Optimal. Leaf size=68 \[ -\frac {c \sqrt {c x^2} \log (a+b x)}{a^2 x}+\frac {c \sqrt {c x^2} \log (x)}{a^2 x}+\frac {c \sqrt {c x^2}}{a x (a+b x)} \]
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Rubi [A] time = 0.02, antiderivative size = 68, 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, 44} \[ -\frac {c \sqrt {c x^2} \log (a+b x)}{a^2 x}+\frac {c \sqrt {c x^2} \log (x)}{a^2 x}+\frac {c \sqrt {c x^2}}{a x (a+b x)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{3/2}}{x^4 (a+b x)^2} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int \frac {1}{x (a+b x)^2} \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx}{x}\\ &=\frac {c \sqrt {c x^2}}{a x (a+b x)}+\frac {c \sqrt {c x^2} \log (x)}{a^2 x}-\frac {c \sqrt {c x^2} \log (a+b x)}{a^2 x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.68 \[ \frac {\left (c x^2\right )^{3/2} (\log (x) (a+b x)-(a+b x) \log (a+b x)+a)}{a^2 x^3 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 47, normalized size = 0.69 \[ \frac {\sqrt {c x^{2}} {\left (a c + {\left (b c x + a c\right )} \log \left (\frac {x}{b x + a}\right )\right )}}{a^{2} b x^{2} + a^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.76 \[ \frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x \ln \relax (x )-b x \ln \left (b x +a \right )+a \ln \relax (x )-a \ln \left (b x +a \right )+a \right )}{\left (b x +a \right ) a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 38, normalized size = 0.56 \[ \frac {c^{\frac {3}{2}}}{a b x + a^{2}} - \frac {c^{\frac {3}{2}} \log \left (b x + a\right )}{a^{2}} + \frac {c^{\frac {3}{2}} \log \relax (x)}{a^{2}} \]
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 \[ \int \frac {{\left (c\,x^2\right )}^{3/2}}{x^4\,{\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{4} \left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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