Optimal. Leaf size=61 \[ -\frac {\sqrt {c x^2}}{a x^2}-\frac {b \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b \sqrt {c x^2} \log (a+b x)}{a^2 x} \]
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Rubi [A]
time = 0.01, antiderivative size = 61, 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, 46}
\begin {gather*} -\frac {b \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b \sqrt {c x^2} \log (a+b x)}{a^2 x}-\frac {\sqrt {c x^2}}{a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 46
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2}}{x^3 (a+b x)} \, dx &=\frac {\sqrt {c x^2} \int \frac {1}{x^2 (a+b x)} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx}{x}\\ &=-\frac {\sqrt {c x^2}}{a x^2}-\frac {b \sqrt {c x^2} \log (x)}{a^2 x}+\frac {b \sqrt {c x^2} \log (a+b x)}{a^2 x}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 32, normalized size = 0.52 \begin {gather*} -\frac {c (a+b x \log (x)-b x \log (a+b x))}{a^2 \sqrt {c x^2}} \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} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.13, size = 33, normalized size = 0.54
method | result | size |
default | \(-\frac {\sqrt {c \,x^{2}}\, \left (b x \ln \left (x \right )-b \ln \left (b x +a \right ) x +a \right )}{a^{2} x^{2}}\) | \(33\) |
risch | \(-\frac {\sqrt {c \,x^{2}}}{a \,x^{2}}+\frac {\sqrt {c \,x^{2}}\, b \ln \left (-b x -a \right )}{x \,a^{2}}-\frac {b \ln \left (x \right ) \sqrt {c \,x^{2}}}{a^{2} x}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 37, normalized size = 0.61 \begin {gather*} \frac {b \sqrt {c} \log \left (b x + a\right )}{a^{2}} - \frac {b \sqrt {c} \log \left (x\right )}{a^{2}} - \frac {\sqrt {c}}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 31, normalized size = 0.51 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x \log \left (\frac {b x + a}{x}\right ) - a\right )}}{a^{2} x^{2}} \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^{3} \left (a + b x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Limit: Max order reached or unable to make series expansion Error: Bad Argument Value} \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}}{x^3\,\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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