Optimal. Leaf size=107 \[ -\frac{a^3 c \sqrt{c x^2}}{b^4}+\frac{a^2 c x \sqrt{c x^2}}{2 b^3}+\frac{a^4 c \sqrt{c x^2} \log (a+b x)}{b^5 x}-\frac{a c x^2 \sqrt{c x^2}}{3 b^2}+\frac{c x^3 \sqrt{c x^2}}{4 b} \]
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Rubi [A] time = 0.0321806, antiderivative size = 107, 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} \[ -\frac{a^3 c \sqrt{c x^2}}{b^4}+\frac{a^2 c x \sqrt{c x^2}}{2 b^3}+\frac{a^4 c \sqrt{c x^2} \log (a+b x)}{b^5 x}-\frac{a c x^2 \sqrt{c x^2}}{3 b^2}+\frac{c x^3 \sqrt{c x^2}}{4 b} \]
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} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{x^4}{a+b x} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (-\frac{a^3}{b^4}+\frac{a^2 x}{b^3}-\frac{a x^2}{b^2}+\frac{x^3}{b}+\frac{a^4}{b^4 (a+b x)}\right ) \, dx}{x}\\ &=-\frac{a^3 c \sqrt{c x^2}}{b^4}+\frac{a^2 c x \sqrt{c x^2}}{2 b^3}-\frac{a c x^2 \sqrt{c x^2}}{3 b^2}+\frac{c x^3 \sqrt{c x^2}}{4 b}+\frac{a^4 c \sqrt{c x^2} \log (a+b x)}{b^5 x}\\ \end{align*}
Mathematica [A] time = 0.013049, size = 64, normalized size = 0.6 \[ \frac{\left (c x^2\right )^{3/2} \left (b x \left (6 a^2 b x-12 a^3-4 a b^2 x^2+3 b^3 x^3\right )+12 a^4 \log (a+b x)\right )}{12 b^5 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 63, normalized size = 0.6 \begin{align*}{\frac{3\,{b}^{4}{x}^{4}-4\,{x}^{3}a{b}^{3}+6\,{x}^{2}{a}^{2}{b}^{2}+12\,{a}^{4}\ln \left ( bx+a \right ) -12\,bx{a}^{3}}{12\,{b}^{5}{x}^{3}} \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 = 1.60599, size = 153, normalized size = 1.43 \begin{align*} \frac{{\left (3 \, b^{4} c x^{4} - 4 \, a b^{3} c x^{3} + 6 \, a^{2} b^{2} c x^{2} - 12 \, a^{3} b c x + 12 \, a^{4} c \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{12 \, b^{5} 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 \left (c x^{2}\right )^{\frac{3}{2}}}{a + b x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05754, size = 109, normalized size = 1.02 \begin{align*} \frac{1}{12} \, c^{\frac{3}{2}}{\left (\frac{12 \, a^{4} \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{5}} - \frac{12 \, a^{4} \log \left ({\left | a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{5}} + \frac{3 \, b^{3} x^{4} \mathrm{sgn}\left (x\right ) - 4 \, a b^{2} x^{3} \mathrm{sgn}\left (x\right ) + 6 \, a^{2} b x^{2} \mathrm{sgn}\left (x\right ) - 12 \, a^{3} 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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