Optimal. Leaf size=61 \[ \frac{a^2 x \log (a+b x)}{b^3 \sqrt{c x^2}}-\frac{a x^2}{b^2 \sqrt{c x^2}}+\frac{x^3}{2 b \sqrt{c x^2}} \]
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Rubi [A] time = 0.0170711, antiderivative size = 61, 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} \[ \frac{a^2 x \log (a+b x)}{b^3 \sqrt{c x^2}}-\frac{a x^2}{b^2 \sqrt{c x^2}}+\frac{x^3}{2 b \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\sqrt{c x^2} (a+b x)} \, dx &=\frac{x \int \frac{x^2}{a+b x} \, dx}{\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}{\sqrt{c x^2}}\\ &=-\frac{a x^2}{b^2 \sqrt{c x^2}}+\frac{x^3}{2 b \sqrt{c x^2}}+\frac{a^2 x \log (a+b x)}{b^3 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0122701, size = 39, normalized size = 0.64 \[ \frac{x \left (2 a^2 \log (a+b x)+b x (b x-2 a)\right )}{2 b^3 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 38, normalized size = 0.6 \begin{align*}{\frac{x \left ({b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -2\,abx \right ) }{2\,{b}^{3}}{\frac{1}{\sqrt{c{x}^{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 = 1.80387, size = 92, normalized size = 1.51 \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 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}}{\sqrt{c x^{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 = 1.06914, size = 89, normalized size = 1.46 \begin{align*} \frac{1}{2} \, \sqrt{c x^{2}}{\left (\frac{x}{b c} - \frac{2 \, a}{b^{2} c}\right )} - \frac{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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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