Optimal. Leaf size=52 \[ \frac{a^2 c \sqrt{c x^2} \log (x)}{x}+2 a b c \sqrt{c x^2}+\frac{1}{2} b^2 c x \sqrt{c x^2} \]
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Rubi [A] time = 0.0097471, antiderivative size = 52, 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} \[ \frac{a^2 c \sqrt{c x^2} \log (x)}{x}+2 a b c \sqrt{c x^2}+\frac{1}{2} b^2 c x \sqrt{c x^2} \]
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}{x^4} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{(a+b x)^2}{x} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (2 a b+\frac{a^2}{x}+b^2 x\right ) \, dx}{x}\\ &=2 a b c \sqrt{c x^2}+\frac{1}{2} b^2 c x \sqrt{c x^2}+\frac{a^2 c \sqrt{c x^2} \log (x)}{x}\\ \end{align*}
Mathematica [A] time = 0.0081543, size = 34, normalized size = 0.65 \[ \frac{\left (c x^2\right )^{3/2} \left (2 a^2 \log (x)+b x (4 a+b x)\right )}{2 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 33, normalized size = 0.6 \begin{align*}{\frac{{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( x \right ) +4\,abx}{2\,{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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50387, size = 81, normalized size = 1.56 \begin{align*} \frac{{\left (b^{2} c x^{2} + 4 \, a b c x + 2 \, a^{2} c \log \left (x\right )\right )} \sqrt{c x^{2}}}{2 \, 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{\left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )^{2}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05001, size = 43, normalized size = 0.83 \begin{align*} \frac{1}{2} \,{\left (b^{2} x^{2} \mathrm{sgn}\left (x\right ) + 4 \, a b x \mathrm{sgn}\left (x\right ) + 2 \, a^{2} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (x\right )\right )} c^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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