Optimal. Leaf size=112 \[ \frac {3 b^2 \sqrt {c x^2} \log (x)}{a^4 x}-\frac {3 b^2 \sqrt {c x^2} \log (a+b x)}{a^4 x}+\frac {b^2 \sqrt {c x^2}}{a^3 x (a+b x)}+\frac {2 b \sqrt {c x^2}}{a^3 x^2}-\frac {\sqrt {c x^2}}{2 a^2 x^3} \]
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Rubi [A] time = 0.04, antiderivative size = 112, 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 \sqrt {c x^2}}{a^3 x (a+b x)}+\frac {3 b^2 \sqrt {c x^2} \log (x)}{a^4 x}-\frac {3 b^2 \sqrt {c x^2} \log (a+b x)}{a^4 x}+\frac {2 b \sqrt {c x^2}}{a^3 x^2}-\frac {\sqrt {c x^2}}{2 a^2 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2}}{x^4 (a+b x)^2} \, dx &=\frac {\sqrt {c x^2} \int \frac {1}{x^3 (a+b x)^2} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (\frac {1}{a^2 x^3}-\frac {2 b}{a^3 x^2}+\frac {3 b^2}{a^4 x}-\frac {b^3}{a^3 (a+b x)^2}-\frac {3 b^3}{a^4 (a+b x)}\right ) \, dx}{x}\\ &=-\frac {\sqrt {c x^2}}{2 a^2 x^3}+\frac {2 b \sqrt {c x^2}}{a^3 x^2}+\frac {b^2 \sqrt {c x^2}}{a^3 x (a+b x)}+\frac {3 b^2 \sqrt {c x^2} \log (x)}{a^4 x}-\frac {3 b^2 \sqrt {c x^2} \log (a+b x)}{a^4 x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 82, normalized size = 0.73 \begin {gather*} \frac {\sqrt {c x^2} \left (a \left (-a^2+3 a b x+6 b^2 x^2\right )+6 b^2 x^2 \log (x) (a+b x)-6 b^2 x^2 (a+b x) \log (a+b x)\right )}{2 a^4 x^3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 77, normalized size = 0.69 \begin {gather*} \sqrt {c x^2} \left (\frac {3 b^2 \log (x)}{a^4 x}-\frac {3 b^2 \log (a+b x)}{a^4 x}+\frac {-a^2+3 a b x+6 b^2 x^2}{2 a^3 x^3 (a+b x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.16, size = 77, normalized size = 0.69 \begin {gather*} \frac {{\left (6 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3} + 6 \, {\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (\frac {x}{b x + a}\right )\right )} \sqrt {c x^{2}}}{2 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} \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 = 0.02, size = 95, normalized size = 0.85 \begin {gather*} \frac {\sqrt {c \,x^{2}}\, \left (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} \ln \relax (x )-6 a \,b^{2} x^{2} \ln \left (b x +a \right )+6 a \,b^{2} x^{2}+3 a^{2} b x -a^{3}\right )}{2 \left (b x +a \right ) a^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 79, normalized size = 0.71 \begin {gather*} \frac {6 \, b^{2} \sqrt {c} x^{2} + 3 \, a b \sqrt {c} x - a^{2} \sqrt {c}}{2 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} - \frac {3 \, b^{2} \sqrt {c} \log \left (b x + a\right )}{a^{4}} + \frac {3 \, b^{2} \sqrt {c} \log \relax (x)}{a^{4}} \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 {\sqrt {c\,x^2}}{x^4\,{\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 {\sqrt {c x^{2}}}{x^{4} \left (a + b x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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