Optimal. Leaf size=30 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^m}}{\sqrt {a}}\right )}{\sqrt {a} m} \]
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Rubi [A]
time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {374, 12, 272,
65, 214} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^m}}{\sqrt {a}}\right )}{\sqrt {a} m} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 214
Rule 272
Rule 374
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a+b (c x)^m}} \, dx &=\frac {\text {Subst}\left (\int \frac {c}{x \sqrt {a+b x^m}} \, dx,x,c x\right )}{c}\\ &=\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^m}} \, dx,x,c x\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,(c x)^m\right )}{m}\\ &=\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b (c x)^m}\right )}{b m}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^m}}{\sqrt {a}}\right )}{\sqrt {a} m}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 30, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^m}}{\sqrt {a}}\right )}{\sqrt {a} m} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.61, size = 25, normalized size = 0.83
method | result | size |
derivativedivides | \(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (c x \right )^{m}}}{\sqrt {a}}\right )}{m \sqrt {a}}\) | \(25\) |
default | \(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (c x \right )^{m}}}{\sqrt {a}}\right )}{m \sqrt {a}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 78, normalized size = 2.60 \begin {gather*} \left [\frac {\log \left (\frac {\left (c x\right )^{m} b - 2 \, \sqrt {\left (c x\right )^{m} b + a} \sqrt {a} + 2 \, a}{\left (c x\right )^{m}}\right )}{\sqrt {a} m}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {\left (c x\right )^{m} b + a} \sqrt {-a}}{a}\right )}{a m}\right ] \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 {a + b \left (c x\right )^{m}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.03 \begin {gather*} \int \frac {1}{x\,\sqrt {a+b\,{\left (c\,x\right )}^m}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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