Optimal. Leaf size=25 \[ \frac{c^2 \sqrt{c x^2} \log (a+b x)}{b x} \]
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Rubi [A] time = 0.0043513, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 31} \[ \frac{c^2 \sqrt{c x^2} \log (a+b x)}{b x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 31
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{5/2}}{x^5 (a+b x)} \, dx &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int \frac{1}{a+b x} \, dx}{x}\\ &=\frac{c^2 \sqrt{c x^2} \log (a+b x)}{b x}\\ \end{align*}
Mathematica [A] time = 0.00404, size = 22, normalized size = 0.88 \[ \frac{\left (c x^2\right )^{5/2} \log (a+b x)}{b x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{b{x}^{5}} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07215, size = 18, normalized size = 0.72 \begin{align*} \frac{c^{\frac{5}{2}} \log \left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8374, size = 49, normalized size = 1.96 \begin{align*} \frac{\sqrt{c x^{2}} c^{2} \log \left (b x + a\right )}{b 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{\left (c x^{2}\right )^{\frac{5}{2}}}{x^{5} \left (a + b x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06493, size = 46, normalized size = 1.84 \begin{align*}{\left (\frac{c^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b} - \frac{c^{2} \log \left ({\left | a \right |}\right ) \mathrm{sgn}\left (x\right )}{b}\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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