Optimal. Leaf size=48 \[ \frac{c^2 \sqrt{c x^2} \log (x)}{a x}-\frac{c^2 \sqrt{c x^2} \log (a+b x)}{a x} \]
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Rubi [A] time = 0.0080137, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {15, 36, 29, 31} \[ \frac{c^2 \sqrt{c x^2} \log (x)}{a x}-\frac{c^2 \sqrt{c x^2} \log (a+b x)}{a x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{5/2}}{x^6 (a+b x)} \, dx &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int \frac{1}{x (a+b x)} \, dx}{x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int \frac{1}{x} \, dx}{a x}-\frac{\left (b c^2 \sqrt{c x^2}\right ) \int \frac{1}{a+b x} \, dx}{a x}\\ &=\frac{c^2 \sqrt{c x^2} \log (x)}{a x}-\frac{c^2 \sqrt{c x^2} \log (a+b x)}{a x}\\ \end{align*}
Mathematica [A] time = 0.0091546, size = 28, normalized size = 0.58 \[ \frac{c^3 x (\log (x)-\log (a+b x))}{a \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 26, normalized size = 0.5 \begin{align*}{\frac{\ln \left ( x \right ) -\ln \left ( bx+a \right ) }{a{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 = 1.04982, size = 32, normalized size = 0.67 \begin{align*} -\frac{c^{\frac{5}{2}} \log \left (b x + a\right )}{a} + \frac{c^{\frac{5}{2}} \log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72324, size = 150, normalized size = 3.12 \begin{align*} \left [\frac{\sqrt{c x^{2}} c^{2} \log \left (\frac{x}{b x + a}\right )}{a x}, \frac{2 \, \sqrt{-c} c^{2} \arctan \left (\frac{\sqrt{c x^{2}}{\left (2 \, b x + a\right )} \sqrt{-c}}{a c x}\right )}{a}\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{\left (c x^{2}\right )^{\frac{5}{2}}}{x^{6} \left (a + b x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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