Optimal. Leaf size=115 \[ -\frac{b^2}{a^3 c \sqrt{c x^2}}-\frac{b^3 x \log (x)}{a^4 c \sqrt{c x^2}}+\frac{b^3 x \log (a+b x)}{a^4 c \sqrt{c x^2}}+\frac{b}{2 a^2 c x \sqrt{c x^2}}-\frac{1}{3 a c x^2 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0283939, antiderivative size = 115, 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} \[ -\frac{b^2}{a^3 c \sqrt{c x^2}}-\frac{b^3 x \log (x)}{a^4 c \sqrt{c x^2}}+\frac{b^3 x \log (a+b x)}{a^4 c \sqrt{c x^2}}+\frac{b}{2 a^2 c x \sqrt{c x^2}}-\frac{1}{3 a c x^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x \left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac{x \int \frac{1}{x^4 (a+b x)} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{1}{a x^4}-\frac{b}{a^2 x^3}+\frac{b^2}{a^3 x^2}-\frac{b^3}{a^4 x}+\frac{b^4}{a^4 (a+b x)}\right ) \, dx}{c \sqrt{c x^2}}\\ &=-\frac{b^2}{a^3 c \sqrt{c x^2}}-\frac{1}{3 a c x^2 \sqrt{c x^2}}+\frac{b}{2 a^2 c x \sqrt{c x^2}}-\frac{b^3 x \log (x)}{a^4 c \sqrt{c x^2}}+\frac{b^3 x \log (a+b x)}{a^4 c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.015498, size = 66, normalized size = 0.57 \[ \frac{c x^2 \left (a \left (-2 a^2+3 a b x-6 b^2 x^2\right )+6 b^3 x^3 \log (a+b x)-6 b^3 x^3 \log (x)\right )}{6 a^4 \left (c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 59, normalized size = 0.5 \begin{align*} -{\frac{6\,{b}^{3}\ln \left ( x \right ){x}^{3}-6\,{b}^{3}\ln \left ( bx+a \right ){x}^{3}+6\,a{b}^{2}{x}^{2}-3\,{a}^{2}bx+2\,{a}^{3}}{6\,{a}^{4}} \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 = 1.04303, size = 93, normalized size = 0.81 \begin{align*} \frac{b^{3} \log \left (b x + a\right )}{a^{4} c^{\frac{3}{2}}} - \frac{b^{3} \log \left (x\right )}{a^{4} c^{\frac{3}{2}}} - \frac{6 \, b^{2} \sqrt{c} x^{2} - 3 \, a b \sqrt{c} x + 2 \, a^{2} \sqrt{c}}{6 \, a^{3} c^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60288, size = 127, normalized size = 1.1 \begin{align*} \frac{{\left (6 \, b^{3} x^{3} \log \left (\frac{b x + a}{x}\right ) - 6 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt{c x^{2}}}{6 \, a^{4} c^{2} x^{4}} \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}{x \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*} \int \frac{1}{\left (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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