Optimal. Leaf size=40 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{\sqrt {a} c^2 (n+2)} \]
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Rubi [A] time = 0.07, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {12, 2029, 206} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{\sqrt {a} c^2 (n+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 2029
Rubi steps
\begin {align*} \int \frac {1}{c^2 x^2 \sqrt {\frac {a}{x^2}+b x^n}} \, dx &=\frac {\int \frac {1}{x^2 \sqrt {\frac {a}{x^2}+b x^n}} \, dx}{c^2}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {1}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{c^2 (2+n)}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{\sqrt {a} c^2 (2+n)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 66, normalized size = 1.65 \begin {gather*} -\frac {2 \sqrt {a+b x^{n+2}} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+2}}}{\sqrt {a}}\right )}{\sqrt {a} c^2 (n+2) x \sqrt {\frac {a}{x^2}+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 64, normalized size = 1.60 \begin {gather*} -\frac {2 x \sqrt {\frac {a}{x^2}+b x^n} \tanh ^{-1}\left (\frac {\sqrt {a+b x^{n+2}}}{\sqrt {a}}\right )}{\sqrt {a} c^2 (n+2) \sqrt {a+b x^{n+2}}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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*} \int \frac {1}{\sqrt {b x^{n} + \frac {a}{x^{2}}} c^{2} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.70, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {b \,x^{n}+\frac {a}{x^{2}}}\, c^{2} x^{2}}\, dx \end {gather*}
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*} \frac {\int \frac {1}{\sqrt {b x^{n} + \frac {a}{x^{2}}} x^{2}}\,{d x}}{c^{2}} \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 = -0.02 \begin {gather*} \int \frac {1}{c^2\,x^2\,\sqrt {b\,x^n+\frac {a}{x^2}}} \,d x \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*} \frac {\int \frac {1}{x^{2} \sqrt {\frac {a}{x^{2}} + b x^{n}}}\, dx}{c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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