Optimal. Leaf size=184 \[ \frac {d (1-4 n) (1-2 n) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{8 a^3 n^2}+\frac {e (1-3 n) (1-n) x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{8 a^3 n^2 (n+1)}-\frac {x \left (d (1-4 n)+e (1-3 n) x^n\right )}{8 a^2 n^2 \left (a+c x^{2 n}\right )}+\frac {x \left (d+e x^n\right )}{4 a n \left (a+c x^{2 n}\right )^2} \]
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Rubi [A] time = 0.10, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {1431, 1418, 245, 364} \[ -\frac {x \left (d (1-4 n)+e (1-3 n) x^n\right )}{8 a^2 n^2 \left (a+c x^{2 n}\right )}+\frac {d (1-4 n) (1-2 n) x \, _2F_1\left (1,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{8 a^3 n^2}+\frac {e (1-3 n) (1-n) x^{n+1} \, _2F_1\left (1,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{8 a^3 n^2 (n+1)}+\frac {x \left (d+e x^n\right )}{4 a n \left (a+c x^{2 n}\right )^2} \]
Antiderivative was successfully verified.
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Rule 245
Rule 364
Rule 1418
Rule 1431
Rubi steps
\begin {align*} \int \frac {d+e x^n}{\left (a+c x^{2 n}\right )^3} \, dx &=\frac {x \left (d+e x^n\right )}{4 a n \left (a+c x^{2 n}\right )^2}-\frac {\int \frac {d (1-4 n)+e (1-3 n) x^n}{\left (a+c x^{2 n}\right )^2} \, dx}{4 a n}\\ &=\frac {x \left (d+e x^n\right )}{4 a n \left (a+c x^{2 n}\right )^2}-\frac {x \left (d (1-4 n)+e (1-3 n) x^n\right )}{8 a^2 n^2 \left (a+c x^{2 n}\right )}+\frac {\int \frac {d (1-4 n) (1-2 n)+e (1-3 n) (1-n) x^n}{a+c x^{2 n}} \, dx}{8 a^2 n^2}\\ &=\frac {x \left (d+e x^n\right )}{4 a n \left (a+c x^{2 n}\right )^2}-\frac {x \left (d (1-4 n)+e (1-3 n) x^n\right )}{8 a^2 n^2 \left (a+c x^{2 n}\right )}+\frac {(d (1-4 n) (1-2 n)) \int \frac {1}{a+c x^{2 n}} \, dx}{8 a^2 n^2}+\frac {(e (1-3 n) (1-n)) \int \frac {x^n}{a+c x^{2 n}} \, dx}{8 a^2 n^2}\\ &=\frac {x \left (d+e x^n\right )}{4 a n \left (a+c x^{2 n}\right )^2}-\frac {x \left (d (1-4 n)+e (1-3 n) x^n\right )}{8 a^2 n^2 \left (a+c x^{2 n}\right )}+\frac {d (1-4 n) (1-2 n) 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 )}{8 a^3 n^2}+\frac {e (1-3 n) (1-n) 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 )}{8 a^3 n^2 (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 83, normalized size = 0.45 \[ \frac {d x \, _2F_1\left (3,\frac {1}{2 n};\frac {1}{2} \left (2+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^3}+\frac {e x^{n+1} \, _2F_1\left (3,\frac {n+1}{2 n};\frac {1}{2} \left (3+\frac {1}{n}\right );-\frac {c x^{2 n}}{a}\right )}{a^3 (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e x^{n} + d}{c^{3} x^{6 \, n} + 3 \, a c^{2} x^{4 \, n} + 3 \, a^{2} c x^{2 \, n} + a^{3}}, x\right ) \]
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 \[ \int \frac {e x^{n} + d}{{\left (c x^{2 \, n} + a\right )}^{3}}\,{d x} \]
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 \[ \int \frac {e \,x^{n}+d}{\left (c \,x^{2 n}+a \right )^{3}}\, dx \]
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 \[ \frac {c e {\left (3 \, n - 1\right )} x x^{3 \, n} + c d {\left (4 \, n - 1\right )} x x^{2 \, n} + a e {\left (5 \, n - 1\right )} x x^{n} + a d {\left (6 \, n - 1\right )} x}{8 \, {\left (a^{2} c^{2} n^{2} x^{4 \, n} + 2 \, a^{3} c n^{2} x^{2 \, n} + a^{4} n^{2}\right )}} + \int \frac {{\left (3 \, n^{2} - 4 \, n + 1\right )} e x^{n} + {\left (8 \, n^{2} - 6 \, n + 1\right )} d}{8 \, {\left (a^{2} c n^{2} x^{2 \, n} + a^{3} n^{2}\right )}}\,{d x} \]
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 \[ \int \frac {d+e\,x^n}{{\left (a+c\,x^{2\,n}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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