Optimal. Leaf size=713 \[ \frac{x \left (c x^n \left (-4 a^2 c e (1-3 n)+a b^2 e+2 a b c d (2-7 n)+b^3 (-d) (1-2 n)\right )-2 a^2 b c e (2-3 n)-4 a^2 c^2 d (1-4 n)+5 a b^2 c d (1-3 n)+a b^3 e-b^4 d (1-2 n)\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (a+b x^n+c x^{2 n}\right )}+\frac{c x \left (-4 a^2 c \left (e \left (3 n^2-4 n+1\right ) \sqrt{b^2-4 a c}+2 c d \left (8 n^2-6 n+1\right )\right )-2 a b c \left (2 a e \left (-3 n^2-n+1\right )-d \left (7 n^2-9 n+2\right ) \sqrt{b^2-4 a c}\right )+b^3 (1-n) \left (a e-d (1-2 n) \sqrt{b^2-4 a c}\right )+a b^2 (1-n) \left (e \sqrt{b^2-4 a c}+6 c d (1-3 n)\right )+b^4 (-d) \left (2 n^2-3 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c x \left (-4 a^2 c \left (e \left (3 n^2-4 n+1\right ) \sqrt{b^2-4 a c}-2 c d \left (8 n^2-6 n+1\right )\right )+2 a b c \left (d \left (7 n^2-9 n+2\right ) \sqrt{b^2-4 a c}+2 a e \left (-3 n^2-n+1\right )\right )-b^3 (1-n) \left (d (1-2 n) \sqrt{b^2-4 a c}+a e\right )+a b^2 (1-n) \left (e \sqrt{b^2-4 a c}-6 c d (1-3 n)\right )+b^4 d \left (2 n^2-3 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{x \left (c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2} \]
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Rubi [A] time = 1.6635, antiderivative size = 713, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1430, 1422, 245} \[ \frac{x \left (c x^n \left (-4 a^2 c e (1-3 n)+a b^2 e+2 a b c d (2-7 n)+b^3 (-d) (1-2 n)\right )-2 a^2 b c e (2-3 n)-4 a^2 c^2 d (1-4 n)+5 a b^2 c d (1-3 n)+a b^3 e-b^4 d (1-2 n)\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (a+b x^n+c x^{2 n}\right )}+\frac{c x \left (-4 a^2 c \left (e \left (3 n^2-4 n+1\right ) \sqrt{b^2-4 a c}+2 c d \left (8 n^2-6 n+1\right )\right )-2 a b c \left (2 a e \left (-3 n^2-n+1\right )-d \left (7 n^2-9 n+2\right ) \sqrt{b^2-4 a c}\right )+b^3 (1-n) \left (a e-d (1-2 n) \sqrt{b^2-4 a c}\right )+a b^2 (1-n) \left (e \sqrt{b^2-4 a c}+6 c d (1-3 n)\right )+b^4 (-d) \left (2 n^2-3 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c x \left (-4 a^2 c \left (e \left (3 n^2-4 n+1\right ) \sqrt{b^2-4 a c}-2 c d \left (8 n^2-6 n+1\right )\right )+2 a b c \left (d \left (7 n^2-9 n+2\right ) \sqrt{b^2-4 a c}+2 a e \left (-3 n^2-n+1\right )\right )-b^3 (1-n) \left (d (1-2 n) \sqrt{b^2-4 a c}+a e\right )+a b^2 (1-n) \left (e \sqrt{b^2-4 a c}-6 c d (1-3 n)\right )+b^4 d \left (2 n^2-3 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 a^2 n^2 \left (b^2-4 a c\right )^2 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{x \left (c x^n (b d-2 a e)-a b e-2 a c d+b^2 d\right )}{2 a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )^2} \]
Antiderivative was successfully verified.
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Rule 1430
Rule 1422
Rule 245
Rubi steps
\begin{align*} \int \frac{d+e x^n}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx &=\frac{x \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}-\frac{\int \frac{-a b e-2 a c d (1-4 n)+b^2 (d-2 d n)+c (b d-2 a e) (1-3 n) x^n}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{2 a \left (b^2-4 a c\right ) n}\\ &=\frac{x \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac{x \left (a b^3 e-4 a^2 c^2 d (1-4 n)+5 a b^2 c d (1-3 n)-2 a^2 b c e (2-3 n)-b^4 d (1-2 n)+c \left (a b^2 e+2 a b c d (2-7 n)-4 a^2 c e (1-3 n)-b^3 d (1-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac{\int \frac{2 a^2 b c e (2-5 n)-a b^3 e (1-n)+b^4 d \left (1-3 n+2 n^2\right )+4 a^2 c^2 d \left (1-6 n+8 n^2\right )-a b^2 c d \left (5-21 n+16 n^2\right )-c \left (a b^2 e+2 a b c d (2-7 n)-4 a^2 c e (1-3 n)-b^3 d (1-2 n)\right ) (1-n) x^n}{a+b x^n+c x^{2 n}} \, dx}{2 a^2 \left (b^2-4 a c\right )^2 n^2}\\ &=\frac{x \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac{x \left (a b^3 e-4 a^2 c^2 d (1-4 n)+5 a b^2 c d (1-3 n)-2 a^2 b c e (2-3 n)-b^4 d (1-2 n)+c \left (a b^2 e+2 a b c d (2-7 n)-4 a^2 c e (1-3 n)-b^3 d (1-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac{\left (c \left (a b^2 \left (\sqrt{b^2-4 a c} e-6 c d (1-3 n)\right ) (1-n)-b^3 \left (a e+\sqrt{b^2-4 a c} d (1-2 n)\right ) (1-n)+b^4 d \left (1-3 n+2 n^2\right )+2 a b c \left (2 a e \left (1-n-3 n^2\right )+\sqrt{b^2-4 a c} d \left (2-9 n+7 n^2\right )\right )-4 a^2 c \left (\sqrt{b^2-4 a c} e \left (1-4 n+3 n^2\right )-2 c d \left (1-6 n+8 n^2\right )\right )\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^n} \, dx}{4 a^2 \left (b^2-4 a c\right )^{5/2} n^2}-\frac{\left (c \left (a b^2 \left (\sqrt{b^2-4 a c} e+6 c d (1-3 n)\right ) (1-n)+b^3 \left (a e-\sqrt{b^2-4 a c} d (1-2 n)\right ) (1-n)-b^4 d \left (1-3 n+2 n^2\right )-2 a b c \left (2 a e \left (1-n-3 n^2\right )-\sqrt{b^2-4 a c} d \left (2-9 n+7 n^2\right )\right )-4 a^2 c \left (\sqrt{b^2-4 a c} e \left (1-4 n+3 n^2\right )+2 c d \left (1-6 n+8 n^2\right )\right )\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^n} \, dx}{4 a^2 \left (b^2-4 a c\right )^{5/2} n^2}\\ &=\frac{x \left (b^2 d-2 a c d-a b e+c (b d-2 a e) x^n\right )}{2 a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac{x \left (a b^3 e-4 a^2 c^2 d (1-4 n)+5 a b^2 c d (1-3 n)-2 a^2 b c e (2-3 n)-b^4 d (1-2 n)+c \left (a b^2 e+2 a b c d (2-7 n)-4 a^2 c e (1-3 n)-b^3 d (1-2 n)\right ) x^n\right )}{2 a^2 \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac{c \left (a b^2 \left (\sqrt{b^2-4 a c} e+6 c d (1-3 n)\right ) (1-n)+b^3 \left (a e-\sqrt{b^2-4 a c} d (1-2 n)\right ) (1-n)-b^4 d \left (1-3 n+2 n^2\right )-2 a b c \left (2 a e \left (1-n-3 n^2\right )-\sqrt{b^2-4 a c} d \left (2-9 n+7 n^2\right )\right )-4 a^2 c \left (\sqrt{b^2-4 a c} e \left (1-4 n+3 n^2\right )+2 c d \left (1-6 n+8 n^2\right )\right )\right ) x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{5/2} \left (b-\sqrt{b^2-4 a c}\right ) n^2}-\frac{c \left (a b^2 \left (\sqrt{b^2-4 a c} e-6 c d (1-3 n)\right ) (1-n)-b^3 \left (a e+\sqrt{b^2-4 a c} d (1-2 n)\right ) (1-n)+b^4 d \left (1-3 n+2 n^2\right )+2 a b c \left (2 a e \left (1-n-3 n^2\right )+\sqrt{b^2-4 a c} d \left (2-9 n+7 n^2\right )\right )-4 a^2 c \left (\sqrt{b^2-4 a c} e \left (1-4 n+3 n^2\right )-2 c d \left (1-6 n+8 n^2\right )\right )\right ) x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right )^{5/2} \left (b+\sqrt{b^2-4 a c}\right ) n^2}\\ \end{align*}
Mathematica [B] time = 6.53474, size = 5848, normalized size = 8.2 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.094, size = 0, normalized size = 0. \begin{align*} \int{\frac{d+e{x}^{n}}{ \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e x^{n} + d}{c^{3} x^{6 \, n} + b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3} + 3 \,{\left (b c^{2} x^{n} + a c^{2}\right )} x^{4 \, n} + 3 \,{\left (b^{2} c x^{2 \, n} + 2 \, a b c x^{n} + a^{2} c\right )} x^{2 \, n}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e x^{n} + d}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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