Optimal. Leaf size=44 \[ \frac {x \log (x)}{a c \sqrt {c x^2}}-\frac {x \log (a+b x)}{a c \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 36, 29, 31} \[ \frac {x \log (x)}{a c \sqrt {c x^2}}-\frac {x \log (a+b x)}{a c \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 29
Rule 31
Rule 36
Rubi steps
\begin {align*} \int \frac {x^2}{\left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac {x \int \frac {1}{x (a+b x)} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \frac {1}{x} \, dx}{a c \sqrt {c x^2}}-\frac {(b x) \int \frac {1}{a+b x} \, dx}{a c \sqrt {c x^2}}\\ &=\frac {x \log (x)}{a c \sqrt {c x^2}}-\frac {x \log (a+b x)}{a c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.61 \[ \frac {x^3 (\log (x)-\log (a+b x))}{a \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 70, normalized size = 1.59 \[ \left [\frac {\sqrt {c x^{2}} \log \left (\frac {x}{b x + a}\right )}{a c^{2} x}, \frac {2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )} \sqrt {-c}}{a c x}\right )}{a c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 63, normalized size = 1.43 \[ \frac {\frac {\log \left ({\left | -{\left (\sqrt {c} x - \sqrt {c x^{2}}\right )} b - 2 \, a \sqrt {c} \right |}\right )}{a \sqrt {c}} - \frac {\log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right )}{a \sqrt {c}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 0.59 \[ \frac {\left (\ln \relax (x )-\ln \left (b x +a \right )\right ) x^{3}}{\left (c \,x^{2}\right )^{\frac {3}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 35, normalized size = 0.80 \[ -\frac {\left (-1\right )^{\frac {2 \, a c x}{b}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{a c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2}{{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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