Optimal. Leaf size=243 \[ -\frac{c x \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{\left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{c x \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\left (\sqrt{b^2-4 a c}+b\right ) \left (a e^2-b d e+c d^2\right )}+\frac{e^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 0.828948, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{c x \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{\left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{c x \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\left (\sqrt{b^2-4 a c}+b\right ) \left (a e^2-b d e+c d^2\right )}+\frac{e^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x^n)*(a + b*x^n + c*x^(2*n))),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x^{n}\right ) \left (a + b x^{n} + c x^{2 n}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(d+e*x**n)/(a+b*x**n+c*x**(2*n)),x)
[Out]
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Mathematica [A] time = 2.40775, size = 379, normalized size = 1.56 \[ \frac{x \left (\frac{2^{-1/n} \left (-c d \sqrt{b^2-4 a c}+b e \sqrt{b^2-4 a c}-2 a c e+b^2 e-b c d\right ) \left (\frac{c x^n}{-\sqrt{b^2-4 a c}+b+2 c x^n}\right )^{-1/n} \, _2F_1\left (-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b-\sqrt{b^2-4 a c}}{2 c x^n+b-\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}+\frac{2^{-1/n} \left (-c d \sqrt{b^2-4 a c}+b e \sqrt{b^2-4 a c}+2 a c e+b^2 (-e)+b c d\right ) \left (\frac{c x^n}{\sqrt{b^2-4 a c}+b+2 c x^n}\right )^{-1/n} \, _2F_1\left (-\frac{1}{n},-\frac{1}{n};\frac{n-1}{n};\frac{b+\sqrt{b^2-4 a c}}{2 c x^n+b+\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}+\frac{2 a e^2 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d}-2 b e+2 c d\right )}{2 a \left (e (a e-b d)+c d^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x^n)*(a + b*x^n + c*x^(2*n))),x]
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Maple [F] time = 0.084, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d+e{x}^{n} \right ) \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(d+e*x^n)/(a+b*x^n+c*x^(2*n)),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}{\left (e x^{n} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^(2*n) + b*x^n + a)*(e*x^n + d)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{a d +{\left (c e x^{n} + c d + b e\right )} x^{2 \, n} +{\left (b d + a e\right )} x^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^(2*n) + b*x^n + a)*(e*x^n + d)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(d+e*x**n)/(a+b*x**n+c*x**(2*n)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2 \, n} + b x^{n} + a\right )}{\left (e x^{n} + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^(2*n) + b*x^n + a)*(e*x^n + d)),x, algorithm="giac")
[Out]