Optimal. Leaf size=89 \[ \frac{a^3 c \sqrt{c x^2}}{b^4 x (a+b x)}+\frac{3 a^2 c \sqrt{c x^2} \log (a+b x)}{b^4 x}-\frac{2 a c \sqrt{c x^2}}{b^3}+\frac{c x \sqrt{c x^2}}{2 b^2} \]
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Rubi [A] time = 0.0285655, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {15, 43} \[ \frac{a^3 c \sqrt{c x^2}}{b^4 x (a+b x)}+\frac{3 a^2 c \sqrt{c x^2} \log (a+b x)}{b^4 x}-\frac{2 a c \sqrt{c x^2}}{b^3}+\frac{c x \sqrt{c x^2}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{3/2}}{(a+b x)^2} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{x^3}{(a+b x)^2} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (-\frac{2 a}{b^3}+\frac{x}{b^2}-\frac{a^3}{b^3 (a+b x)^2}+\frac{3 a^2}{b^3 (a+b x)}\right ) \, dx}{x}\\ &=-\frac{2 a c \sqrt{c x^2}}{b^3}+\frac{c x \sqrt{c x^2}}{2 b^2}+\frac{a^3 c \sqrt{c x^2}}{b^4 x (a+b x)}+\frac{3 a^2 c \sqrt{c x^2} \log (a+b x)}{b^4 x}\\ \end{align*}
Mathematica [A] time = 0.0176026, size = 71, normalized size = 0.8 \[ \frac{\left (c x^2\right )^{3/2} \left (-4 a^2 b x+6 a^2 (a+b x) \log (a+b x)+2 a^3-3 a b^2 x^2+b^3 x^3\right )}{2 b^4 x^3 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 76, normalized size = 0.9 \begin{align*}{\frac{{b}^{3}{x}^{3}+6\,\ln \left ( bx+a \right ) x{a}^{2}b-3\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) -4\,{a}^{2}bx+2\,{a}^{3}}{2\,{x}^{3}{b}^{4} \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 = 1.27045, size = 170, normalized size = 1.91 \begin{align*} \frac{{\left (b^{3} c x^{3} - 3 \, a b^{2} c x^{2} - 4 \, a^{2} b c x + 2 \, a^{3} c + 6 \,{\left (a^{2} b c x + a^{3} c\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{2 \,{\left (b^{5} x^{2} + a b^{4} 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{\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 = 1.0887, size = 108, normalized size = 1.21 \begin{align*} \frac{1}{2} \, c^{\frac{3}{2}}{\left (\frac{6 \, a^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{4}} + \frac{2 \, a^{3} \mathrm{sgn}\left (x\right )}{{\left (b x + a\right )} b^{4}} - \frac{2 \,{\left (3 \, a^{2} \log \left ({\left | a \right |}\right ) + a^{2}\right )} \mathrm{sgn}\left (x\right )}{b^{4}} + \frac{b^{2} x^{2} \mathrm{sgn}\left (x\right ) - 4 \, a b x \mathrm{sgn}\left (x\right )}{b^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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