Optimal. Leaf size=1707 \[ \text{result too large to display} \]
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Rubi [A] time = 5.25273, antiderivative size = 1707, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1436, 1430, 1422, 245} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Rule 1436
Rule 1430
Rule 1422
Rule 245
Rubi steps
\begin{align*} \int \frac{\left (d+e x^n\right )^3}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx &=\int \left (\frac{c^2 d^3-3 a c d e^2+a b e^3+\left (3 c^2 d^2 e-3 b c d e^2+b^2 e^3-a c e^3\right ) x^n}{c^2 \left (a+b x^n+c x^{2 n}\right )^3}+\frac{e^2 \left (3 c d-b e+c e x^n\right )}{c^2 \left (a+b x^n+c x^{2 n}\right )^2}\right ) \, dx\\ &=\frac{\int \frac{c^2 d^3-3 a c d e^2+a b e^3+\left (3 c^2 d^2 e-3 b c d e^2+b^2 e^3-a c e^3\right ) x^n}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx}{c^2}+\frac{e^2 \int \frac{3 c d-b e+c e x^n}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{c^2}\\ &=\frac{x \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^n\right )}{2 a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac{e^2 x \left (3 b^2 c d-6 a c^2 d-b^3 e+a b c e+c \left (3 b c d-b^2 e-2 a c e\right ) x^n\right )}{a c^2 \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac{\int \frac{-a b c e \left (3 c d^2+a e^2 (1-8 n)\right )-2 a c^2 d \left (c d^2-3 a e^2\right ) (1-4 n)-2 a b^3 e^3 n+b^2 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-c \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) (1-3 n) x^n}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{2 a c^2 \left (b^2-4 a c\right ) n}-\frac{e^2 \int \frac{-a b c e-2 a c (3 c d-b e) (1-2 n)+b^2 (3 c d-b e) (1-n)+c \left (3 b c d-b^2 e-2 a c e\right ) (1-n) x^n}{a+b x^n+c x^{2 n}} \, dx}{a c^2 \left (b^2-4 a c\right ) n}\\ &=\frac{x \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^n\right )}{2 a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac{e^2 x \left (3 b^2 c d-6 a c^2 d-b^3 e+a b c e+c \left (3 b c d-b^2 e-2 a c e\right ) x^n\right )}{a c^2 \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac{x \left (a b^2 c^2 d \left (3 a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )+4 a^2 c^3 d \left (c d^2-3 a e^2\right ) (1-4 n)-2 a b^5 e^3 n+2 a^2 b c^2 e \left (3 c d^2 (2-3 n)-5 a e^2 n\right )-3 a b^3 c e \left (c d^2-3 a e^2 n\right )+b^4 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )+c \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right ) x^n\right )}{2 a^2 c^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 b^5 e^3 (1-n) n+b^4 c d (1-n) \left (c d^2 (1-2 n)+6 a e^2 n\right )+2 a^2 b c^2 e \left (3 c d^2 (2-5 n)-a e^2 (7-16 n) n\right )-a b^3 c e \left (3 c d^2 (1-n)-2 a e^2 (5-8 n) n\right )+4 a^2 c^3 d \left (c d^2-3 a e^2\right ) \left (1-6 n+8 n^2\right )-a b^2 c^2 d \left (c d^2 \left (5-21 n+16 n^2\right )-3 a e^2 \left (1-11 n+16 n^2\right )\right )+c (1-n) \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right ) x^n}{a+b x^n+c x^{2 n}} \, dx}{2 a^2 c^2 \left (b^2-4 a c\right )^2 n^2}-\frac{\left (e^2 \left (\frac{2 a b c e (2-5 n)-12 a c^2 d (1-2 n)+3 b^2 c d (1-n)-b^3 e (1-n)}{\sqrt{b^2-4 a c}}+\left (3 b c d-b^2 e-2 a c e\right ) (1-n)\right )\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^n} \, dx}{2 a c \left (b^2-4 a c\right ) n}+\frac{\left (e^2 \left (b c \left (2 a e (2-5 n)-3 \sqrt{b^2-4 a c} d (1-n)\right )-2 a c \left (6 c d (1-2 n)-\sqrt{b^2-4 a c} e (1-n)\right )+b^2 \left (3 c d+\sqrt{b^2-4 a c} e\right ) (1-n)-b^3 (e-e n)\right )\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^n} \, dx}{2 a c \left (b^2-4 a c\right )^{3/2} n}\\ &=\frac{x \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^n\right )}{2 a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac{e^2 x \left (3 b^2 c d-6 a c^2 d-b^3 e+a b c e+c \left (3 b c d-b^2 e-2 a c e\right ) x^n\right )}{a c^2 \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac{x \left (a b^2 c^2 d \left (3 a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )+4 a^2 c^3 d \left (c d^2-3 a e^2\right ) (1-4 n)-2 a b^5 e^3 n+2 a^2 b c^2 e \left (3 c d^2 (2-3 n)-5 a e^2 n\right )-3 a b^3 c e \left (c d^2-3 a e^2 n\right )+b^4 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )+c \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right ) x^n\right )}{2 a^2 c^2 \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac{e^2 \left (\frac{2 a b c e (2-5 n)-12 a c^2 d (1-2 n)+3 b^2 c d (1-n)-b^3 e (1-n)}{\sqrt{b^2-4 a c}}+\left (3 b c d-b^2 e-2 a c e\right ) (1-n)\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 )}{a c \left (b^2-4 a c\right ) \left (b-\sqrt{b^2-4 a c}\right ) n}+\frac{e^2 \left (b c \left (2 a e (2-5 n)-3 \sqrt{b^2-4 a c} d (1-n)\right )-2 a c \left (6 c d (1-2 n)-\sqrt{b^2-4 a c} e (1-n)\right )+b^2 \left (3 c d+\sqrt{b^2-4 a c} e\right ) (1-n)-b^3 (e-e n)\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 )}{a c \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt{b^2-4 a c}\right ) n}+\frac{\left ((1-n) \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right )-\frac{2 a b^5 e^3 (1-n) n-b^4 c d (1-n) \left (c d^2 (1-2 n)+6 a e^2 n\right )-8 a^2 c^3 d \left (c d^2-3 a e^2\right ) \left (1-6 n+8 n^2\right )+6 a b^2 c^2 d \left (c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 n^2\right )\right )-4 a^2 b c^2 e \left (3 c d^2 \left (1-n-3 n^2\right )+a e^2 \left (1-11 n+19 n^2\right )\right )+a b^3 c e \left (3 c d^2 (1-n)+a e^2 \left (1-19 n+30 n^2\right )\right )}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^n} \, dx}{4 a^2 c \left (b^2-4 a c\right )^2 n^2}+\frac{\left ((1-n) \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right )+\frac{2 a b^5 e^3 (1-n) n-b^4 c d (1-n) \left (c d^2 (1-2 n)+6 a e^2 n\right )-8 a^2 c^3 d \left (c d^2-3 a e^2\right ) \left (1-6 n+8 n^2\right )+6 a b^2 c^2 d \left (c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 n^2\right )\right )-4 a^2 b c^2 e \left (3 c d^2 \left (1-n-3 n^2\right )+a e^2 \left (1-11 n+19 n^2\right )\right )+a b^3 c e \left (3 c d^2 (1-n)+a e^2 \left (1-19 n+30 n^2\right )\right )}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^n} \, dx}{4 a^2 c \left (b^2-4 a c\right )^2 n^2}\\ &=\frac{x \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^n\right )}{2 a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )^2}+\frac{e^2 x \left (3 b^2 c d-6 a c^2 d-b^3 e+a b c e+c \left (3 b c d-b^2 e-2 a c e\right ) x^n\right )}{a c^2 \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac{x \left (a b^2 c^2 d \left (3 a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )+4 a^2 c^3 d \left (c d^2-3 a e^2\right ) (1-4 n)-2 a b^5 e^3 n+2 a^2 b c^2 e \left (3 c d^2 (2-3 n)-5 a e^2 n\right )-3 a b^3 c e \left (c d^2-3 a e^2 n\right )+b^4 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )+c \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right ) x^n\right )}{2 a^2 c^2 \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}-\frac{e^2 \left (\frac{2 a b c e (2-5 n)-12 a c^2 d (1-2 n)+3 b^2 c d (1-n)-b^3 e (1-n)}{\sqrt{b^2-4 a c}}+\left (3 b c d-b^2 e-2 a c e\right ) (1-n)\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 )}{a c \left (b^2-4 a c\right ) \left (b-\sqrt{b^2-4 a c}\right ) n}+\frac{\left ((1-n) \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right )-\frac{2 a b^5 e^3 (1-n) n-b^4 c d (1-n) \left (c d^2 (1-2 n)+6 a e^2 n\right )-8 a^2 c^3 d \left (c d^2-3 a e^2\right ) \left (1-6 n+8 n^2\right )+6 a b^2 c^2 d \left (c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 n^2\right )\right )-4 a^2 b c^2 e \left (3 c d^2 \left (1-n-3 n^2\right )+a e^2 \left (1-11 n+19 n^2\right )\right )+a b^3 c e \left (3 c d^2 (1-n)+a e^2 \left (1-19 n+30 n^2\right )\right )}{\sqrt{b^2-4 a c}}\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 c \left (b^2-4 a c\right )^2 \left (b-\sqrt{b^2-4 a c}\right ) n^2}+\frac{e^2 \left (b c \left (2 a e (2-5 n)-3 \sqrt{b^2-4 a c} d (1-n)\right )-2 a c \left (6 c d (1-2 n)-\sqrt{b^2-4 a c} e (1-n)\right )+b^2 \left (3 c d+\sqrt{b^2-4 a c} e\right ) (1-n)-b^3 (e-e n)\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 )}{a c \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt{b^2-4 a c}\right ) n}+\frac{\left ((1-n) \left (4 a^2 c^2 e \left (3 c d^2-a e^2\right ) (1-3 n)-2 a b^4 e^3 n-2 a b c^2 d \left (c d^2 (2-7 n)+3 a e^2 n\right )+b^3 c d \left (c d^2 (1-2 n)+6 a e^2 n\right )-a b^2 c e \left (3 c d^2-a e^2 (1+2 n)\right )\right )+\frac{2 a b^5 e^3 (1-n) n-b^4 c d (1-n) \left (c d^2 (1-2 n)+6 a e^2 n\right )-8 a^2 c^3 d \left (c d^2-3 a e^2\right ) \left (1-6 n+8 n^2\right )+6 a b^2 c^2 d \left (c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 n^2\right )\right )-4 a^2 b c^2 e \left (3 c d^2 \left (1-n-3 n^2\right )+a e^2 \left (1-11 n+19 n^2\right )\right )+a b^3 c e \left (3 c d^2 (1-n)+a e^2 \left (1-19 n+30 n^2\right )\right )}{\sqrt{b^2-4 a c}}\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 c \left (b^2-4 a c\right )^2 \left (b+\sqrt{b^2-4 a c}\right ) n^2}\\ \end{align*}
Mathematica [B] time = 7.75207, size = 9060, normalized size = 5.31 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.104, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d+e{x}^{n} \right ) ^{3}}{ \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^{3} x^{3 \, n} + 3 \, d e^{2} x^{2 \, n} + 3 \, d^{2} e x^{n} + d^{3}}{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{{\left (e x^{n} + d\right )}^{3}}{{\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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