Optimal. Leaf size=141 \[ \frac {e x^{n+1} \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a c (n+1)}+\frac {d x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {3 d e^2 x}{c}+\frac {e^3 x^{n+1}}{c (n+1)} \]
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Rubi [A] time = 0.15, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1425, 1418, 245, 364} \[ \frac {e x^{n+1} \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a c (n+1)}+\frac {d x \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a c}+\frac {3 d e^2 x}{c}+\frac {e^3 x^{n+1}}{c (n+1)} \]
Antiderivative was successfully verified.
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Rule 245
Rule 364
Rule 1418
Rule 1425
Rubi steps
\begin {align*} \int \frac {\left (d+e x^n\right )^3}{a+c x^{2 n}} \, dx &=\int \left (\frac {3 d e^2}{c}+\frac {e^3 x^n}{c}+\frac {c d^3-3 a d e^2+\left (3 c d^2 e-a e^3\right ) x^n}{c \left (a+c x^{2 n}\right )}\right ) \, dx\\ &=\frac {3 d e^2 x}{c}+\frac {e^3 x^{1+n}}{c (1+n)}+\frac {\int \frac {c d^3-3 a d e^2+\left (3 c d^2 e-a e^3\right ) x^n}{a+c x^{2 n}} \, dx}{c}\\ &=\frac {3 d e^2 x}{c}+\frac {e^3 x^{1+n}}{c (1+n)}+\frac {\left (d \left (c d^2-3 a e^2\right )\right ) \int \frac {1}{a+c x^{2 n}} \, dx}{c}+\frac {\left (e \left (3 c d^2-a e^2\right )\right ) \int \frac {x^n}{a+c x^{2 n}} \, dx}{c}\\ &=\frac {3 d e^2 x}{c}+\frac {e^3 x^{1+n}}{c (1+n)}+\frac {d \left (c d^2-3 a e^2\right ) 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 c}+\frac {e \left (3 c d^2-a e^2\right ) 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 c (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 127, normalized size = 0.90 \[ \frac {x \left (d (n+1) \left (c d^2-3 a e^2\right ) \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )+e \left (x^n \left (3 c d^2-a e^2\right ) \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )+a e \left (3 d (n+1)+e x^n\right )\right )\right )}{a c (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{3} x^{3 \, n} + 3 \, d e^{2} x^{2 \, n} + 3 \, d^{2} e x^{n} + d^{3}}{c x^{2 \, n} + a}, x\right ) \]
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 \[ \int \frac {{\left (e x^{n} + d\right )}^{3}}{c x^{2 \, n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \,x^{n}+d \right )^{3}}{c \,x^{2 n}+a}\, dx \]
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 \[ \frac {3 \, d e^{2} {\left (n + 1\right )} x + e^{3} x x^{n}}{c {\left (n + 1\right )}} - \int -\frac {c d^{3} - 3 \, a d e^{2} + {\left (3 \, c d^{2} e - a e^{3}\right )} x^{n}}{c^{2} x^{2 \, n} + a c}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x^n\right )}^3}{a+c\,x^{2\,n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 10.97, size = 337, normalized size = 2.39 \[ - \frac {3 d e^{2} x \Phi \left (\frac {a x^{- 2 n} e^{i \pi }}{c}, 1, \frac {e^{i \pi }}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{4 c n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {d^{3} x \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2 n}\right )}{4 a n^{2} \Gamma \left (1 + \frac {1}{2 n}\right )} + \frac {3 d^{2} e x x^{n} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 a n \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} + \frac {3 d^{2} e x x^{n} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{2 n}\right )}{4 a n^{2} \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )} + \frac {3 e^{3} x x^{3 n} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {3}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )}{4 a n \Gamma \left (\frac {5}{2} + \frac {1}{2 n}\right )} + \frac {e^{3} x x^{3 n} \Phi \left (\frac {c x^{2 n} e^{i \pi }}{a}, 1, \frac {3}{2} + \frac {1}{2 n}\right ) \Gamma \left (\frac {3}{2} + \frac {1}{2 n}\right )}{4 a n^{2} \Gamma \left (\frac {5}{2} + \frac {1}{2 n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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