Optimal. Leaf size=51 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}}{\sqrt {a}}\right )}{\sqrt {a} m n p q} \]
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Rubi [A]
time = 0.46, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {374, 12, 272,
65, 214} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}}{\sqrt {a}}\right )}{\sqrt {a} m n p q} \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 \left (e (f x)^m\right )^n\right )^p\right )^q}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b \left (c \left (d (e x)^n\right )^p\right )^q}} \, dx,x,(f x)^m\right )}{m}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b \left (c (d x)^p\right )^q}} \, dx,x,\left (e (f x)^m\right )^n\right )}{m n}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b (c x)^q}} \, dx,x,\left (d \left (e (f x)^m\right )^n\right )^p\right )}{m n p}\\ &=\frac {\text {Subst}\left (\int \frac {c}{x \sqrt {a+b x^q}} \, dx,x,c \left (d \left (e (f x)^m\right )^n\right )^p\right )}{c m n p}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^q}} \, dx,x,c \left (d \left (e (f x)^m\right )^n\right )^p\right )}{m n p}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q\right )}{m n p q}\\ &=\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}\right )}{b m n p q}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}}{\sqrt {a}}\right )}{\sqrt {a} m n p q}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 51, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}}{\sqrt {a}}\right )}{\sqrt {a} m n p q} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 46, normalized size = 0.90
method | result | size |
derivativedivides | \(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (c \left (d \left (e \left (f x \right )^{m}\right )^{n}\right )^{p}\right )^{q}}}{\sqrt {a}}\right )}{m n p q \sqrt {a}}\) | \(46\) |
default | \(-\frac {2 \arctanh \left (\frac {\sqrt {a +b \left (c \left (d \left (e \left (f x \right )^{m}\right )^{n}\right )^{p}\right )^{q}}}{\sqrt {a}}\right )}{m n p q \sqrt {a}}\) | \(46\) |
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 = 0.36, size = 174, normalized size = 3.41 \begin {gather*} \left [\frac {\log \left ({\left (b e^{\left (m n p q \log \left (f x\right ) + n p q + p q \log \left (d\right ) + q \log \left (c\right )\right )} - 2 \, \sqrt {b e^{\left (m n p q \log \left (f x\right ) + n p q + p q \log \left (d\right ) + q \log \left (c\right )\right )} + a} \sqrt {a} + 2 \, a\right )} e^{\left (-m n p q \log \left (f x\right ) - n p q - p q \log \left (d\right ) - q \log \left (c\right )\right )}\right )}{\sqrt {a} m n p q}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b e^{\left (m n p q \log \left (f x\right ) + n p q + p q \log \left (d\right ) + q \log \left (c\right )\right )} + a} \sqrt {-a}}{a}\right )}{a m n p q}\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 \left (f x\right )^{m}\right )^{n}\right )^{p}\right )^{q}}}\, 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\,{\left (f\,x\right )}^m\right )}^n\right )}^p\right )}^q}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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