Optimal. Leaf size=70 \[ \frac{a^2 x \log (a+b x)}{b^3 c \sqrt{c x^2}}-\frac{a x^2}{b^2 c \sqrt{c x^2}}+\frac{x^3}{2 b c \sqrt{c x^2}} \]
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Rubi [A] time = 0.0200026, antiderivative size = 70, 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} \[ \frac{a^2 x \log (a+b x)}{b^3 c \sqrt{c x^2}}-\frac{a x^2}{b^2 c \sqrt{c x^2}}+\frac{x^3}{2 b c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac{x \int \frac{x^2}{a+b x} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \left (-\frac{a}{b^2}+\frac{x}{b}+\frac{a^2}{b^2 (a+b x)}\right ) \, dx}{c \sqrt{c x^2}}\\ &=-\frac{a x^2}{b^2 c \sqrt{c x^2}}+\frac{x^3}{2 b c \sqrt{c x^2}}+\frac{a^2 x \log (a+b x)}{b^3 c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0078899, size = 41, normalized size = 0.59 \[ \frac{x^3 \left (2 a^2 \log (a+b x)+b x (b x-2 a)\right )}{2 b^3 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 40, normalized size = 0.6 \begin{align*}{\frac{{x}^{3} \left ({b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -2\,abx \right ) }{2\,{b}^{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 [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 = 1.56188, size = 95, normalized size = 1.36 \begin{align*} \frac{{\left (b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{2 \, b^{3} 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^{5}}{\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 = 1.07358, size = 95, normalized size = 1.36 \begin{align*} \frac{\sqrt{c x^{2}}{\left (\frac{x}{b c} - \frac{2 \, a}{b^{2} c}\right )} - \frac{2 \, a^{2} \log \left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{b^{3} \sqrt{c}}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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