Optimal. Leaf size=44 \[ \frac {c^2 \sqrt {c x^2}}{b}-\frac {a c^2 \sqrt {c x^2} \log (a+b x)}{b^2 x} \]
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Rubi [A] time = 0.01, antiderivative size = 44, 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 {c^2 \sqrt {c x^2}}{b}-\frac {a c^2 \sqrt {c x^2} \log (a+b x)}{b^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{5/2}}{x^4 (a+b x)} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \frac {x}{a+b x} \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx}{x}\\ &=\frac {c^2 \sqrt {c x^2}}{b}-\frac {a c^2 \sqrt {c x^2} \log (a+b x)}{b^2 x}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 30, normalized size = 0.68 \begin {gather*} \frac {c^3 x (b x-a \log (a+b x))}{b^2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 33, normalized size = 0.75 \begin {gather*} \left (c x^2\right )^{5/2} \left (\frac {1}{b x^4}-\frac {a \log (a+b x)}{b^2 x^5}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 33, normalized size = 0.75 \begin {gather*} \frac {{\left (b c^{2} x - a c^{2} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{b^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 46, normalized size = 1.05 \begin {gather*} {\left (\frac {c^{2} x \mathrm {sgn}\relax (x)}{b} - \frac {a c^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{2}} + \frac {a c^{2} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{2}}\right )} \sqrt {c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.66 \begin {gather*} -\frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \left (a \ln \left (b x +a \right )-b x \right )}{b^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.51, size = 77, normalized size = 1.75 \begin {gather*} -\frac {\left (-1\right )^{\frac {2 \, c x}{b}} a c^{\frac {5}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{2}} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a c^{\frac {5}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2}} + \frac {\sqrt {c x^{2}} c^{2}}{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 = -0.02 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{5/2}}{x^4\,\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 {5}{2}}}{x^{4} \left (a + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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