Optimal. Leaf size=37 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c (d x)^m\right )^n}}{\sqrt {a}}\right )}{\sqrt {a} m n} \]
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Rubi [A]
time = 0.12, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {374, 12, 272,
65, 214} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c (d x)^m\right )^n}}{\sqrt {a}}\right )}{\sqrt {a} m n} \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 \left (c (d x)^m\right )^n}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b (c x)^n}} \, dx,x,(d x)^m\right )}{m}\\ &=\frac {\text {Subst}\left (\int \frac {c}{x \sqrt {a+b x^n}} \, dx,x,c (d x)^m\right )}{c m}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^n}} \, dx,x,c (d x)^m\right )}{m}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\left (c (d x)^m\right )^n\right )}{m n}\\ &=\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \left (c (d x)^m\right )^n}\right )}{b m n}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c (d x)^m\right )^n}}{\sqrt {a}}\right )}{\sqrt {a} m n}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 37, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c (d x)^m\right )^n}}{\sqrt {a}}\right )}{\sqrt {a} m n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 32, normalized size = 0.86
method | result | size |
derivativedivides | \(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (c \left (d x \right )^{m}\right )^{n}}}{\sqrt {a}}\right )}{m n \sqrt {a}}\) | \(32\) |
default | \(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (c \left (d x \right )^{m}\right )^{n}}}{\sqrt {a}}\right )}{m n \sqrt {a}}\) | \(32\) |
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 = 0.40, size = 116, normalized size = 3.14 \begin {gather*} \left [\frac {\log \left ({\left (b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} - 2 \, \sqrt {b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} + a} \sqrt {a} + 2 \, a\right )} e^{\left (-m n \log \left (d x\right ) - n \log \left (c\right )\right )}\right )}{\sqrt {a} m n}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b e^{\left (m n \log \left (d x\right ) + n \log \left (c\right )\right )} + a} \sqrt {-a}}{a}\right )}{a m n}\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 \left (d x\right )^{m}\right )^{n}}}\, 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\,{\left (d\,x\right )}^m\right )}^n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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