Optimal. Leaf size=35 \[ \frac{a x \log (x)}{c \sqrt{c x^2}}+\frac{b x^2}{c \sqrt{c x^2}} \]
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Rubi [A] time = 0.005395, antiderivative size = 35, 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} \[ \frac{a x \log (x)}{c \sqrt{c x^2}}+\frac{b x^2}{c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{a+b x}{x} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \left (b+\frac{a}{x}\right ) \, dx}{c \sqrt{c x^2}}\\ &=\frac{b x^2}{c \sqrt{c x^2}}+\frac{a x \log (x)}{c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0039511, size = 21, normalized size = 0.6 \[ \frac{x^3 (a \log (x)+b x)}{\left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 20, normalized size = 0.6 \begin{align*}{{x}^{3} \left ( bx+a\ln \left ( x \right ) \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 = 1.07308, size = 31, normalized size = 0.89 \begin{align*} \frac{b x^{2}}{\sqrt{c x^{2}} c} + \frac{a \log \left (x\right )}{c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60319, size = 51, normalized size = 1.46 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x + a \log \left (x\right )\right )}}{c^{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{x^{2} \left (a + b x\right )}{\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 = 1.06013, size = 54, normalized size = 1.54 \begin{align*} -\frac{\frac{a \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{\sqrt{c}} - \frac{\sqrt{c x^{2}} b}{c}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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