Optimal. Leaf size=23 \[ \frac{x \log (a+b x)}{b c \sqrt{c x^2}} \]
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Rubi [A] time = 0.0039192, antiderivative size = 23, 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} \[ \frac{x \log (a+b x)}{b c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 31
Rubi steps
\begin{align*} \int \frac{x^3}{\left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac{x \int \frac{1}{a+b x} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \log (a+b x)}{b c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0037252, size = 22, normalized size = 0.96 \[ \frac{x^3 \log (a+b x)}{b \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 21, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}\ln \left ( bx+a \right ) }{b} \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 = 1.44449, size = 49, normalized size = 2.13 \begin{align*} \frac{\sqrt{c x^{2}} \log \left (b x + a\right )}{b 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^{3}}{\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 = 1.07928, size = 47, normalized size = 2.04 \begin{align*} -\frac{\log \left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{b c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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