Optimal. Leaf size=115 \[ -\frac {b^3 x \log (x)}{a^4 c \sqrt {c x^2}}+\frac {b^3 x \log (a+b x)}{a^4 c \sqrt {c x^2}}-\frac {b^2}{a^3 c \sqrt {c x^2}}+\frac {b}{2 a^2 c x \sqrt {c x^2}}-\frac {1}{3 a c x^2 \sqrt {c x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 115, 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}{a^3 c \sqrt {c x^2}}-\frac {b^3 x \log (x)}{a^4 c \sqrt {c x^2}}+\frac {b^3 x \log (a+b x)}{a^4 c \sqrt {c x^2}}+\frac {b}{2 a^2 c x \sqrt {c x^2}}-\frac {1}{3 a c x^2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin {align*} \int \frac {1}{x \left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac {x \int \frac {1}{x^4 (a+b x)} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {1}{a x^4}-\frac {b}{a^2 x^3}+\frac {b^2}{a^3 x^2}-\frac {b^3}{a^4 x}+\frac {b^4}{a^4 (a+b x)}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {b^2}{a^3 c \sqrt {c x^2}}-\frac {1}{3 a c x^2 \sqrt {c x^2}}+\frac {b}{2 a^2 c x \sqrt {c x^2}}-\frac {b^3 x \log (x)}{a^4 c \sqrt {c x^2}}+\frac {b^3 x \log (a+b x)}{a^4 c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 66, normalized size = 0.57 \begin {gather*} \frac {c x^2 \left (a \left (-2 a^2+3 a b x-6 b^2 x^2\right )+6 b^3 x^3 \log (a+b x)-6 b^3 x^3 \log (x)\right )}{6 a^4 \left (c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 66, normalized size = 0.57 \begin {gather*} \frac {-\frac {b^3 x^3 \log (x)}{a^4}+\frac {b^3 x^3 \log (a+b x)}{a^4}+\frac {-2 a^2+3 a b x-6 b^2 x^2}{6 a^3}}{\left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 58, normalized size = 0.50 \begin {gather*} \frac {{\left (6 \, b^{3} x^{3} \log \left (\frac {b x + a}{x}\right ) - 6 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt {c x^{2}}}{6 \, a^{4} c^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c x^{2}\right )^{\frac {3}{2}} {\left (b x + a\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 59, normalized size = 0.51 \begin {gather*} -\frac {6 b^{3} x^{3} \ln \relax (x )-6 b^{3} x^{3} \ln \left (b x +a \right )+6 a \,b^{2} x^{2}-3 a^{2} b x +2 a^{3}}{6 \left (c \,x^{2}\right )^{\frac {3}{2}} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 69, normalized size = 0.60 \begin {gather*} \frac {b^{3} \log \left (b x + a\right )}{a^{4} c^{\frac {3}{2}}} - \frac {b^{3} \log \relax (x)}{a^{4} c^{\frac {3}{2}}} - \frac {6 \, b^{2} \sqrt {c} x^{2} - 3 \, a b \sqrt {c} x + 2 \, a^{2} \sqrt {c}}{6 \, a^{3} c^{2} x^{3}} \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 {1}{x\,{\left (c\,x^2\right )}^{3/2}\,\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 {1}{x \left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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