Optimal. Leaf size=49 \[ \frac {2 \sqrt {a+b (c x)^n}}{n}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{n} \]
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Rubi [A] time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {367, 12, 266, 50, 63, 208} \[ \frac {2 \sqrt {a+b (c x)^n}}{n}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{n} \]
Antiderivative was successfully verified.
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Rule 12
Rule 50
Rule 63
Rule 208
Rule 266
Rule 367
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b (c x)^n}}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {c \sqrt {a+b x^n}}{x} \, dx,x,c x\right )}{c}\\ &=\operatorname {Subst}\left (\int \frac {\sqrt {a+b x^n}}{x} \, dx,x,c x\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,(c x)^n\right )}{n}\\ &=\frac {2 \sqrt {a+b (c x)^n}}{n}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,(c x)^n\right )}{n}\\ &=\frac {2 \sqrt {a+b (c x)^n}}{n}+\frac {(2 a) \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 \sqrt {a+b (c x)^n}}{n}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 0.96 \[ \frac {2 \sqrt {a+b (c x)^n}-2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b (c x)^n}}{\sqrt {a}}\right )}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 103, normalized size = 2.10 \[ \left [\frac {\sqrt {a} \log \left (\frac {\left (c x\right )^{n} b - 2 \, \sqrt {\left (c x\right )^{n} b + a} \sqrt {a} + 2 \, a}{\left (c x\right )^{n}}\right ) + 2 \, \sqrt {\left (c x\right )^{n} b + a}}{n}, \frac {2 \, {\left (\sqrt {-a} \arctan \left (\frac {\sqrt {\left (c x\right )^{n} b + a} \sqrt {-a}}{a}\right ) + \sqrt {\left (c x\right )^{n} b + a}\right )}}{n}\right ] \]
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 \[ \int \frac {\sqrt {\left (c x\right )^{n} b + a}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 40, normalized size = 0.82 \[ \frac {-2 \sqrt {a}\, \arctanh \left (\frac {\sqrt {b \left (c x \right )^{n}+a}}{\sqrt {a}}\right )+2 \sqrt {b \left (c x \right )^{n}+a}}{n} \]
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 \[ \int \frac {\sqrt {\left (c x\right )^{n} b + a}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a+b\,{\left (c\,x\right )}^n}}{x} \,d x \]
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 \[ \int \frac {\sqrt {a + b \left (c x\right )^{n}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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