Optimal. Leaf size=63 \[ -\frac {b x \log (x)}{a^2 c \sqrt {c x^2}}+\frac {b x \log (a+b x)}{a^2 c \sqrt {c x^2}}-\frac {1}{a c \sqrt {c x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 44} \begin {gather*} -\frac {b x \log (x)}{a^2 c \sqrt {c x^2}}+\frac {b x \log (a+b x)}{a^2 c \sqrt {c x^2}}-\frac {1}{a c \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin {align*} \int \frac {x}{\left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac {x \int \frac {1}{x^2 (a+b x)} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {1}{a c \sqrt {c x^2}}-\frac {b x \log (x)}{a^2 c \sqrt {c x^2}}+\frac {b x \log (a+b x)}{a^2 c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 0.56 \begin {gather*} \frac {x^2 (b x \log (a+b x)-a-b x \log (x))}{a^2 \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 44, normalized size = 0.70 \begin {gather*} \frac {-\frac {b x^3 \log (x)}{a^2}+\frac {b x^3 \log (a+b x)}{a^2}-\frac {x^2}{a}}{\left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 34, normalized size = 0.54 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x \log \left (\frac {b x + a}{x}\right ) - a\right )}}{a^{2} c^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 91, normalized size = 1.44 \begin {gather*} -\frac {\frac {b \log \left ({\left | -{\left (\sqrt {c} x - \sqrt {c x^{2}}\right )} b - 2 \, a \sqrt {c} \right |}\right )}{a^{2} c} - \frac {b \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right )}{a^{2} c} - \frac {2}{{\left (\sqrt {c} x - \sqrt {c x^{2}}\right )} a \sqrt {c}}}{\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 33, normalized size = 0.52 \begin {gather*} -\frac {\left (b x \ln \relax (x )-b x \ln \left (b x +a \right )+a \right ) x^{2}}{\left (c \,x^{2}\right )^{\frac {3}{2}} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.48, size = 51, normalized size = 0.81 \begin {gather*} \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} b \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{a^{2} c^{\frac {3}{2}}} - \frac {1}{\sqrt {c x^{2}} a c} \end {gather*}
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 \begin {gather*} \int \frac {x}{{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )} \,d x \end {gather*}
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 \begin {gather*} \int \frac {x}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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