Optimal. Leaf size=111 \[ -\frac{a^4 c \sqrt{c x^2}}{b^5 x (a+b x)}+\frac{3 a^2 c \sqrt{c x^2}}{b^4}-\frac{4 a^3 c \sqrt{c x^2} \log (a+b x)}{b^5 x}-\frac{a c x \sqrt{c x^2}}{b^3}+\frac{c x^2 \sqrt{c x^2}}{3 b^2} \]
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Rubi [A] time = 0.0373899, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 43} \[ -\frac{a^4 c \sqrt{c x^2}}{b^5 x (a+b x)}+\frac{3 a^2 c \sqrt{c x^2}}{b^4}-\frac{4 a^3 c \sqrt{c x^2} \log (a+b x)}{b^5 x}-\frac{a c x \sqrt{c x^2}}{b^3}+\frac{c x^2 \sqrt{c x^2}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x \left (c x^2\right )^{3/2}}{(a+b x)^2} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{x^4}{(a+b x)^2} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (\frac{3 a^2}{b^4}-\frac{2 a x}{b^3}+\frac{x^2}{b^2}+\frac{a^4}{b^4 (a+b x)^2}-\frac{4 a^3}{b^4 (a+b x)}\right ) \, dx}{x}\\ &=\frac{3 a^2 c \sqrt{c x^2}}{b^4}-\frac{a c x \sqrt{c x^2}}{b^3}+\frac{c x^2 \sqrt{c x^2}}{3 b^2}-\frac{a^4 c \sqrt{c x^2}}{b^5 x (a+b x)}-\frac{4 a^3 c \sqrt{c x^2} \log (a+b x)}{b^5 x}\\ \end{align*}
Mathematica [A] time = 0.0196553, size = 82, normalized size = 0.74 \[ \frac{\left (c x^2\right )^{3/2} \left (6 a^2 b^2 x^2+9 a^3 b x-12 a^3 (a+b x) \log (a+b x)-3 a^4-2 a b^3 x^3+b^4 x^4\right )}{3 b^5 x^3 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 88, normalized size = 0.8 \begin{align*} -{\frac{-{b}^{4}{x}^{4}+2\,{x}^{3}a{b}^{3}+12\,\ln \left ( bx+a \right ) x{a}^{3}b-6\,{x}^{2}{a}^{2}{b}^{2}+12\,{a}^{4}\ln \left ( bx+a \right ) -9\,bx{a}^{3}+3\,{a}^{4}}{3\,{b}^{5}{x}^{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 = 1.37743, size = 196, normalized size = 1.77 \begin{align*} \frac{{\left (b^{4} c x^{4} - 2 \, a b^{3} c x^{3} + 6 \, a^{2} b^{2} c x^{2} + 9 \, a^{3} b c x - 3 \, a^{4} c - 12 \,{\left (a^{3} b c x + a^{4} c\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{3 \,{\left (b^{6} x^{2} + a b^{5} x\right )}} \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 \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 [A] time = 1.0597, size = 130, normalized size = 1.17 \begin{align*} -\frac{1}{3} \, c^{\frac{3}{2}}{\left (\frac{12 \, a^{3} \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{5}} + \frac{3 \, a^{4} \mathrm{sgn}\left (x\right )}{{\left (b x + a\right )} b^{5}} - \frac{3 \,{\left (4 \, a^{3} \log \left ({\left | a \right |}\right ) + a^{3}\right )} \mathrm{sgn}\left (x\right )}{b^{5}} - \frac{b^{4} x^{3} \mathrm{sgn}\left (x\right ) - 3 \, a b^{3} x^{2} \mathrm{sgn}\left (x\right ) + 9 \, a^{2} b^{2} x \mathrm{sgn}\left (x\right )}{b^{6}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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