Optimal. Leaf size=47 \[ \frac{4 \sqrt{a x^2+b x^3}}{b^2 x}-\frac{2 x^2}{b \sqrt{a x^2+b x^3}} \]
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Rubi [A] time = 0.0567734, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2015, 1588} \[ \frac{4 \sqrt{a x^2+b x^3}}{b^2 x}-\frac{2 x^2}{b \sqrt{a x^2+b x^3}} \]
Antiderivative was successfully verified.
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Rule 2015
Rule 1588
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a x^2+b x^3\right )^{3/2}} \, dx &=-\frac{2 x^2}{b \sqrt{a x^2+b x^3}}+\frac{2 \int \frac{x}{\sqrt{a x^2+b x^3}} \, dx}{b}\\ &=-\frac{2 x^2}{b \sqrt{a x^2+b x^3}}+\frac{4 \sqrt{a x^2+b x^3}}{b^2 x}\\ \end{align*}
Mathematica [A] time = 0.0124749, size = 26, normalized size = 0.55 \[ \frac{2 x (2 a+b x)}{b^2 \sqrt{x^2 (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 34, normalized size = 0.7 \begin{align*} 2\,{\frac{ \left ( bx+a \right ) \left ( bx+2\,a \right ){x}^{3}}{{b}^{2} \left ( b{x}^{3}+a{x}^{2} \right ) ^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11266, size = 26, normalized size = 0.55 \begin{align*} \frac{2 \,{\left (b x + 2 \, a\right )}}{\sqrt{b x + a} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.82862, size = 74, normalized size = 1.57 \begin{align*} \frac{2 \, \sqrt{b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )}}{b^{3} x^{2} + a b^{2} x} \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^{4}}{\left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13415, size = 38, normalized size = 0.81 \begin{align*} \frac{2 \,{\left (\frac{1}{b} + \frac{2 \, a}{b^{2} x}\right )}}{\sqrt{\frac{b}{x} + \frac{a}{x^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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