Optimal. Leaf size=45 \[ \frac{x^2}{b c \sqrt{c x^2}}-\frac{a x \log (a+b x)}{b^2 c \sqrt{c x^2}} \]
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Rubi [A] time = 0.0131412, antiderivative size = 45, 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, 43} \[ \frac{x^2}{b c \sqrt{c x^2}}-\frac{a x \log (a+b x)}{b^2 c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^4}{\left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac{x \int \frac{x}{a+b x} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{1}{b}-\frac{a}{b (a+b x)}\right ) \, dx}{c \sqrt{c x^2}}\\ &=\frac{x^2}{b c \sqrt{c x^2}}-\frac{a x \log (a+b x)}{b^2 c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0065016, size = 29, normalized size = 0.64 \[ \frac{x^3 (b x-a \log (a+b x))}{b^2 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 29, normalized size = 0.6 \begin{align*} -{\frac{{x}^{3} \left ( a\ln \left ( bx+a \right ) -bx \right ) }{{b}^{2}} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70125, size = 65, normalized size = 1.44 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x - a \log \left (b x + a\right )\right )}}{b^{2} c^{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 (c x^{2}\right )^{\frac{3}{2}} \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 = 1.08184, size = 73, normalized size = 1.62 \begin{align*} \frac{\frac{a \log \left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{b^{2} \sqrt{c}} + \frac{\sqrt{c x^{2}}}{b c}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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