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 = 0.17, 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 = {367, 12, 266, 63, 208} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c (d x)^m\right )^n}}{\sqrt {a}}\right )}{\sqrt {a} m n} \]
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 (d x)^m\right )^n}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b (c x)^n}} \, dx,x,(d x)^m\right )}{m}\\ &=\frac {\operatorname {Subst}\left (\int \frac {c}{x \sqrt {a+b x^n}} \, dx,x,c (d x)^m\right )}{c m}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x^n}} \, dx,x,c (d x)^m\right )}{m}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\left (c (d x)^m\right )^n\right )}{m n}\\ &=\frac {2 \operatorname {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 = 0.07, size = 37, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c (d x)^m\right )^n}}{\sqrt {a}}\right )}{\sqrt {a} m n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 116, normalized size = 3.14 \[ \left [\frac {\log \left ({\left (b e^{\left (m n \log \left (d x\right ) + n \log \relax (c)\right )} - 2 \, \sqrt {b e^{\left (m n \log \left (d x\right ) + n \log \relax (c)\right )} + a} \sqrt {a} + 2 \, a\right )} e^{\left (-m n \log \left (d x\right ) - n \log \relax (c)\right )}\right )}{\sqrt {a} m n}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b e^{\left (m n \log \left (d x\right ) + n \log \relax (c)\right )} + a} \sqrt {-a}}{a}\right )}{a m n}\right ] \]
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 \[ \int \frac {1}{\sqrt {\left (\left (d x\right )^{m} c\right )^{n} b + a} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 32, normalized size = 0.86 \[ -\frac {2 \arctanh \left (\frac {\sqrt {b \left (c \left (d x \right )^{m}\right )^{n}+a}}{\sqrt {a}}\right )}{\sqrt {a}\, m n} \]
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 \[ \int \frac {1}{\sqrt {\left (\left (d x\right )^{m} c\right )^{n} b + a} x}\,{d x} \]
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 \[ \int \frac {1}{x\,\sqrt {a+b\,{\left (c\,{\left (d\,x\right )}^m\right )}^n}} \,d x \]
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 \[ \int \frac {1}{x \sqrt {a + b \left (c \left (d x\right )^{m}\right )^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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