Optimal. Leaf size=67 \[ \frac {a^2 c^2 \sqrt {c x^2} \log (a+b x)}{b^3 x}-\frac {a c^2 \sqrt {c x^2}}{b^2}+\frac {c^2 x \sqrt {c x^2}}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 67, 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} \[ \frac {a^2 c^2 \sqrt {c x^2} \log (a+b x)}{b^3 x}-\frac {a c^2 \sqrt {c x^2}}{b^2}+\frac {c^2 x \sqrt {c x^2}}{2 b} \]
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^3 (a+b x)} \, dx &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \frac {x^2}{a+b x} \, dx}{x}\\ &=\frac {\left (c^2 \sqrt {c x^2}\right ) \int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx}{x}\\ &=-\frac {a c^2 \sqrt {c x^2}}{b^2}+\frac {c^2 x \sqrt {c x^2}}{2 b}+\frac {a^2 c^2 \sqrt {c x^2} \log (a+b x)}{b^3 x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 42, normalized size = 0.63 \[ \frac {c^3 x \left (2 a^2 \log (a+b x)+b x (b x-2 a)\right )}{2 b^3 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 48, normalized size = 0.72 \[ \frac {{\left (b^{2} c^{2} x^{2} - 2 \, a b c^{2} x + 2 \, a^{2} c^{2} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{2 \, b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 66, normalized size = 0.99 \[ \frac {1}{2} \, {\left (\frac {2 \, a^{2} c^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{3}} - \frac {2 \, a^{2} c^{2} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{3}} + \frac {b c^{2} x^{2} \mathrm {sgn}\relax (x) - 2 \, a c^{2} x \mathrm {sgn}\relax (x)}{b^{2}}\right )} \sqrt {c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 40, normalized size = 0.60 \[ \frac {\left (c \,x^{2}\right )^{\frac {5}{2}} \left (b^{2} x^{2}+2 a^{2} \ln \left (b x +a \right )-2 a b x \right )}{2 b^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.50, size = 97, normalized size = 1.45 \[ \frac {\left (-1\right )^{\frac {2 \, c x}{b}} a^{2} c^{\frac {5}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{3}} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{2} c^{\frac {5}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{3}} + \frac {\sqrt {c x^{2}} c^{2} x}{2 \, b} - \frac {\sqrt {c x^{2}} a c^{2}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,x^2\right )}^{5/2}}{x^3\,\left (a+b\,x\right )} \,d x \]
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 \[ \int \frac {\left (c x^{2}\right )^{\frac {5}{2}}}{x^{3} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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