Optimal. Leaf size=91 \[ -\frac {2 b c \sqrt {c x^2} \log (x)}{a^3 x}+\frac {2 b c \sqrt {c x^2} \log (a+b x)}{a^3 x}-\frac {b c \sqrt {c x^2}}{a^2 x (a+b x)}-\frac {c \sqrt {c x^2}}{a^2 x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 91, 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} \begin {gather*} -\frac {b c \sqrt {c x^2}}{a^2 x (a+b x)}-\frac {2 b c \sqrt {c x^2} \log (x)}{a^3 x}+\frac {2 b c \sqrt {c x^2} \log (a+b x)}{a^3 x}-\frac {c \sqrt {c x^2}}{a^2 x^2} \end {gather*}
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^5 (a+b x)^2} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int \frac {1}{x^2 (a+b x)^2} \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (\frac {1}{a^2 x^2}-\frac {2 b}{a^3 x}+\frac {b^2}{a^2 (a+b x)^2}+\frac {2 b^2}{a^3 (a+b x)}\right ) \, dx}{x}\\ &=-\frac {c \sqrt {c x^2}}{a^2 x^2}-\frac {b c \sqrt {c x^2}}{a^2 x (a+b x)}-\frac {2 b c \sqrt {c x^2} \log (x)}{a^3 x}+\frac {2 b c \sqrt {c x^2} \log (a+b x)}{a^3 x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 59, normalized size = 0.65 \begin {gather*} -\frac {c^2 (a (a+2 b x)+2 b x \log (x) (a+b x)-2 b x (a+b x) \log (a+b x))}{a^3 \sqrt {c x^2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 59, normalized size = 0.65 \begin {gather*} \left (c x^2\right )^{3/2} \left (-\frac {2 b \log (x)}{a^3 x^3}+\frac {2 b \log (a+b x)}{a^3 x^3}+\frac {-a-2 b x}{a^2 x^4 (a+b x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.16, size = 65, normalized size = 0.71 \begin {gather*} -\frac {{\left (2 \, a b c x + a^{2} c - 2 \, {\left (b^{2} c x^{2} + a b c x\right )} \log \left (\frac {b x + a}{x}\right )\right )} \sqrt {c x^{2}}}{a^{3} b x^{3} + a^{4} x^{2}} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 0.81 \begin {gather*} -\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (2 b^{2} x^{2} \ln \relax (x )-2 b^{2} x^{2} \ln \left (b x +a \right )+2 a b x \ln \relax (x )-2 a b x \ln \left (b x +a \right )+2 a b x +a^{2}\right )}{\left (b x +a \right ) a^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 58, normalized size = 0.64 \begin {gather*} \frac {2 \, b c^{\frac {3}{2}} \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b c^{\frac {3}{2}} \log \relax (x)}{a^{3}} - \frac {2 \, b c^{\frac {3}{2}} x + a c^{\frac {3}{2}}}{a^{2} b x^{2} + a^{3} x} \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 = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{3/2}}{x^5\,{\left (a+b\,x\right )}^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 {\left (c x^{2}\right )^{\frac {3}{2}}}{x^{5} \left (a + b x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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