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 = 0.66, 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 = {367, 12, 266, 63, 208} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}}{\sqrt {a}}\right )}{\sqrt {a} m n p q} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 266
Rule 367
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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 {\operatorname {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 \operatorname {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 = 0.26, size = 51, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c \left (d \left (e (f x)^m\right )^n\right )^p\right )^q}}{\sqrt {a}}\right )}{\sqrt {a} m n p q} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 182, normalized size = 3.57 \[ \left [\frac {\log \left ({\left (b e^{\left (m n p q \log \left (f x\right ) + n p q \log \relax (e) + p q \log \relax (d) + q \log \relax (c)\right )} - 2 \, \sqrt {b e^{\left (m n p q \log \left (f x\right ) + n p q \log \relax (e) + p q \log \relax (d) + q \log \relax (c)\right )} + a} \sqrt {a} + 2 \, a\right )} e^{\left (-m n p q \log \left (f x\right ) - n p q \log \relax (e) - p q \log \relax (d) - q \log \relax (c)\right )}\right )}{\sqrt {a} m n p q}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b e^{\left (m n p q \log \left (f x\right ) + n p q \log \relax (e) + p q \log \relax (d) + q \log \relax (c)\right )} + a} \sqrt {-a}}{a}\right )}{a m n p q}\right ] \]
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 \[ \int \frac {1}{\sqrt {\left (\left (\left (\left (f x\right )^{m} e\right )^{n} d\right )^{p} c\right )^{q} b + a} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 46, normalized size = 0.90 \[ -\frac {2 \arctanh \left (\frac {\sqrt {b \left (c \left (d \left (e \left (f x \right )^{m}\right )^{n}\right )^{p}\right )^{q}+a}}{\sqrt {a}}\right )}{\sqrt {a}\, m n p q} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {1}{x\,\sqrt {a+b\,{\left (c\,{\left (d\,{\left (e\,{\left (f\,x\right )}^m\right )}^n\right )}^p\right )}^q}} \,d x \]
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 \[ \int \frac {1}{x \sqrt {a + b \left (c \left (d \left (e \left (f x\right )^{m}\right )^{n}\right )^{p}\right )^{q}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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