Optimal. Leaf size=33 \[ \frac {b x \log (x)}{c^2 \sqrt {c x^2}}-\frac {a}{c^2 \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 33, 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, 43} \begin {gather*} \frac {b x \log (x)}{c^2 \sqrt {c x^2}}-\frac {a}{c^2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {x^3 (a+b x)}{\left (c x^2\right )^{5/2}} \, dx &=\frac {x \int \frac {a+b x}{x^2} \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a}{x^2}+\frac {b}{x}\right ) \, dx}{c^2 \sqrt {c x^2}}\\ &=-\frac {a}{c^2 \sqrt {c x^2}}+\frac {b x \log (x)}{c^2 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.67 \begin {gather*} \frac {b x \log (x)-a}{c^2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 24, normalized size = 0.73 \begin {gather*} \frac {b x^5 \log (x)-a x^4}{\left (c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 23, normalized size = 0.70 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x \log \relax (x) - a\right )}}{c^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 47, normalized size = 1.42 \begin {gather*} -\frac {b \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right ) - \frac {2 \, a \sqrt {c}}{\sqrt {c} x - \sqrt {c x^{2}}}}{c^{\frac {5}{2}}} \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 = 0.64 \begin {gather*} \frac {\left (b x \ln \relax (x )-a \right ) x^{4}}{\left (c \,x^{2}\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 24, normalized size = 0.73 \begin {gather*} -\frac {a x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}} c} + \frac {b \log \relax (x)}{c^{\frac {5}{2}}} \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 = -0.03 \begin {gather*} \int \frac {x^3\,\left (a+b\,x\right )}{{\left (c\,x^2\right )}^{5/2}} \,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^{3} \left (a + b x\right )}{\left (c x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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