Optimal. Leaf size=56 \[ -\frac {a^2}{c^2 \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c^2 \sqrt {c x^2}}+\frac {b^2 x^2}{c^2 \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 56, 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, 43} \begin {gather*} -\frac {a^2}{c^2 \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c^2 \sqrt {c x^2}}+\frac {b^2 x^2}{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)^2}{\left (c x^2\right )^{5/2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^2} \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {x \int \left (b^2+\frac {a^2}{x^2}+\frac {2 a b}{x}\right ) \, dx}{c^2 \sqrt {c x^2}}\\ &=-\frac {a^2}{c^2 \sqrt {c x^2}}+\frac {b^2 x^2}{c^2 \sqrt {c x^2}}+\frac {2 a b x \log (x)}{c^2 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.59 \begin {gather*} \frac {-a^2+2 a b x \log (x)+b^2 x^2}{c^2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 35, normalized size = 0.62 \begin {gather*} \frac {-a^2 x^4+2 a b x^5 \log (x)+b^2 x^6}{\left (c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 34, normalized size = 0.61 \begin {gather*} \frac {{\left (b^{2} x^{2} + 2 \, a b x \log \relax (x) - a^{2}\right )} \sqrt {c x^{2}}}{c^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 65, normalized size = 1.16 \begin {gather*} \frac {\sqrt {c x^{2}} b^{2}}{c^{3}} - \frac {2 \, {\left (a b \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right ) - \frac {a^{2} \sqrt {c}}{\sqrt {c} x - \sqrt {c x^{2}}}\right )}}{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 = 32, normalized size = 0.57 \begin {gather*} \frac {\left (2 a b x \ln \relax (x )+b^{2} x^{2}-a^{2}\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 = 1.45, size = 45, normalized size = 0.80 \begin {gather*} \frac {b^{2} x^{4}}{\left (c x^{2}\right )^{\frac {3}{2}} c} - \frac {a^{2} x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}} c} + \frac {2 \, a 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 = -0.02 \begin {gather*} \int \frac {x^3\,{\left (a+b\,x\right )}^2}{{\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 )^{2}}{\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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