Optimal. Leaf size=76 \[ \frac {x \left (c+d x^{2 n}\right )^p \left (\frac {d x^{2 n}}{c}+1\right )^{-p} F_1\left (\frac {1}{2 n};1,-p;\frac {1}{2} \left (2+\frac {1}{n}\right );\frac {b^2 x^{2 n}}{a^2},-\frac {d x^{2 n}}{c}\right )}{a^2} \]
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Rubi [A] time = 0.06, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {517, 430, 429} \[ \frac {x \left (c+d x^{2 n}\right )^p \left (\frac {d x^{2 n}}{c}+1\right )^{-p} F_1\left (\frac {1}{2 n};1,-p;\frac {1}{2} \left (2+\frac {1}{n}\right );\frac {b^2 x^{2 n}}{a^2},-\frac {d x^{2 n}}{c}\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 429
Rule 430
Rule 517
Rubi steps
\begin {align*} \int \frac {\left (c+d x^{2 n}\right )^p}{\left (a-b x^n\right ) \left (a+b x^n\right )} \, dx &=\int \frac {\left (c+d x^{2 n}\right )^p}{a^2-b^2 x^{2 n}} \, dx\\ &=\left (\left (c+d x^{2 n}\right )^p \left (1+\frac {d x^{2 n}}{c}\right )^{-p}\right ) \int \frac {\left (1+\frac {d x^{2 n}}{c}\right )^p}{a^2-b^2 x^{2 n}} \, dx\\ &=\frac {x \left (c+d x^{2 n}\right )^p \left (1+\frac {d x^{2 n}}{c}\right )^{-p} F_1\left (\frac {1}{2 n};1,-p;\frac {1}{2} \left (2+\frac {1}{n}\right );\frac {b^2 x^{2 n}}{a^2},-\frac {d x^{2 n}}{c}\right )}{a^2}\\ \end {align*}
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Mathematica [B] time = 0.43, size = 258, normalized size = 3.39 \[ \frac {a^2 c (2 n+1) x \left (c+d x^{2 n}\right )^p F_1\left (\frac {1}{2 n};-p,1;1+\frac {1}{2 n};-\frac {d x^{2 n}}{c},\frac {b^2 x^{2 n}}{a^2}\right )}{\left (a^2-b^2 x^{2 n}\right ) \left (2 a^2 d n p x^{2 n} F_1\left (1+\frac {1}{2 n};1-p,1;2+\frac {1}{2 n};-\frac {d x^{2 n}}{c},\frac {b^2 x^{2 n}}{a^2}\right )+2 b^2 c n x^{2 n} F_1\left (1+\frac {1}{2 n};-p,2;2+\frac {1}{2 n};-\frac {d x^{2 n}}{c},\frac {b^2 x^{2 n}}{a^2}\right )+a^2 c (2 n+1) F_1\left (\frac {1}{2 n};-p,1;1+\frac {1}{2 n};-\frac {d x^{2 n}}{c},\frac {b^2 x^{2 n}}{a^2}\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (d x^{2 \, n} + c\right )}^{p}}{b^{2} x^{2 \, n} - a^{2}}, x\right ) \]
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 \[ \int -\frac {{\left (d x^{2 \, n} + c\right )}^{p}}{{\left (b x^{n} + a\right )} {\left (b x^{n} - a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.22, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{2 n}+c \right )^{p}}{\left (-b \,x^{n}+a \right ) \left (b \,x^{n}+a \right )}\, dx \]
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 \[ -\int \frac {{\left (d x^{2 \, n} + c\right )}^{p}}{{\left (b x^{n} + a\right )} {\left (b x^{n} - a\right )}}\,{d x} \]
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 \[ -\int -\frac {{\left (c+d\,x^{2\,n}\right )}^p}{a^2-b^2\,x^{2\,n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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