Optimal. Leaf size=56 \[ -\frac{a^2}{c \sqrt{c x^2}}+\frac{2 a b x \log (x)}{c \sqrt{c x^2}}+\frac{b^2 x^2}{c \sqrt{c x^2}} \]
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Rubi [A] time = 0.0120212, antiderivative size = 56, 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} \[ -\frac{a^2}{c \sqrt{c x^2}}+\frac{2 a b x \log (x)}{c \sqrt{c x^2}}+\frac{b^2 x^2}{c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{(a+b x)^2}{x^2} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \left (b^2+\frac{a^2}{x^2}+\frac{2 a b}{x}\right ) \, dx}{c \sqrt{c x^2}}\\ &=-\frac{a^2}{c \sqrt{c x^2}}+\frac{b^2 x^2}{c \sqrt{c x^2}}+\frac{2 a b x \log (x)}{c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0055392, size = 33, normalized size = 0.59 \[ \frac{x^2 \left (-a^2+2 a b x \log (x)+b^2 x^2\right )}{\left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.6 \begin{align*}{{x}^{2} \left ( 2\,ab\ln \left ( x \right ) x+{b}^{2}{x}^{2}-{a}^{2} \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 [A] time = 1.06537, size = 57, normalized size = 1.02 \begin{align*} \frac{b^{2} x^{2}}{\sqrt{c x^{2}} c} + \frac{2 \, a b \log \left (x\right )}{c^{\frac{3}{2}}} - \frac{a^{2}}{\sqrt{c x^{2}} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53961, size = 76, normalized size = 1.36 \begin{align*} \frac{{\left (b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}\right )} \sqrt{c x^{2}}}{c^{2} x^{2}} \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 (a + b x\right )^{2}}{\left (c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07066, size = 93, normalized size = 1.66 \begin{align*} \frac{\frac{\sqrt{c x^{2}} b^{2}}{c} - \frac{2 \,{\left (a b \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right ) - \frac{a^{2} \sqrt{c}}{\sqrt{c} x - \sqrt{c x^{2}}}\right )}}{\sqrt{c}}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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