Optimal. Leaf size=47 \[ \frac {a \sqrt {c x^2}}{b^2 x (a+b x)}+\frac {\sqrt {c x^2} \log (a+b x)}{b^2 x} \]
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Rubi [A]
time = 0.01, antiderivative size = 47, 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, 45}
\begin {gather*} \frac {a \sqrt {c x^2}}{b^2 x (a+b x)}+\frac {\sqrt {c x^2} \log (a+b x)}{b^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2}}{(a+b x)^2} \, dx &=\frac {\sqrt {c x^2} \int \frac {x}{(a+b x)^2} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx}{x}\\ &=\frac {a \sqrt {c x^2}}{b^2 x (a+b x)}+\frac {\sqrt {c x^2} \log (a+b x)}{b^2 x}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 36, normalized size = 0.77 \begin {gather*} \frac {c x (a+(a+b x) \log (a+b x))}{b^2 \sqrt {c x^2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 41, normalized size = 0.87
method | result | size |
default | \(\frac {\sqrt {c \,x^{2}}\, \left (b \ln \left (b x +a \right ) x +a \ln \left (b x +a \right )+a \right )}{x \,b^{2} \left (b x +a \right )}\) | \(41\) |
risch | \(\frac {a \sqrt {c \,x^{2}}}{b^{2} x \left (b x +a \right )}+\frac {\ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b^{2} x}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 79, normalized size = 1.68 \begin {gather*} \frac {\left (-1\right )^{\frac {2 \, c x}{b}} \sqrt {c} \log \left (\frac {2 \, c x}{b}\right )}{b^{2}} + \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} \sqrt {c} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2}} - \frac {\sqrt {c x^{2}}}{b^{2} x + a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 38, normalized size = 0.81 \begin {gather*} \frac {\sqrt {c x^{2}} {\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )}}{b^{3} x^{2} + a b^{2} 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}}}{\left (a + b x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 48, normalized size = 1.02 \begin {gather*} \sqrt {c} \left (\frac {a \mathrm {sign}\left (x\right )}{b^{2} \left (b x+a\right )}+\frac {\mathrm {sign}\left (x\right ) \ln \left |b x+a\right |}{b^{2}}+\frac {\left (-\ln \left |a\right |-1\right ) \mathrm {sign}\left (x\right )}{b^{2}}\right ) \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 = -0.02 \begin {gather*} \int \frac {\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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