Optimal. Leaf size=49 \[ \frac {a c \sqrt {c x^2}}{b^2 x (a+b x)}+\frac {c \sqrt {c x^2} \log (a+b x)}{b^2 x} \]
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Rubi [A] time = 0.01, antiderivative size = 49, 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} \[ \frac {a c \sqrt {c x^2}}{b^2 x (a+b x)}+\frac {c \sqrt {c x^2} \log (a+b x)}{b^2 x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{3/2}}{x^2 (a+b x)^2} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int \frac {x}{(a+b x)^2} \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx}{x}\\ &=\frac {a c \sqrt {c x^2}}{b^2 x (a+b x)}+\frac {c \sqrt {c x^2} \log (a+b x)}{b^2 x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 0.78 \[ \frac {c^2 x ((a+b x) \log (a+b x)+a)}{b^2 \sqrt {c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 43, normalized size = 0.88 \[ \frac {\sqrt {c x^{2}} {\left (a c + {\left (b c x + a c\right )} \log \left (b x + a\right )\right )}}{b^{3} x^{2} + a b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 46, normalized size = 0.94 \[ -c^{\frac {3}{2}} {\left (\frac {{\left (\log \left ({\left | a \right |}\right ) + 1\right )} \mathrm {sgn}\relax (x)}{b^{2}} - \frac {\log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{2}} - \frac {a \mathrm {sgn}\relax (x)}{{\left (b x + a\right )} b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 41, normalized size = 0.84 \[ \frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x \ln \left (b x +a \right )+a \ln \left (b x +a \right )+a \right )}{\left (b x +a \right ) b^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.48, size = 80, normalized size = 1.63 \[ \frac {\left (-1\right )^{\frac {2 \, c x}{b}} c^{\frac {3}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{2}} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} c^{\frac {3}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2}} - \frac {\sqrt {c x^{2}} c}{b^{2} x + a b} \]
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 \[ \int \frac {{\left (c\,x^2\right )}^{3/2}}{x^2\,{\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{2} \left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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