Optimal. Leaf size=44 \[ \frac{c \sqrt{c x^2} \log (x)}{a x}-\frac{c \sqrt{c x^2} \log (a+b x)}{a x} \]
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Rubi [A] time = 0.0068632, antiderivative size = 44, 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} \[ \frac{c \sqrt{c x^2} \log (x)}{a x}-\frac{c \sqrt{c x^2} \log (a+b x)}{a x} \]
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 )^{3/2}}{x^4 (a+b x)} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{1}{x (a+b x)} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{1}{x} \, dx}{a x}-\frac{\left (b c \sqrt{c x^2}\right ) \int \frac{1}{a+b x} \, dx}{a x}\\ &=\frac{c \sqrt{c x^2} \log (x)}{a x}-\frac{c \sqrt{c x^2} \log (a+b x)}{a x}\\ \end{align*}
Mathematica [A] time = 0.0076032, size = 27, normalized size = 0.61 \[ \frac{\left (c x^2\right )^{3/2} (\log (x)-\log (a+b x))}{a x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 26, normalized size = 0.6 \begin{align*}{\frac{\ln \left ( x \right ) -\ln \left ( bx+a \right ) }{a{x}^{3}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03084, size = 32, normalized size = 0.73 \begin{align*} -\frac{c^{\frac{3}{2}} \log \left (b x + a\right )}{a} + \frac{c^{\frac{3}{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 = 1.59278, size = 144, normalized size = 3.27 \begin{align*} \left [\frac{\sqrt{c x^{2}} c \log \left (\frac{x}{b x + a}\right )}{a x}, \frac{2 \, \sqrt{-c} c \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{3}{2}}}{x^{4} \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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