Optimal. Leaf size=76 \[ \frac {x \left (c+d x^{2 n}\right )^p \left (1+\frac {d x^{2 n}}{c}\right )^{-p} F_1\left (\frac {1}{2 n};1,-p;\frac {1}{2} \left (2+\frac {1}{n}\right );\frac {b^2 x^{2 n}}{a^2},-\frac {d x^{2 n}}{c}\right )}{a^2} \]
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Rubi [A]
time = 0.04, 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 = {531, 441, 440}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rule 531
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] Leaf count is larger than twice the leaf count of optimal. \(258\) vs. \(2(76)=152\).
time = 0.32, size = 258, normalized size = 3.39 \begin {gather*} \frac {a^2 c (1+2 n) 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 (1+2 n) 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 )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\left (c +d \,x^{2 n}\right )^{p}}{\left (a -b \,x^{n}\right ) \left (a +b \,x^{n}\right )}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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 = -0.01 \begin {gather*} -\int -\frac {{\left (c+d\,x^{2\,n}\right )}^p}{a^2-b^2\,x^{2\,n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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