Optimal. Leaf size=68 \[ -\frac{x \log (a+b x)}{a^2 c \sqrt{c x^2}}+\frac{x \log (x)}{a^2 c \sqrt{c x^2}}+\frac{x}{a c \sqrt{c x^2} (a+b x)} \]
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Rubi [A] time = 0.0167383, antiderivative size = 68, 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, 44} \[ -\frac{x \log (a+b x)}{a^2 c \sqrt{c x^2}}+\frac{x \log (x)}{a^2 c \sqrt{c x^2}}+\frac{x}{a c \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin{align*} \int \frac{x^2}{\left (c x^2\right )^{3/2} (a+b x)^2} \, dx &=\frac{x \int \frac{1}{x (a+b x)^2} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx}{c \sqrt{c x^2}}\\ &=\frac{x}{a c \sqrt{c x^2} (a+b x)}+\frac{x \log (x)}{a^2 c \sqrt{c x^2}}-\frac{x \log (a+b x)}{a^2 c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0134991, size = 46, normalized size = 0.68 \[ \frac{x^3 (\log (x) (a+b x)-(a+b x) \log (a+b x)+a)}{a^2 \left (c x^2\right )^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 52, normalized size = 0.8 \begin{align*}{\frac{{x}^{3} \left ( b\ln \left ( x \right ) x-b\ln \left ( bx+a \right ) x+a\ln \left ( x \right ) -a\ln \left ( bx+a \right ) +a \right ) }{{a}^{2} \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 = 1.38343, size = 100, normalized size = 1.47 \begin{align*} \frac{\sqrt{c x^{2}}{\left ({\left (b x + a\right )} \log \left (\frac{x}{b x + a}\right ) + a\right )}}{a^{2} b c^{2} x^{2} + a^{3} 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 (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 [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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