Optimal. Leaf size=72 \[ \frac {2}{a c^4 (n+2) x \sqrt {\frac {a}{x^2}+b x^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{a^{3/2} c^4 (n+2)} \]
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Rubi [A] time = 0.16, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2030, 2029, 206} \[ \frac {2}{a c^4 (n+2) x \sqrt {\frac {a}{x^2}+b x^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{a^{3/2} c^4 (n+2)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 2029
Rule 2030
Rubi steps
\begin {align*} \int \frac {1}{c^4 x^4 \left (\frac {a}{x^2}+b x^n\right )^{3/2}} \, dx &=\frac {\int \frac {1}{x^4 \left (\frac {a}{x^2}+b x^n\right )^{3/2}} \, dx}{c^4}\\ &=\frac {2}{a c^4 (2+n) x \sqrt {\frac {a}{x^2}+b x^n}}+\frac {\int \frac {1}{x^2 \sqrt {\frac {a}{x^2}+b x^n}} \, dx}{a c^4}\\ &=\frac {2}{a c^4 (2+n) x \sqrt {\frac {a}{x^2}+b x^n}}-\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 )}{a c^4 (2+n)}\\ &=\frac {2}{a c^4 (2+n) x \sqrt {\frac {a}{x^2}+b x^n}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a}}{x \sqrt {\frac {a}{x^2}+b x^n}}\right )}{a^{3/2} c^4 (2+n)}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 51, normalized size = 0.71 \[ \frac {2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b x^{n+2}}{a}+1\right )}{a c^4 (n+2) x \sqrt {\frac {a}{x^2}+b x^n}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {1}{{\left (b x^{n} + \frac {a}{x^{2}}\right )}^{\frac {3}{2}} c^{4} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.72, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{n}+\frac {a}{x^{2}}\right )^{\frac {3}{2}} c^{4} x^{4}}\, dx \]
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 \[ \frac {\int \frac {1}{{\left (b x^{n} + \frac {a}{x^{2}}\right )}^{\frac {3}{2}} x^{4}}\,{d x}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{c^4\,x^4\,{\left (b\,x^n+\frac {a}{x^2}\right )}^{3/2}} \,d x \]
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 \[ \frac {\int \frac {1}{a x^{2} \sqrt {\frac {a}{x^{2}} + b x^{n}} + b x^{4} x^{n} \sqrt {\frac {a}{x^{2}} + b x^{n}}}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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