Optimal. Leaf size=368 \[ -\frac {c x \left (-2 c e \left (d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c^2 d^2\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\left (-b \sqrt {b^2-4 a c}-4 a c+b^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac {c x \left (-2 c e \left (-d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+2 c^2 d^2\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\left (b \sqrt {b^2-4 a c}-4 a c+b^2\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac {e^2 x (2 c d-b e) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d \left (a e^2-b d e+c d^2\right )^2}+\frac {e^2 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 0.71, antiderivative size = 368, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1424, 245, 1422} \[ -\frac {c x \left (-2 c e \left (d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c^2 d^2\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )}{\left (-b \sqrt {b^2-4 a c}-4 a c+b^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac {c x \left (-2 c e \left (-d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+2 c^2 d^2\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{\left (b \sqrt {b^2-4 a c}-4 a c+b^2\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac {e^2 x (2 c d-b e) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d \left (a e^2-b d e+c d^2\right )^2}+\frac {e^2 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 245
Rule 1422
Rule 1424
Rubi steps
\begin {align*} \int \frac {1}{\left (d+e x^n\right )^2 \left (a+b x^n+c x^{2 n}\right )} \, dx &=\int \left (\frac {e^2}{\left (c d^2-b d e+a e^2\right ) \left (d+e x^n\right )^2}-\frac {e^2 (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right )^2 \left (d+e x^n\right )}+\frac {c^2 d^2-2 b c d e+b^2 e^2-a c e^2-\left (2 c^2 d e-b c e^2\right ) x^n}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x^n+c x^{2 n}\right )}\right ) \, dx\\ &=\frac {\int \frac {c^2 d^2-2 b c d e+b^2 e^2-a c e^2-\left (2 c^2 d e-b c e^2\right ) x^n}{a+b x^n+c x^{2 n}} \, dx}{\left (c d^2-b d e+a e^2\right )^2}+\frac {\left (e^2 (2 c d-b e)\right ) \int \frac {1}{d+e x^n} \, dx}{\left (c d^2-b d e+a e^2\right )^2}+\frac {e^2 \int \frac {1}{\left (d+e x^n\right )^2} \, dx}{c d^2-b d e+a e^2}\\ &=\frac {e^2 (2 c d-b e) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d \left (c d^2-b d e+a e^2\right )^2}+\frac {e^2 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (c d^2-b d e+a e^2\right )}-\frac {\left (c \left (2 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right )\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{2 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2}+\frac {\left (c \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right )\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^n} \, dx}{2 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2}\\ &=\frac {c \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\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 )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2}-\frac {c \left (2 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\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 )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )^2}+\frac {e^2 (2 c d-b e) x \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d \left (c d^2-b d e+a e^2\right )^2}+\frac {e^2 x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2 \left (c d^2-b d e+a e^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.89, size = 327, normalized size = 0.89 \[ \frac {x \left (\frac {c \left (-2 c e \left (d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c^2 d^2\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};\frac {2 c x^n}{\sqrt {b^2-4 a c}-b}\right )}{b \sqrt {b^2-4 a c}+4 a c-b^2}+\frac {c \left (2 c e \left (-d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}-b\right )-2 c^2 d^2\right ) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{b \sqrt {b^2-4 a c}-4 a c+b^2}+\frac {e^2 \left (e (a e-b d)+c d^2\right ) \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d^2}+\frac {e^2 (2 c d-b e) \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {e x^n}{d}\right )}{d}\right )}{\left (e (a e-b d)+c d^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b e^{2} x^{3 \, n} + a d^{2} + {\left (c e^{2} x^{2 \, n} + 2 \, c d e x^{n} + c d^{2}\right )} x^{2 \, n} + {\left (2 \, b d e + a e^{2}\right )} x^{2 \, n} + {\left (b d^{2} + 2 \, a d e\right )} x^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (c x^{2 \, n} + b x^{n} + a\right )} {\left (e x^{n} + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \,x^{n}+d \right )^{2} \left (b \,x^{n}+c \,x^{2 n}+a \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {e^{2} x}{c d^{4} n - b d^{3} e n + a d^{2} e^{2} n + {\left (c d^{3} e n - b d^{2} e^{2} n + a d e^{3} n\right )} x^{n}} + {\left (c d^{2} e^{2} {\left (3 \, n - 1\right )} - b d e^{3} {\left (2 \, n - 1\right )} + a e^{4} {\left (n - 1\right )}\right )} \int \frac {1}{c^{2} d^{6} n - 2 \, b c d^{5} e n + b^{2} d^{4} e^{2} n + a^{2} d^{2} e^{4} n + 2 \, {\left (c d^{4} e^{2} n - b d^{3} e^{3} n\right )} a + {\left (c^{2} d^{5} e n - 2 \, b c d^{4} e^{2} n + b^{2} d^{3} e^{3} n + a^{2} d e^{5} n + 2 \, {\left (c d^{3} e^{3} n - b d^{2} e^{4} n\right )} a\right )} x^{n}}\,{d x} + \int \frac {c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - a c e^{2} - {\left (2 \, c^{2} d e - b c e^{2}\right )} x^{n}}{a^{3} e^{4} + 2 \, {\left (c d^{2} e^{2} - b d e^{3}\right )} a^{2} + {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} a + {\left (c^{3} d^{4} - 2 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2} + a^{2} c e^{4} + 2 \, {\left (c^{2} d^{2} e^{2} - b c d e^{3}\right )} a\right )} x^{2 \, n} + {\left (b c^{2} d^{4} - 2 \, b^{2} c d^{3} e + b^{3} d^{2} e^{2} + a^{2} b e^{4} + 2 \, {\left (b c d^{2} e^{2} - b^{2} d e^{3}\right )} a\right )} x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (d+e\,x^n\right )}^2\,\left (a+b\,x^n+c\,x^{2\,n}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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