Optimal. Leaf size=64 \[ -\frac{b c \sqrt{c x^2} \log (x)}{a^2 x}+\frac{b c \sqrt{c x^2} \log (a+b x)}{a^2 x}-\frac{c \sqrt{c x^2}}{a x^2} \]
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Rubi [A] time = 0.016311, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 44} \[ -\frac{b c \sqrt{c x^2} \log (x)}{a^2 x}+\frac{b c \sqrt{c x^2} \log (a+b x)}{a^2 x}-\frac{c \sqrt{c x^2}}{a x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{3/2}}{x^5 (a+b x)} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{1}{x^2 (a+b x)} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx}{x}\\ &=-\frac{c \sqrt{c x^2}}{a x^2}-\frac{b c \sqrt{c x^2} \log (x)}{a^2 x}+\frac{b c \sqrt{c x^2} \log (a+b x)}{a^2 x}\\ \end{align*}
Mathematica [A] time = 0.0103781, size = 34, normalized size = 0.53 \[ -\frac{c^2 (-b x \log (a+b x)+a+b x \log (x))}{a^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 33, normalized size = 0.5 \begin{align*} -{\frac{b\ln \left ( x \right ) x-b\ln \left ( bx+a \right ) x+a}{{a}^{2}{x}^{4}} \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 = 1.05236, size = 50, normalized size = 0.78 \begin{align*} \frac{b c^{\frac{3}{2}} \log \left (b x + a\right )}{a^{2}} - \frac{b c^{\frac{3}{2}} \log \left (x\right )}{a^{2}} - \frac{c^{\frac{3}{2}}}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55071, size = 73, normalized size = 1.14 \begin{align*} \frac{{\left (b c x \log \left (\frac{b x + a}{x}\right ) - a c\right )} \sqrt{c x^{2}}}{a^{2} x^{2}} \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^{5} \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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