3.72 \(\int \frac{1}{\left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )} \, dx\)

Optimal. Leaf size=243 \[ -\frac{c x \left (2 c d-e \left (\sqrt{b^2-4 a c}+b\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )}{\left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{c x \left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{\left (\sqrt{b^2-4 a c}+b\right ) \left (a e^2-b d e+c d^2\right )}+\frac{e^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d \left (a e^2-b d e+c d^2\right )} \]

[Out]

-((c*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-
1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c
*d^2 - b*d*e + a*e^2))) - (c*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*x*Hypergeomet
ric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^
2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)) + (e^2*x*Hypergeometric2F1[1, n^(-1), 1 + n
^(-1), -((e*x^n)/d)])/(d*(c*d^2 - b*d*e + a*e^2))

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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]

-((c*(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-
1), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])])/((b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c])*(c
*d^2 - b*d*e + a*e^2))) - (c*(e + (2*c*d - b*e)/Sqrt[b^2 - 4*a*c])*x*Hypergeomet
ric2F1[1, n^(-1), 1 + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c])])/((b + Sqrt[b^
2 - 4*a*c])*(c*d^2 - b*d*e + a*e^2)) + (e^2*x*Hypergeometric2F1[1, n^(-1), 1 + n
^(-1), -((e*x^n)/d)])/(d*(c*d^2 - b*d*e + a*e^2))

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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]

Integral(1/((d + e*x**n)*(a + b*x**n + c*x**(2*n))), x)

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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]

[Out]

(x*(2*c*d - 2*b*e + (2*a*e^2*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/
d)])/d + ((-(b*c*d) - c*Sqrt[b^2 - 4*a*c]*d + b^2*e - 2*a*c*e + b*Sqrt[b^2 - 4*a
*c]*e)*Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n, (b - Sqrt[b^2 - 4*a*c])/(
b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)])/(2^n^(-1)*Sqrt[b^2 - 4*a*c]*((c*x^n)/(b - Sqr
t[b^2 - 4*a*c] + 2*c*x^n))^n^(-1)) + ((b*c*d - c*Sqrt[b^2 - 4*a*c]*d - b^2*e + 2
*a*c*e + b*Sqrt[b^2 - 4*a*c]*e)*Hypergeometric2F1[-n^(-1), -n^(-1), (-1 + n)/n,
(b + Sqrt[b^2 - 4*a*c])/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)])/(2^n^(-1)*Sqrt[b^2 -
4*a*c]*((c*x^n)/(b + Sqrt[b^2 - 4*a*c] + 2*c*x^n))^n^(-1))))/(2*a*(c*d^2 + e*(-
(b*d) + a*e)))

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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]

int(1/(d+e*x^n)/(a+b*x^n+c*x^(2*n)),x)

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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]

integrate(1/((c*x^(2*n) + b*x^n + a)*(e*x^n + d)), x)

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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]

integral(1/(a*d + (c*e*x^n + c*d + b*e)*x^(2*n) + (b*d + a*e)*x^n), x)

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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]

Timed 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]

integrate(1/((c*x^(2*n) + b*x^n + a)*(e*x^n + d)), x)