Optimal. Leaf size=89 \[ \frac {b^2 x \log (x)}{a^3 c \sqrt {c x^2}}-\frac {b^2 x \log (a+b x)}{a^3 c \sqrt {c x^2}}+\frac {b}{a^2 c \sqrt {c x^2}}-\frac {1}{2 a c x \sqrt {c x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 44} \[ \frac {b^2 x \log (x)}{a^3 c \sqrt {c x^2}}-\frac {b^2 x \log (a+b x)}{a^3 c \sqrt {c x^2}}+\frac {b}{a^2 c \sqrt {c x^2}}-\frac {1}{2 a c x \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin {align*} \int \frac {1}{\left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac {x \int \frac {1}{x^3 (a+b x)} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \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}{c \sqrt {c x^2}}\\ &=\frac {b}{a^2 c \sqrt {c x^2}}-\frac {1}{2 a c x \sqrt {c x^2}}+\frac {b^2 x \log (x)}{a^3 c \sqrt {c x^2}}-\frac {b^2 x \log (a+b x)}{a^3 c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 51, normalized size = 0.57 \[ \frac {x \left (-2 b^2 x^2 \log (a+b x)-a (a-2 b x)+2 b^2 x^2 \log (x)\right )}{2 a^3 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 47, normalized size = 0.53 \[ \frac {{\left (2 \, b^{2} x^{2} \log \left (\frac {x}{b x + a}\right ) + 2 \, a b x - a^{2}\right )} \sqrt {c x^{2}}}{2 \, a^{3} c^{2} x^{3}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 49, normalized size = 0.55 \[ \frac {\left (2 b^{2} x^{2} \ln \relax (x )-2 b^{2} x^{2} \ln \left (b x +a \right )+2 a b x -a^{2}\right ) x}{2 \left (c \,x^{2}\right )^{\frac {3}{2}} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.54, size = 65, normalized size = 0.73 \[ -\frac {\left (-1\right )^{\frac {2 \, a c x}{b}} b^{2} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{a^{3} c^{\frac {3}{2}}} + \frac {b}{\sqrt {c x^{2}} a^{2} c} - \frac {1}{2 \, a c^{\frac {3}{2}} x^{2}} \]
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 \[ \int \frac {1}{{\left (c\,x^2\right )}^{3/2}\,\left (a+b\,x\right )} \,d x \]
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 \[ \int \frac {1}{\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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