Optimal. Leaf size=32 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {-a+b (c x)^n}}{\sqrt {a}}\right )}{\sqrt {a} n} \]
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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {374, 12, 272,
65, 211} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {\sqrt {b (c x)^n-a}}{\sqrt {a}}\right )}{\sqrt {a} n} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 211
Rule 272
Rule 374
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {-a+b (c x)^n}} \, dx &=\frac {\text {Subst}\left (\int \frac {c}{x \sqrt {-a+b x^n}} \, dx,x,c x\right )}{c}\\ &=\text {Subst}\left (\int \frac {1}{x \sqrt {-a+b x^n}} \, dx,x,c x\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {-a+b x}} \, dx,x,(c x)^n\right )}{n}\\ &=\frac {2 \text {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 \tan ^{-1}\left (\frac {\sqrt {-a+b (c x)^n}}{\sqrt {a}}\right )}{\sqrt {a} n}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 32, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {-a+b (c x)^n}}{\sqrt {a}}\right )}{\sqrt {a} n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.65, size = 27, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {2 \arctan \left (\frac {\sqrt {-a +b \left (c x \right )^{n}}}{\sqrt {a}}\right )}{n \sqrt {a}}\) | \(27\) |
default | \(\frac {2 \arctan \left (\frac {\sqrt {-a +b \left (c x \right )^{n}}}{\sqrt {a}}\right )}{n \sqrt {a}}\) | \(27\) |
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 = 0.37, size = 80, normalized size = 2.50 \begin {gather*} \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 )}{a n}, \frac {2 \, \arctan \left (\frac {\sqrt {\left (c x\right )^{n} b - a}}{\sqrt {a}}\right )}{\sqrt {a} n}\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 x\right )^{n}}}\, 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 = -0.03 \begin {gather*} \int \frac {1}{x\,\sqrt {b\,{\left (c\,x\right )}^n-a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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