Optimal. Leaf size=738 \[ -\frac{2 b c^2 e (2-n) x^{n+2} \, _2F_1\left (1,\frac{n+2}{n};2 \left (1+\frac{1}{n}\right );-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a n (n+2) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}+\frac{2 b c^2 e (2-n) x^{n+2} \, _2F_1\left (1,\frac{n+2}{n};2 \left (1+\frac{1}{n}\right );-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a n (n+2) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}-\frac{c d x \left (-b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c d x \left (b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{d x \left (-2 a c+b^2+b c x^n\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )}-\frac{c e x^2 \left (4 a c (1-n)-b^2 (2-n)\right ) \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c e x^2 \left (4 a c (1-n)-b^2 (2-n)\right ) \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{e x^2 \left (-2 a c+b^2+b c x^n\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )} \]
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Rubi [A] time = 1.33704, antiderivative size = 738, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {1796, 1345, 1422, 245, 1384, 1560, 1383, 364} \[ -\frac{2 b c^2 e (2-n) x^{n+2} \, _2F_1\left (1,\frac{n+2}{n};2 \left (1+\frac{1}{n}\right );-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a n (n+2) \left (b^2-4 a c\right )^{3/2} \left (b-\sqrt{b^2-4 a c}\right )}+\frac{2 b c^2 e (2-n) x^{n+2} \, _2F_1\left (1,\frac{n+2}{n};2 \left (1+\frac{1}{n}\right );-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a n (n+2) \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}-\frac{c d x \left (-b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c d x \left (b (1-n) \sqrt{b^2-4 a c}+4 a c (1-2 n)+b^2 (-(1-n))\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{d x \left (-2 a c+b^2+b c x^n\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )}-\frac{c e x^2 \left (4 a c (1-n)-b^2 (2-n)\right ) \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}-\frac{c e x^2 \left (4 a c (1-n)-b^2 (2-n)\right ) \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a n \left (b^2-4 a c\right ) \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{e x^2 \left (-2 a c+b^2+b c x^n\right )}{a n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )} \]
Antiderivative was successfully verified.
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Rule 1796
Rule 1345
Rule 1422
Rule 245
Rule 1384
Rule 1560
Rule 1383
Rule 364
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx &=\int \left (\frac{d}{\left (a+b x^n+c x^{2 n}\right )^2}+\frac{e x}{\left (a+b x^n+c x^{2 n}\right )^2}\right ) \, dx\\ &=d \int \frac{1}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx+e \int \frac{x}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx\\ &=\frac{d x \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{e x^2 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac{d \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 \left (b^2-4 a c\right ) n}-\frac{e \int \frac{x \left (-4 a c (1-n)+b^2 (2-n)+b c (2-n) x^n\right )}{a+b x^n+c x^{2 n}} \, dx}{a \left (b^2-4 a c\right ) n}\\ &=\frac{d x \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{e x^2 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}-\frac{e \int \left (-\frac{b^2 \left (1-\frac{4 a c (-1+n)}{b^2 (-2+n)}\right ) (-2+n) x}{a+b x^n+c x^{2 n}}-\frac{b c (-2+n) x^{1+n}}{a+b x^n+c x^{2 n}}\right ) \, dx}{a \left (b^2-4 a c\right ) n}+\frac{\left (c d \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 (c d \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{d x \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{e x^2 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{c d \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{c d \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 (e \left (4 a c (1-n)-b^2 (2-n)\right )\right ) \int \frac{x}{a+b x^n+c x^{2 n}} \, dx}{a \left (b^2-4 a c\right ) n}-\frac{(b c e (2-n)) \int \frac{x^{1+n}}{a+b x^n+c x^{2 n}} \, dx}{a \left (b^2-4 a c\right ) n}\\ &=\frac{d x \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{e x^2 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{c d \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{c d \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 (2 c e \left (4 a c (1-n)-b^2 (2-n)\right )\right ) \int \frac{x}{b-\sqrt{b^2-4 a c}+2 c x^n} \, dx}{a \left (b^2-4 a c\right )^{3/2} n}-\frac{\left (2 c e \left (4 a c (1-n)-b^2 (2-n)\right )\right ) \int \frac{x}{b+\sqrt{b^2-4 a c}+2 c x^n} \, dx}{a \left (b^2-4 a c\right )^{3/2} n}-\frac{\left (2 b c^2 e (2-n)\right ) \int \frac{x^{1+n}}{b-\sqrt{b^2-4 a c}+2 c x^n} \, dx}{a \left (b^2-4 a c\right )^{3/2} n}+\frac{\left (2 b c^2 e (2-n)\right ) \int \frac{x^{1+n}}{b+\sqrt{b^2-4 a c}+2 c x^n} \, dx}{a \left (b^2-4 a c\right )^{3/2} n}\\ &=\frac{d x \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{e x^2 \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) n \left (a+b x^n+c x^{2 n}\right )}+\frac{c d \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{c d \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{c e \left (4 a c (1-n)-b^2 (2-n)\right ) x^2 \, _2F_1\left (1,\frac{2}{n};\frac{2+n}{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{c e \left (4 a c (1-n)-b^2 (2-n)\right ) x^2 \, _2F_1\left (1,\frac{2}{n};\frac{2+n}{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{2 b c^2 e (2-n) x^{2+n} \, _2F_1\left (1,\frac{2+n}{n};2 \left (1+\frac{1}{n}\right );-\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 (2+n)}+\frac{2 b c^2 e (2-n) x^{2+n} \, _2F_1\left (1,\frac{2+n}{n};2 \left (1+\frac{1}{n}\right );-\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 (2+n)}\\ \end{align*}
Mathematica [B] time = 4.17041, size = 1878, normalized size = 2.54 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\frac{ex+d}{ \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{2}}}\, 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*} \frac{{\left (b^{2} e - 2 \, a c e\right )} x^{2} +{\left (b c e x^{2} + b c d x\right )} x^{n} +{\left (b^{2} d - 2 \, a c d\right )} x}{a^{2} b^{2} n - 4 \, a^{3} c n +{\left (a b^{2} c n - 4 \, a^{2} c^{2} n\right )} x^{2 \, n} +{\left (a b^{3} n - 4 \, a^{2} b c n\right )} x^{n}} - \int \frac{2 \, a c d{\left (2 \, n - 1\right )} - b^{2} d{\left (n - 1\right )} -{\left (b c e{\left (n - 2\right )} x + b c d{\left (n - 1\right )}\right )} x^{n} +{\left (4 \, a c e{\left (n - 1\right )} - b^{2} e{\left (n - 2\right )}\right )} x}{a^{2} b^{2} n - 4 \, a^{3} c n +{\left (a b^{2} c n - 4 \, a^{2} c^{2} n\right )} x^{2 \, n} +{\left (a b^{3} n - 4 \, a^{2} b c n\right )} x^{n}}\,{d x} \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 + d}{c^{2} x^{4 \, n} + b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} + 2 \,{\left (b c x^{n} + a 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 + d}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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