Optimal. Leaf size=83 \[ \frac {d x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a}+\frac {e x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a (n+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1418, 245, 364} \[ \frac {d x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a}+\frac {e x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a (n+1)} \]
Antiderivative was successfully verified.
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Rule 245
Rule 364
Rule 1418
Rubi steps
\begin {align*} \int \frac {d+e x^n}{a+c x^{2 n}} \, dx &=d \int \frac {1}{a+c x^{2 n}} \, dx+e \int \frac {x^n}{a+c x^{2 n}} \, dx\\ &=\frac {d x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a}+\frac {e x^{1+n} \, _2F_1\left (1,\frac {1+n}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 83, normalized size = 1.00 \[ \frac {d x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a}+\frac {e x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e x^{n} + d}{c x^{2 \, n} + a}, x\right ) \]
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 \[ \int \frac {e x^{n} + d}{c x^{2 \, n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {e \,x^{n}+d}{c \,x^{2 n}+a}\, dx \]
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 \[ \int \frac {e x^{n} + d}{c x^{2 \, n} + a}\,{d x} \]
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 \[ \int \frac {d+e\,x^n}{a+c\,x^{2\,n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.54, size = 153, normalized size = 1.84 \[ \frac {d x \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{4 a n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {e x x^{n} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 a n \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} + \frac {e x x^{n} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 a n^{2} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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