Optimal. Leaf size=44 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}}{\sqrt {a}}\right )}{\sqrt {a} m n p} \]
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Rubi [A]
time = 0.26, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {374, 12, 272,
65, 214} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}}{\sqrt {a}}\right )}{\sqrt {a} m n p} \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 \left (d (e x)^m\right )^n\right )^p}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b \left (c (d x)^n\right )^p}} \, dx,x,(e x)^m\right )}{m}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b (c x)^p}} \, dx,x,\left (d (e x)^m\right )^n\right )}{m n}\\ &=\frac {\text {Subst}\left (\int \frac {c}{x \sqrt {a+b x^p}} \, dx,x,c \left (d (e x)^m\right )^n\right )}{c m n}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^p}} \, dx,x,c \left (d (e x)^m\right )^n\right )}{m n}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\left (c \left (d (e x)^m\right )^n\right )^p\right )}{m n p}\\ &=\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}\right )}{b m n p}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}}{\sqrt {a}}\right )}{\sqrt {a} m n p}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 44, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c \left (d (e x)^m\right )^n\right )^p}}{\sqrt {a}}\right )}{\sqrt {a} m n p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 39, normalized size = 0.89
method | result | size |
derivativedivides | \(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (c \left (d \left (e x \right )^{m}\right )^{n}\right )^{p}}}{\sqrt {a}}\right )}{m n p \sqrt {a}}\) | \(39\) |
default | \(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (c \left (d \left (e x \right )^{m}\right )^{n}\right )^{p}}}{\sqrt {a}}\right )}{m n p \sqrt {a}}\) | \(39\) |
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 = 151, normalized size = 3.43 \begin {gather*} \left [\frac {\log \left ({\left (b e^{\left (m n p \log \left (x e\right ) + n p \log \left (d\right ) + p \log \left (c\right )\right )} - 2 \, \sqrt {b e^{\left (m n p \log \left (x e\right ) + n p \log \left (d\right ) + p \log \left (c\right )\right )} + a} \sqrt {a} + 2 \, a\right )} e^{\left (-m n p \log \left (x e\right ) - n p \log \left (d\right ) - p \log \left (c\right )\right )}\right )}{\sqrt {a} m n p}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b e^{\left (m n p \log \left (x e\right ) + n p \log \left (d\right ) + p \log \left (c\right )\right )} + a} \sqrt {-a}}{a}\right )}{a m n p}\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 \left (e x\right )^{m}\right )^{n}\right )^{p}}}\, 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 = -0.02 \begin {gather*} \int \frac {1}{x\,\sqrt {a+b\,{\left (c\,{\left (d\,{\left (e\,x\right )}^m\right )}^n\right )}^p}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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