Optimal. Leaf size=87 \[ -\frac {2 b \sqrt {c x^2} \log (x)}{a^3 x}+\frac {2 b \sqrt {c x^2} \log (a+b x)}{a^3 x}-\frac {b \sqrt {c x^2}}{a^2 x (a+b x)}-\frac {\sqrt {c x^2}}{a^2 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 87, 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} \[ -\frac {b \sqrt {c x^2}}{a^2 x (a+b x)}-\frac {2 b \sqrt {c x^2} \log (x)}{a^3 x}+\frac {2 b \sqrt {c x^2} \log (a+b x)}{a^3 x}-\frac {\sqrt {c x^2}}{a^2 x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2}}{x^3 (a+b x)^2} \, dx &=\frac {\sqrt {c x^2} \int \frac {1}{x^2 (a+b x)^2} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \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 {\sqrt {c x^2}}{a^2 x^2}-\frac {b \sqrt {c x^2}}{a^2 x (a+b x)}-\frac {2 b \sqrt {c x^2} \log (x)}{a^3 x}+\frac {2 b \sqrt {c x^2} \log (a+b x)}{a^3 x}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 57, normalized size = 0.66 \[ -\frac {c (a (a+2 b x)+2 b x \log (x) (a+b x)-2 b x (a+b x) \log (a+b x))}{a^3 \sqrt {c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 60, normalized size = 0.69 \[ -\frac {{\left (2 \, a b x + a^{2} - 2 \, {\left (b^{2} x^{2} + a b x\right )} \log \left (\frac {b x + a}{x}\right )\right )} \sqrt {c x^{2}}}{a^{3} b x^{3} + a^{4} x^{2}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 0.85 \[ -\frac {\sqrt {c \,x^{2}}\, \left (2 b^{2} x^{2} \ln \relax (x )-2 b^{2} x^{2} \ln \left (b x +a \right )+2 a b x \ln \relax (x )-2 a b x \ln \left (b x +a \right )+2 a b x +a^{2}\right )}{\left (b x +a \right ) a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 58, normalized size = 0.67 \[ -\frac {2 \, b \sqrt {c} x + a \sqrt {c}}{a^{2} b x^{2} + a^{3} x} + \frac {2 \, b \sqrt {c} \log \left (b x + a\right )}{a^{3}} - \frac {2 \, b \sqrt {c} \log \relax (x)}{a^{3}} \]
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 \[ \int \frac {\sqrt {c\,x^2}}{x^3\,{\left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \frac {\sqrt {c x^{2}}}{x^{3} \left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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