Optimal. Leaf size=87 \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (1-\frac {b^4 x^{4 n}}{a^4}\right )^{-p} \, _2F_1\left (\frac {1}{4 n},-p;\frac {1}{4} \left (4+\frac {1}{n}\right );\frac {b^4 x^{4 n}}{a^4}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {519, 253, 246, 245} \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (1-\frac {b^4 x^{4 n}}{a^4}\right )^{-p} \, _2F_1\left (\frac {1}{4 n},-p;\frac {1}{4} \left (4+\frac {1}{n}\right );\frac {b^4 x^{4 n}}{a^4}\right ) \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rule 253
Rule 519
Rubi steps
\begin {align*} \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \, dx &=\left (\left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2-b^2 x^{2 n}\right )^{-p}\right ) \int \left (a^2-b^2 x^{2 n}\right )^p \left (a^2+b^2 x^{2 n}\right )^p \, dx\\ &=\left (\left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (a^4-b^4 x^{4 n}\right )^{-p}\right ) \int \left (a^4-b^4 x^{4 n}\right )^p \, dx\\ &=\left (\left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (1-\frac {b^4 x^{4 n}}{a^4}\right )^{-p}\right ) \int \left (1-\frac {b^4 x^{4 n}}{a^4}\right )^p \, dx\\ &=x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (1-\frac {b^4 x^{4 n}}{a^4}\right )^{-p} \, _2F_1\left (\frac {1}{4 n},-p;\frac {1}{4} \left (4+\frac {1}{n}\right );\frac {b^4 x^{4 n}}{a^4}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 87, normalized size = 1.00 \[ x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (1-\frac {b^4 x^{4 n}}{a^4}\right )^{-p} \, _2F_1\left (\frac {1}{4 n},-p;1+\frac {1}{4 n};\frac {b^4 x^{4 n}}{a^4}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.31, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} x^{2 \, n} + a^{2}\right )}^{p} {\left (b x^{n} + a\right )}^{p} {\left (-b x^{n} + a\right )}^{p}, x\right ) \]
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 \[ \int {\left (b^{2} x^{2 \, n} + a^{2}\right )}^{p} {\left (b x^{n} + a\right )}^{p} {\left (-b x^{n} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.50, size = 0, normalized size = 0.00 \[ \int \left (-b \,x^{n}+a \right )^{p} \left (b \,x^{n}+a \right )^{p} \left (b^{2} x^{2 n}+a^{2}\right )^{p}\, dx \]
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 \[ \int {\left (b^{2} x^{2 \, n} + a^{2}\right )}^{p} {\left (b x^{n} + a\right )}^{p} {\left (-b x^{n} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,x^n\right )}^p\,{\left (a-b\,x^n\right )}^p\,{\left (a^2+b^2\,x^{2\,n}\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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