Optimal. Leaf size=23 \[ \frac {c \sqrt {c x^2} \log (a+b x)}{b x} \]
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Rubi [A] time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 31} \begin {gather*} \frac {c \sqrt {c x^2} \log (a+b x)}{b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 31
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{3/2}}{x^3 (a+b x)} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int \frac {1}{a+b x} \, dx}{x}\\ &=\frac {c \sqrt {c x^2} \log (a+b x)}{b x}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 0.96 \begin {gather*} \frac {\left (c x^2\right )^{3/2} \log (a+b x)}{b x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 22, normalized size = 0.96 \begin {gather*} \frac {\left (c x^2\right )^{3/2} \log (a+b x)}{b x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 21, normalized size = 0.91 \begin {gather*} \frac {\sqrt {c x^{2}} c \log \left (b x + a\right )}{b x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 28, normalized size = 1.22 \begin {gather*} c^{\frac {3}{2}} {\left (\frac {\log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b} - \frac {\log \left ({\left | a \right |}\right ) \mathrm {sgn}\relax (x)}{b}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.91 \begin {gather*} \frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \ln \left (b x +a \right )}{b \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 13, normalized size = 0.57 \begin {gather*} \frac {c^{\frac {3}{2}} \log \left (b x + a\right )}{b} \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 = -0.04 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{3/2}}{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 {\left (c x^{2}\right )^{\frac {3}{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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