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 = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.97, size = 35, normalized size = 1.52 \[ \frac {2 \, \sqrt {b x^{2} + a x} x}{3 \, {\left (a b^{2} x^{2} + 2 \, a^{2} b x + a^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 89, normalized size = 3.87 \[ \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} b^{\frac {3}{2}} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a b + a^{2} \sqrt {b}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a\right )}^{3} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 25, normalized size = 1.09 \[ \frac {2 \left (b x +a \right ) x^{4}}{3 \left (b \,x^{2}+a x \right )^{\frac {5}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.39, size = 74, normalized size = 3.22 \[ -\frac {x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} - \frac {a x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} + \frac {2 \, x}{3 \, \sqrt {b x^{2} + a x} a b} + \frac {1}{3 \, \sqrt {b x^{2} + a x} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 24, normalized size = 1.04 \[ \frac {2\,x\,\sqrt {b\,x^2+a\,x}}{3\,a\,{\left (a+b\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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