Optimal. Leaf size=59 \[ \frac {x}{a \sqrt {c x^2} (a+b x)}+\frac {x \log (x)}{a^2 \sqrt {c x^2}}-\frac {x \log (a+b x)}{a^2 \sqrt {c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 59, 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, 46}
\begin {gather*} -\frac {x \log (a+b x)}{a^2 \sqrt {c x^2}}+\frac {x \log (x)}{a^2 \sqrt {c x^2}}+\frac {x}{a \sqrt {c x^2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 46
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {c x^2} (a+b x)^2} \, dx &=\frac {x \int \frac {1}{x (a+b x)^2} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {x}{a \sqrt {c x^2} (a+b x)}+\frac {x \log (x)}{a^2 \sqrt {c x^2}}-\frac {x \log (a+b x)}{a^2 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 44, normalized size = 0.75 \begin {gather*} \frac {x (a+(a+b x) \log (x)-(a+b x) \log (a+b x))}{a^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 = 0.13, size = 50, normalized size = 0.85
method | result | size |
default | \(\frac {x \left (b x \ln \left (x \right )-b \ln \left (b x +a \right ) x +a \ln \left (x \right )-a \ln \left (b x +a \right )+a \right )}{\sqrt {c \,x^{2}}\, a^{2} \left (b x +a \right )}\) | \(50\) |
risch | \(\frac {x}{a \left (b x +a \right ) \sqrt {c \,x^{2}}}+\frac {x \ln \left (-x \right )}{\sqrt {c \,x^{2}}\, a^{2}}-\frac {x \ln \left (b x +a \right )}{a^{2} \sqrt {c \,x^{2}}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 61, normalized size = 1.03 \begin {gather*} -\frac {\sqrt {c x^{2}} b}{a^{2} b c x + a^{3} c} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{a^{2} \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 44, normalized size = 0.75 \begin {gather*} \frac {\sqrt {c x^{2}} {\left ({\left (b x + a\right )} \log \left (\frac {x}{b x + a}\right ) + a\right )}}{a^{2} b c x^{2} + a^{3} c 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 {1}{\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 [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}{\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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