Optimal. Leaf size=167 \[ \frac {x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1}{2 n};-p,1;\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d}-\frac {e x^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1+n}{2 n};-p,1;\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (1+n)} \]
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Rubi [A]
time = 0.10, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1452, 441, 440,
525, 524} \begin {gather*} \frac {x \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {1}{2 n};-p,1;\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d}-\frac {e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} F_1\left (\frac {n+1}{2 n};-p,1;\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rule 524
Rule 525
Rule 1452
Rubi steps
\begin {align*} \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx &=\int \left (\frac {d \left (a+c x^{2 n}\right )^p}{d^2-e^2 x^{2 n}}+\frac {e x^n \left (a+c x^{2 n}\right )^p}{-d^2+e^2 x^{2 n}}\right ) \, dx\\ &=d \int \frac {\left (a+c x^{2 n}\right )^p}{d^2-e^2 x^{2 n}} \, dx+e \int \frac {x^n \left (a+c x^{2 n}\right )^p}{-d^2+e^2 x^{2 n}} \, dx\\ &=\left (d \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {c x^{2 n}}{a}\right )^p}{d^2-e^2 x^{2 n}} \, dx+\left (e \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p}\right ) \int \frac {x^n \left (1+\frac {c x^{2 n}}{a}\right )^p}{-d^2+e^2 x^{2 n}} \, dx\\ &=\frac {x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1}{2 n};-p,1;\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d}-\frac {e x^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} F_1\left (\frac {1+n}{2 n};-p,1;\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (1+n)}\\ \end {align*}
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Mathematica [F]
time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+c x^{2 n}\right )^p}{d+e x^n} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (a +c \,x^{2 n}\right )^{p}}{d +e \,x^{n}}\, dx\]
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 (a+c\,x^{2\,n}\right )}^p}{d+e\,x^n} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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