Optimal. Leaf size=54 \[ -\frac {1}{a \sqrt {c x^2}}-\frac {b x \log (x)}{a^2 \sqrt {c x^2}}+\frac {b x \log (a+b x)}{a^2 \sqrt {c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 54, 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 x \log (x)}{a^2 \sqrt {c x^2}}+\frac {b x \log (a+b x)}{a^2 \sqrt {c x^2}}-\frac {1}{a \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 46
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {c x^2} (a+b x)} \, dx &=\frac {x \int \frac {1}{x^2 (a+b x)} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx}{\sqrt {c x^2}}\\ &=-\frac {1}{a \sqrt {c x^2}}-\frac {b x \log (x)}{a^2 \sqrt {c x^2}}+\frac {b x \log (a+b x)}{a^2 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 36, normalized size = 0.67 \begin {gather*} \frac {c x^2 (-a-b x \log (x)+b x \log (a+b x))}{a^2 \left (c x^2\right )^{3/2}} \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 while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.13, size = 30, normalized size = 0.56
method | result | size |
default | \(-\frac {b x \ln \left (x \right )-b \ln \left (b x +a \right ) x +a}{\sqrt {c \,x^{2}}\, a^{2}}\) | \(30\) |
risch | \(-\frac {1}{a \sqrt {c \,x^{2}}}+\frac {x b \ln \left (-b x -a \right )}{\sqrt {c \,x^{2}}\, a^{2}}-\frac {b x \ln \left (x \right )}{a^{2} \sqrt {c \,x^{2}}}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 37, normalized size = 0.69 \begin {gather*} \frac {b \log \left (b x + a\right )}{a^{2} \sqrt {c}} - \frac {b \log \left (x\right )}{a^{2} \sqrt {c}} - \frac {1}{a \sqrt {c} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.29, size = 34, normalized size = 0.63 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x \log \left (\frac {b x + a}{x}\right ) - a\right )}}{a^{2} c 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 {1}{x \sqrt {c x^{2}} \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 {1}{x\,\sqrt {c\,x^2}\,\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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