Optimal. Leaf size=84 \[ \frac{b^2 \sqrt{c x^2} \log (x)}{a^3 x}-\frac{b^2 \sqrt{c x^2} \log (a+b x)}{a^3 x}+\frac{b \sqrt{c x^2}}{a^2 x^2}-\frac{\sqrt{c x^2}}{2 a x^3} \]
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Rubi [A] time = 0.0226037, antiderivative size = 84, 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} \[ \frac{b^2 \sqrt{c x^2} \log (x)}{a^3 x}-\frac{b^2 \sqrt{c x^2} \log (a+b x)}{a^3 x}+\frac{b \sqrt{c x^2}}{a^2 x^2}-\frac{\sqrt{c x^2}}{2 a x^3} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin{align*} \int \frac{\sqrt{c x^2}}{x^4 (a+b x)} \, dx &=\frac{\sqrt{c x^2} \int \frac{1}{x^3 (a+b x)} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (\frac{1}{a x^3}-\frac{b}{a^2 x^2}+\frac{b^2}{a^3 x}-\frac{b^3}{a^3 (a+b x)}\right ) \, dx}{x}\\ &=-\frac{\sqrt{c x^2}}{2 a x^3}+\frac{b \sqrt{c x^2}}{a^2 x^2}+\frac{b^2 \sqrt{c x^2} \log (x)}{a^3 x}-\frac{b^2 \sqrt{c x^2} \log (a+b x)}{a^3 x}\\ \end{align*}
Mathematica [A] time = 0.0146526, size = 53, normalized size = 0.63 \[ \frac{\sqrt{c x^2} \left (-2 b^2 x^2 \log (a+b x)-a (a-2 b x)+2 b^2 x^2 \log (x)\right )}{2 a^3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 51, normalized size = 0.6 \begin{align*}{\frac{2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,abx-{a}^{2}}{2\,{a}^{3}{x}^{3}}\sqrt{c{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09557, size = 70, normalized size = 0.83 \begin{align*} -\frac{b^{2} \sqrt{c} \log \left (b x + a\right )}{a^{3}} + \frac{b^{2} \sqrt{c} \log \left (x\right )}{a^{3}} + \frac{2 \, b \sqrt{c} x - a \sqrt{c}}{2 \, a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62491, size = 97, normalized size = 1.15 \begin{align*} \frac{{\left (2 \, b^{2} x^{2} \log \left (\frac{x}{b x + a}\right ) + 2 \, a b x - a^{2}\right )} \sqrt{c x^{2}}}{2 \, a^{3} x^{3}} \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{\sqrt{c x^{2}}}{x^{4} \left (a + b x\right )}\, 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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