b^{2}} + \frac{2 \, x}{3 \, \sqrt{b x^{2} + a x} a b} + \frac{1}{3 \, \sqrt{b x^{2} + a x} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84234, size = 74, normalized size = 3.22 \begin{align*} \frac{2 \, \sqrt{b x^{2} + a x} x}{3 \,{\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15981, size = 120, normalized size = 5.22 \begin{align*} \frac{2 \,{\left (3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} b^{\frac{3}{2}} + 3 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a b + a^{2} \sqrt{b}\right )}}{3 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} \sqrt{b} + a\right )}^{3} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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