Optimal. Leaf size=88 \[ \frac {b^2 c \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 c \sqrt {c x^2} \log (a+b x)}{a^3 x}+\frac {b c \sqrt {c x^2}}{a^2 x^2}-\frac {c \sqrt {c x^2}}{2 a x^3} \]
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Rubi [A] time = 0.02, antiderivative size = 88, 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^2 c \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 c \sqrt {c x^2} \log (a+b x)}{a^3 x}+\frac {b c \sqrt {c x^2}}{a^2 x^2}-\frac {c \sqrt {c x^2}}{2 a x^3} \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^6 (a+b x)} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int \frac {1}{x^3 (a+b x)} \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx}{x}\\ &=-\frac {c \sqrt {c x^2}}{2 a x^3}+\frac {b c \sqrt {c x^2}}{a^2 x^2}+\frac {b^2 c \sqrt {c x^2} \log (x)}{a^3 x}-\frac {b^2 c \sqrt {c x^2} \log (a+b x)}{a^3 x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 53, normalized size = 0.60 \begin {gather*} \frac {\left (c x^2\right )^{3/2} \left (-2 b^2 x^2 \log (a+b x)-a (a-2 b x)+2 b^2 x^2 \log (x)\right )}{2 a^3 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 58, normalized size = 0.66 \begin {gather*} \left (c x^2\right )^{3/2} \left (\frac {b^2 \log (x)}{a^3 x^3}-\frac {b^2 \log (a+b x)}{a^3 x^3}+\frac {2 b x-a}{2 a^2 x^5}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 47, normalized size = 0.53 \begin {gather*} \frac {{\left (2 \, b^{2} c x^{2} \log \left (\frac {x}{b x + a}\right ) + 2 \, a b c x - a^{2} c\right )} \sqrt {c x^{2}}}{2 \, a^{3} x^{3}} \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 = 0.00, size = 51, normalized size = 0.58 \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 -a^{2}\right )}{2 a^{3} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 52, normalized size = 0.59 \begin {gather*} -\frac {b^{2} c^{\frac {3}{2}} \log \left (b x + a\right )}{a^{3}} + \frac {b^{2} c^{\frac {3}{2}} \log \relax (x)}{a^{3}} + \frac {2 \, b c^{\frac {3}{2}} x - a c^{\frac {3}{2}}}{2 \, a^{2} x^{2}} \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^6\,\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 {3}{2}}}{x^{6} \left (a + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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