Optimal. Leaf size=328 \[ \frac{c (d x)^{m+1} \left (\frac{4 a c (m-2 n+1)-b^2 (m-n+1)}{\sqrt{b^2-4 a c}}-b (m-n+1)\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a d (m+1) n \left (b^2-4 a c\right ) \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (d x)^{m+1} \left (b (m-n+1) \sqrt{b^2-4 a c}+4 a c (m-2 n+1)+b^2 (-(m-n+1))\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a d (m+1) n \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(d x)^{m+1} \left (-2 a c+b^2+b c x^n\right )}{a d n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )} \]
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Rubi [A] time = 0.961283, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1384, 1560, 364} \[ \frac{c (d x)^{m+1} \left (\frac{4 a c (m-2 n+1)-b^2 (m-n+1)}{\sqrt{b^2-4 a c}}-b (m-n+1)\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{a d (m+1) n \left (b^2-4 a c\right ) \left (b-\sqrt{b^2-4 a c}\right )}-\frac{c (d x)^{m+1} \left (b (m-n+1) \sqrt{b^2-4 a c}+4 a c (m-2 n+1)+b^2 (-(m-n+1))\right ) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{a d (m+1) n \left (b^2-4 a c\right )^{3/2} \left (\sqrt{b^2-4 a c}+b\right )}+\frac{(d x)^{m+1} \left (-2 a c+b^2+b c x^n\right )}{a d n \left (b^2-4 a c\right ) \left (a+b x^n+c x^{2 n}\right )} \]
Antiderivative was successfully verified.
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Rule 1384
Rule 1560
Rule 364
Rubi steps
\begin{align*} \int \frac{(d x)^m}{\left (a+b x^n+c x^{2 n}\right )^2} \, dx &=\frac{(d x)^{1+m} \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) d n \left (a+b x^n+c x^{2 n}\right )}-\frac{\int \frac{(d x)^m \left (-2 a c (1+m-2 n)+b^2 (1+m-n)+b c (1+m-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)^{1+m} \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) d n \left (a+b x^n+c x^{2 n}\right )}-\frac{\int \left (\frac{\left (b c (1+m-n)+\frac{c \left (b^2-4 a c+b^2 m-4 a c m-b^2 n+8 a c n\right )}{\sqrt{b^2-4 a c}}\right ) (d x)^m}{b-\sqrt{b^2-4 a c}+2 c x^n}+\frac{\left (b c (1+m-n)-\frac{c \left (b^2-4 a c+b^2 m-4 a c m-b^2 n+8 a c n\right )}{\sqrt{b^2-4 a c}}\right ) (d x)^m}{b+\sqrt{b^2-4 a c}+2 c x^n}\right ) \, dx}{a \left (b^2-4 a c\right ) n}\\ &=\frac{(d x)^{1+m} \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) d n \left (a+b x^n+c x^{2 n}\right )}+\frac{\left (c \left (4 a c (1+m-2 n)-b^2 (1+m-n)-b \sqrt{b^2-4 a c} (1+m-n)\right )\right ) \int \frac{(d x)^m}{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 (c \left (4 a c (1+m-2 n)-b^2 (1+m-n)+b \sqrt{b^2-4 a c} (1+m-n)\right )\right ) \int \frac{(d x)^m}{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)^{1+m} \left (b^2-2 a c+b c x^n\right )}{a \left (b^2-4 a c\right ) d n \left (a+b x^n+c x^{2 n}\right )}+\frac{c \left (4 a c (1+m-2 n)-b^2 (1+m-n)-b \sqrt{b^2-4 a c} (1+m-n)\right ) (d x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+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 ) d (1+m) n}-\frac{c \left (4 a c (1+m-2 n)-b^2 (1+m-n)+b \sqrt{b^2-4 a c} (1+m-n)\right ) (d x)^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+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 ) d (1+m) n}\\ \end{align*}
Mathematica [B] time = 2.28882, size = 1511, normalized size = 4.61 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{ \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{b c d^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )} +{\left (b^{2} d^{m} - 2 \, a c d^{m}\right )} x x^{m}}{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{b c d^{m}{\left (m - n + 1\right )} e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )} +{\left (b^{2} d^{m}{\left (m - n + 1\right )} - 2 \, a c d^{m}{\left (m - 2 \, n + 1\right )}\right )} x^{m}}{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{\left (d x\right )^{m}}{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{\left (d x\right )^{m}}{{\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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