Optimal. Leaf size=30 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{\sqrt {a} n} \]
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Rubi [A] time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {367, 12, 266, 63, 208} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{\sqrt {a} n} \]
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 (c x)^n}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {c}{x \sqrt {a+b x^n}} \, dx,x,c x\right )}{c}\\ &=\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x^n}} \, dx,x,c x\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,(c x)^n\right )}{n}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b (c x)^n}\right )}{b n}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{\sqrt {a} n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{\sqrt {a} n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 78, normalized size = 2.60 \[ \left [\frac {\log \left (\frac {\left (c x\right )^{n} b - 2 \, \sqrt {\left (c x\right )^{n} b + a} \sqrt {a} + 2 \, a}{\left (c x\right )^{n}}\right )}{\sqrt {a} n}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {\left (c x\right )^{n} b + a} \sqrt {-a}}{a}\right )}{a n}\right ] \]
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 \[ \int \frac {1}{\sqrt {\left (c x\right )^{n} b + a} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.83 \[ -\frac {2 \arctanh \left (\frac {\sqrt {b \left (c x \right )^{n}+a}}{\sqrt {a}}\right )}{\sqrt {a}\, n} \]
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 \[ \int \frac {1}{\sqrt {\left (c x\right )^{n} b + a} x}\,{d x} \]
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 \[ \int \frac {1}{x\,\sqrt {a+b\,{\left (c\,x\right )}^n}} \,d x \]
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 \[ \int \frac {1}{x \sqrt {a + b \left (c x\right )^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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