Optimal. Leaf size=68 \[ -\frac{c \sqrt{c x^2} \log (a+b x)}{a^2 x}+\frac{c \sqrt{c x^2} \log (x)}{a^2 x}+\frac{c \sqrt{c x^2}}{a x (a+b x)} \]
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Rubi [A] time = 0.0177966, 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} \[ -\frac{c \sqrt{c x^2} \log (a+b x)}{a^2 x}+\frac{c \sqrt{c x^2} \log (x)}{a^2 x}+\frac{c \sqrt{c x^2}}{a x (a+b x)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{3/2}}{x^4 (a+b x)^2} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{1}{x (a+b x)^2} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx}{x}\\ &=\frac{c \sqrt{c x^2}}{a x (a+b x)}+\frac{c \sqrt{c x^2} \log (x)}{a^2 x}-\frac{c \sqrt{c x^2} \log (a+b x)}{a^2 x}\\ \end{align*}
Mathematica [A] time = 0.0155185, size = 46, normalized size = 0.68 \[ \frac{\left (c x^2\right )^{3/2} (\log (x) (a+b x)-(a+b x) \log (a+b x)+a)}{a^2 x^3 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 52, normalized size = 0.8 \begin{align*}{\frac{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}{{a}^{2}{x}^{3} \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 [A] time = 1.04309, size = 51, normalized size = 0.75 \begin{align*} \frac{c^{\frac{3}{2}}}{a b x + a^{2}} - \frac{c^{\frac{3}{2}} \log \left (b x + a\right )}{a^{2}} + \frac{c^{\frac{3}{2}} \log \left (x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31839, size = 97, normalized size = 1.43 \begin{align*} \frac{\sqrt{c x^{2}}{\left (a c +{\left (b c x + a c\right )} \log \left (\frac{x}{b x + a}\right )\right )}}{a^{2} b x^{2} + a^{3} 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}}}{x^{4} \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: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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