Optimal. Leaf size=80 \[ \frac{a^2 \sqrt{c x^2}}{b^3}-\frac{a^3 \sqrt{c x^2} \log (a+b x)}{b^4 x}-\frac{a x \sqrt{c x^2}}{2 b^2}+\frac{x^2 \sqrt{c x^2}}{3 b} \]
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Rubi [A] time = 0.0249259, antiderivative size = 80, 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} \[ \frac{a^2 \sqrt{c x^2}}{b^3}-\frac{a^3 \sqrt{c x^2} \log (a+b x)}{b^4 x}-\frac{a x \sqrt{c x^2}}{2 b^2}+\frac{x^2 \sqrt{c x^2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2 \sqrt{c x^2}}{a+b x} \, dx &=\frac{\sqrt{c x^2} \int \frac{x^3}{a+b x} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (\frac{a^2}{b^3}-\frac{a x}{b^2}+\frac{x^2}{b}-\frac{a^3}{b^3 (a+b x)}\right ) \, dx}{x}\\ &=\frac{a^2 \sqrt{c x^2}}{b^3}-\frac{a x \sqrt{c x^2}}{2 b^2}+\frac{x^2 \sqrt{c x^2}}{3 b}-\frac{a^3 \sqrt{c x^2} \log (a+b x)}{b^4 x}\\ \end{align*}
Mathematica [A] time = 0.0152084, size = 52, normalized size = 0.65 \[ \frac{c x \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 52, normalized size = 0.7 \begin{align*} -{\frac{-2\,{b}^{3}{x}^{3}+3\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) -6\,{a}^{2}bx}{6\,x{b}^{4}}\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 = 1.49981, size = 113, normalized size = 1.41 \begin{align*} \frac{{\left (2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{6 \, b^{4} 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^{2} \sqrt{c x^{2}}}{a + b x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08084, size = 93, normalized size = 1.16 \begin{align*} -\frac{1}{6} \, \sqrt{c}{\left (\frac{6 \, a^{3} \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{4}} - \frac{6 \, a^{3} \log \left ({\left | a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{4}} - \frac{2 \, b^{2} x^{3} \mathrm{sgn}\left (x\right ) - 3 \, a b x^{2} \mathrm{sgn}\left (x\right ) + 6 \, a^{2} x \mathrm{sgn}\left (x\right )}{b^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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