Optimal. Leaf size=61 \[ -\frac {b \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b \sqrt {c x^2} \log (a+b x)}{a^2 x}-\frac {\sqrt {c x^2}}{a x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 44} \begin {gather*} -\frac {b \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b \sqrt {c x^2} \log (a+b x)}{a^2 x}-\frac {\sqrt {c x^2}}{a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2}}{x^3 (a+b x)} \, dx &=\frac {\sqrt {c x^2} \int \frac {1}{x^2 (a+b x)} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx}{x}\\ &=-\frac {\sqrt {c x^2}}{a x^2}-\frac {b \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b \sqrt {c x^2} \log (a+b x)}{a^2 x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 0.52 \begin {gather*} -\frac {c (-b x \log (a+b x)+a+b x \log (x))}{a^2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 44, normalized size = 0.72 \begin {gather*} \sqrt {c x^2} \left (-\frac {b \log (x)}{a^2 x}+\frac {b \log (a+b x)}{a^2 x}-\frac {1}{a x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 31, normalized size = 0.51 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x \log \left (\frac {b x + a}{x}\right ) - a\right )}}{a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = 0.54 \begin {gather*} -\frac {\sqrt {c \,x^{2}}\, \left (b x \ln \relax (x )-b x \ln \left (b x +a \right )+a \right )}{a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 37, normalized size = 0.61 \begin {gather*} \frac {b \sqrt {c} \log \left (b x + a\right )}{a^{2}} - \frac {b \sqrt {c} \log \relax (x)}{a^{2}} - \frac {\sqrt {c}}{a x} \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 {\sqrt {c\,x^2}}{x^3\,\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 {\sqrt {c x^{2}}}{x^{3} \left (a + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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