Optimal. Leaf size=89 \[ \frac{b^2 x \log (x)}{a^3 c \sqrt{c x^2}}-\frac{b^2 x \log (a+b x)}{a^3 c \sqrt{c x^2}}+\frac{b}{a^2 c \sqrt{c x^2}}-\frac{1}{2 a c x \sqrt{c x^2}} \]
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Rubi [A] time = 0.0222743, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {15, 44} \[ \frac{b^2 x \log (x)}{a^3 c \sqrt{c x^2}}-\frac{b^2 x \log (a+b x)}{a^3 c \sqrt{c x^2}}+\frac{b}{a^2 c \sqrt{c x^2}}-\frac{1}{2 a c x \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac{x \int \frac{1}{x^3 (a+b x)} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \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}{c \sqrt{c x^2}}\\ &=\frac{b}{a^2 c \sqrt{c x^2}}-\frac{1}{2 a c x \sqrt{c x^2}}+\frac{b^2 x \log (x)}{a^3 c \sqrt{c x^2}}-\frac{b^2 x \log (a+b x)}{a^3 c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.006718, size = 51, normalized size = 0.57 \[ \frac{x \left (-2 b^2 x^2 \log (a+b x)-a (a-2 b x)+2 b^2 x^2 \log (x)\right )}{2 a^3 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 49, normalized size = 0.6 \begin{align*}{\frac{x \left ( 2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,abx-{a}^{2} \right ) }{2\,{a}^{3}} \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 = 1.49573, size = 103, normalized size = 1.16 \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} c^{2} 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{1}{\left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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