Optimal. Leaf size=1191 \[ \text{result too large to display} \]
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Rubi [A] time = 4.00994, antiderivative size = 1191, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {1436, 1430, 1422, 245, 1345} \[ -\frac{\left (-(1-n) b^2-\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 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 ) e^2}{a \left (b^2-4 a c\right ) \left (b^2-\sqrt{b^2-4 a c} b-4 a c\right ) n}-\frac{\left (-(1-n) b^2+\sqrt{b^2-4 a c} (1-n) b+4 a c (1-2 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 ) e^2}{a \left (b^2-4 a c\right ) \left (b^2+\sqrt{b^2-4 a c} b-4 a c\right ) n}+\frac{x \left (b c x^n+b^2-2 a c\right ) e^2}{a c \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )}-\frac{\left ((1-n) \left (-\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right )+\frac{-(1-n) \left (c (1-2 n) d^2+2 a e^2 n\right ) b^4+2 a c d e (1-n) b^3+2 a c \left (3 c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (15 n^2-10 n+1\right )\right ) b^2-8 a^2 c^2 d e \left (-3 n^2-n+1\right ) b-8 a^2 c^2 \left (c d^2-a e^2\right ) \left (8 n^2-6 n+1\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 \left (b^2-4 a c\right )^2 \left (b-\sqrt{b^2-4 a c}\right ) n^2}-\frac{\left ((1-n) \left (-\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right )-\frac{-(1-n) \left (c (1-2 n) d^2+2 a e^2 n\right ) b^4+2 a c d e (1-n) b^3+2 a c \left (3 c d^2 \left (3 n^2-4 n+1\right )-a e^2 \left (15 n^2-10 n+1\right )\right ) b^2-8 a^2 c^2 d e \left (-3 n^2-n+1\right ) b-8 a^2 c^2 \left (c d^2-a e^2\right ) \left (8 n^2-6 n+1\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 \left (b^2-4 a c\right )^2 \left (b+\sqrt{b^2-4 a c}\right ) n^2}+\frac{x \left (c \left (-\left (c (1-2 n) d^2+2 a e^2 n\right ) b^3+2 a c d e b^2+2 a c \left (c (2-7 n) d^2+a e^2 n\right ) b-8 a^2 c^2 d e (1-3 n)\right ) x^n+2 a b^3 c d e-a b^2 c \left (a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )-4 a^2 c^2 \left (c d^2-a e^2\right ) (1-4 n)-4 a^2 b c^2 d e (2-3 n)-b^4 \left (c (1-2 n) d^2+2 a e^2 n\right )\right )}{2 a^2 c \left (b^2-4 a c\right )^2 n^2 \left (b x^n+c x^{2 n}+a\right )}+\frac{x \left (\left (b c d^2-4 a c e d+a b e^2\right ) x^n+b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )\right )}{2 a \left (b^2-4 a c\right ) n \left (b x^n+c x^{2 n}+a\right )^2} \]
Antiderivative was successfully verified.
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Rule 1436
Rule 1430
Rule 1422
Rule 245
Rule 1345
Rubi steps
\begin{align*} \int \frac{\left (d+e x^n\right )^2}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx &=\int \left (\frac{c d^2-a e^2+\left (2 c d e-b e^2\right ) x^n}{c \left (a+b x^n+c x^{2 n}\right )^3}+\frac{e^2}{c \left (a+b x^n+c x^{2 n}\right )^2}\right ) \, dx\\ &=\frac{\int \frac{c d^2-a e^2+\left (2 c d e-b e^2\right ) x^n}{\left (a+b x^n+c x^{2 n}\right )^3} \, dx}{c}+\frac{e^2 \int \frac{1}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{c}\\ &=\frac{x \left (b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )+\left (b c d^2-4 a c d e+a b e^2\right ) 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{e^2 x \left (b^2-2 a c+b c x^n\right )}{a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac{\int \frac{-2 a b c d e-2 a c \left (c d^2-a e^2\right ) (1-4 n)+b^2 \left (c d^2 (1-2 n)+2 a e^2 n\right )+c \left (b c d^2-4 a c d e+a b e^2\right ) (1-3 n) x^n}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx}{2 a c \left (b^2-4 a c\right ) n}-\frac{e^2 \int \frac{b^2-2 a c-\left (b^2-4 a c\right ) n+b c (1-n) x^n}{a+b x^n+c x^{2 n}} \, dx}{a c \left (b^2-4 a c\right ) n}\\ &=\frac{x \left (b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )+\left (b c d^2-4 a c d e+a b e^2\right ) 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{e^2 x \left (b^2-2 a c+b c x^n\right )}{a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{x \left (2 a b^3 c d e-a b^2 c \left (a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )-4 a^2 c^2 \left (c d^2-a e^2\right ) (1-4 n)-4 a^2 b c^2 d e (2-3 n)-b^4 \left (c d^2 (1-2 n)+2 a e^2 n\right )+c \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right ) x^n\right )}{2 a^2 c \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac{\int \frac{4 a^2 b c^2 d e (2-5 n)-2 a b^3 c d e (1-n)+b^4 (1-n) \left (c d^2 (1-2 n)+2 a e^2 n\right )+4 a^2 c^2 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )-a b^2 c \left (c d^2 \left (5-21 n+16 n^2\right )-a e^2 \left (1-11 n+16 n^2\right )\right )-c (1-n) \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right ) x^n}{a+b x^n+c x^{2 n}} \, dx}{2 a^2 c \left (b^2-4 a c\right )^2 n^2}+\frac{\left (e^2 \left (4 a c (1-2 n)-b^2 (1-n)-b \sqrt{b^2-4 a c} (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 \left (b^2-4 a c\right )^{3/2} n}-\frac{\left (e^2 \left (4 a c (1-2 n)-b^2 (1-n)+b \sqrt{b^2-4 a c} (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 \left (b^2-4 a c\right )^{3/2} n}\\ &=\frac{x \left (b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )+\left (b c d^2-4 a c d e+a b e^2\right ) 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{e^2 x \left (b^2-2 a c+b c x^n\right )}{a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{x \left (2 a b^3 c d e-a b^2 c \left (a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )-4 a^2 c^2 \left (c d^2-a e^2\right ) (1-4 n)-4 a^2 b c^2 d e (2-3 n)-b^4 \left (c d^2 (1-2 n)+2 a e^2 n\right )+c \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right ) x^n\right )}{2 a^2 c \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac{e^2 \left (4 a c (1-2 n)-b^2 (1-n)-b \sqrt{b^2-4 a c} (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 \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right ) n}-\frac{e^2 \left (4 a c (1-2 n)-b^2 (1-n)+b \sqrt{b^2-4 a c} (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 \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt{b^2-4 a c}\right ) n}-\frac{\left ((1-n) \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right )-\frac{2 a b^3 c d e (1-n)-b^4 (1-n) \left (c d^2 (1-2 n)+2 a e^2 n\right )-8 a^2 b c^2 d e \left (1-n-3 n^2\right )-8 a^2 c^2 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )+2 a b^2 c \left (3 c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 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 \left (b^2-4 a c\right )^2 n^2}-\frac{\left ((1-n) \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right )+\frac{2 a b^3 c d e (1-n)-b^4 (1-n) \left (c d^2 (1-2 n)+2 a e^2 n\right )-8 a^2 b c^2 d e \left (1-n-3 n^2\right )-8 a^2 c^2 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )+2 a b^2 c \left (3 c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 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 \left (b^2-4 a c\right )^2 n^2}\\ &=\frac{x \left (b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )+\left (b c d^2-4 a c d e+a b e^2\right ) 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{e^2 x \left (b^2-2 a c+b c x^n\right )}{a c \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{x \left (2 a b^3 c d e-a b^2 c \left (a e^2 (1-9 n)-5 c d^2 (1-3 n)\right )-4 a^2 c^2 \left (c d^2-a e^2\right ) (1-4 n)-4 a^2 b c^2 d e (2-3 n)-b^4 \left (c d^2 (1-2 n)+2 a e^2 n\right )+c \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right ) x^n\right )}{2 a^2 c \left (b^2-4 a c\right )^2 n^2 \left (a+b x^n+c x^{2 n}\right )}+\frac{e^2 \left (4 a c (1-2 n)-b^2 (1-n)-b \sqrt{b^2-4 a c} (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 \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right ) n}-\frac{\left ((1-n) \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right )+\frac{2 a b^3 c d e (1-n)-b^4 (1-n) \left (c d^2 (1-2 n)+2 a e^2 n\right )-8 a^2 b c^2 d e \left (1-n-3 n^2\right )-8 a^2 c^2 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )+2 a b^2 c \left (3 c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 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 \left (b^2-4 a c\right )^2 \left (b-\sqrt{b^2-4 a c}\right ) n^2}-\frac{e^2 \left (4 a c (1-2 n)-b^2 (1-n)+b \sqrt{b^2-4 a c} (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 \left (b^2-4 a c\right )^{3/2} \left (b+\sqrt{b^2-4 a c}\right ) n}-\frac{\left ((1-n) \left (2 a b^2 c d e-8 a^2 c^2 d e (1-3 n)+2 a b c \left (c d^2 (2-7 n)+a e^2 n\right )-b^3 \left (c d^2 (1-2 n)+2 a e^2 n\right )\right )-\frac{2 a b^3 c d e (1-n)-b^4 (1-n) \left (c d^2 (1-2 n)+2 a e^2 n\right )-8 a^2 b c^2 d e \left (1-n-3 n^2\right )-8 a^2 c^2 \left (c d^2-a e^2\right ) \left (1-6 n+8 n^2\right )+2 a b^2 c \left (3 c d^2 \left (1-4 n+3 n^2\right )-a e^2 \left (1-10 n+15 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 \left (b^2-4 a c\right )^2 \left (b+\sqrt{b^2-4 a c}\right ) n^2}\\ \end{align*}
Mathematica [B] time = 6.83908, size = 7402, normalized size = 6.21 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.1, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d+e{x}^{n} \right ) ^{2}}{ \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^{2} x^{2 \, n} + 2 \, d e x^{n} + d^{2}}{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 )}^{2}}{{\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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