Optimal. Leaf size=78 \[ -\frac {2 b x \log (x)}{a^3 \sqrt {c x^2}}+\frac {2 b x \log (a+b x)}{a^3 \sqrt {c x^2}}-\frac {b x}{a^2 \sqrt {c x^2} (a+b x)}-\frac {1}{a^2 \sqrt {c x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 78, 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 x}{a^2 \sqrt {c x^2} (a+b x)}-\frac {2 b x \log (x)}{a^3 \sqrt {c x^2}}+\frac {2 b x \log (a+b x)}{a^3 \sqrt {c x^2}}-\frac {1}{a^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 \sqrt {c x^2} (a+b x)^2} \, dx &=\frac {x \int \frac {1}{x^2 (a+b x)^2} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \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}{\sqrt {c x^2}}\\ &=-\frac {1}{a^2 \sqrt {c x^2}}-\frac {b x}{a^2 \sqrt {c x^2} (a+b x)}-\frac {2 b x \log (x)}{a^3 \sqrt {c x^2}}+\frac {2 b x \log (a+b x)}{a^3 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 0.77 \begin {gather*} \frac {c x^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 \left (c x^2\right )^{3/2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 68, normalized size = 0.87 \begin {gather*} \sqrt {c x^2} \left (-\frac {2 b \log (x)}{a^3 c x}+\frac {2 b \log (a+b x)}{a^3 c x}+\frac {-a-2 b x}{a^2 c x^2 (a+b x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 62, normalized size = 0.79 \begin {gather*} -\frac {{\left (2 \, a b x + a^{2} - 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \left (\frac {b x + a}{x}\right )\right )} \sqrt {c x^{2}}}{a^{3} b c x^{3} + a^{4} c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 126, normalized size = 1.62 \begin {gather*} \frac {b {\left (\frac {2 \, \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{3} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} + \frac {1}{{\left (b x + a\right )} a^{2} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} - \frac {1}{a^{3} {\left (\frac {a}{b x + a} - 1\right )} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )}\right )}}{\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 71, normalized size = 0.91 \begin {gather*} -\frac {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}}{\sqrt {c \,x^{2}}\, \left (b x +a \right ) a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 57, normalized size = 0.73 \begin {gather*} -\frac {2 \, b x + a}{a^{2} b \sqrt {c} x^{2} + a^{3} \sqrt {c} x} + \frac {2 \, b \log \left (b x + a\right )}{a^{3} \sqrt {c}} - \frac {2 \, b \log \relax (x)}{a^{3} \sqrt {c}} \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\,\sqrt {c\,x^2}\,{\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 {1}{x \sqrt {c x^{2}} \left (a + b x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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