Optimal. Leaf size=73 \[ -\frac{a^2 x}{b^3 c \sqrt{c x^2} (a+b x)}-\frac{2 a x \log (a+b x)}{b^3 c \sqrt{c x^2}}+\frac{x^2}{b^2 c \sqrt{c x^2}} \]
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Rubi [A] time = 0.02105, antiderivative size = 73, 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} \[ -\frac{a^2 x}{b^3 c \sqrt{c x^2} (a+b x)}-\frac{2 a x \log (a+b x)}{b^3 c \sqrt{c x^2}}+\frac{x^2}{b^2 c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx &=\frac{x \int \frac{x^2}{(a+b x)^2} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{1}{b^2}+\frac{a^2}{b^2 (a+b x)^2}-\frac{2 a}{b^2 (a+b x)}\right ) \, dx}{c \sqrt{c x^2}}\\ &=\frac{x^2}{b^2 c \sqrt{c x^2}}-\frac{a^2 x}{b^3 c \sqrt{c x^2} (a+b x)}-\frac{2 a x \log (a+b x)}{b^3 c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0142128, size = 54, normalized size = 0.74 \[ \frac{x^3 \left (-a^2+a b x-2 a (a+b x) \log (a+b x)+b^2 x^2\right )}{b^3 \left (c x^2\right )^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 62, normalized size = 0.9 \begin{align*} -{\frac{{x}^{3} \left ( 2\,\ln \left ( bx+a \right ) xab-{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -abx+{a}^{2} \right ) }{{b}^{3} \left ( bx+a \right ) } \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 = 1.21525, size = 130, normalized size = 1.78 \begin{align*} \frac{{\left (b^{2} x^{2} + a b x - a^{2} - 2 \,{\left (a b x + a^{2}\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{b^{4} c^{2} x^{2} + a 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 )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\left (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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