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

Optimal. Leaf size=152 \[ -\frac{c e x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )}+\frac{c d x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )}+\frac{e^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d \left (a e^2+c d^2\right )} \]

[Out]

(c*d*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^
2 + a*e^2)) + (e^2*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*
(c*d^2 + a*e^2)) - (c*e*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1
))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)*(1 + n))

_______________________________________________________________________________________

Rubi [A]  time = 0.224354, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{c e x^{n+1} \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a (n+1) \left (a e^2+c d^2\right )}+\frac{c d x \, _2F_1\left (1,\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )}{a \left (a e^2+c d^2\right )}+\frac{e^2 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )}{d \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x^n)*(a + c*x^(2*n))),x]

[Out]

(c*d*x*Hypergeometric2F1[1, 1/(2*n), (2 + n^(-1))/2, -((c*x^(2*n))/a)])/(a*(c*d^
2 + a*e^2)) + (e^2*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)])/(d*
(c*d^2 + a*e^2)) - (c*e*x^(1 + n)*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1
))/2, -((c*x^(2*n))/a)])/(a*(c*d^2 + a*e^2)*(1 + n))

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{2 n}\right ) \left (d + e x^{n}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d+e*x**n)/(a+c*x**(2*n)),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.153087, size = 131, normalized size = 0.86 \[ \frac{x \left (c d^2 (n+1) \, _2F_1\left (1,\frac{1}{2 n};1+\frac{1}{2 n};-\frac{c x^{2 n}}{a}\right )+e \left (a e (n+1) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{e x^n}{d}\right )-c d x^n \, _2F_1\left (1,\frac{n+1}{2 n};\frac{1}{2} \left (3+\frac{1}{n}\right );-\frac{c x^{2 n}}{a}\right )\right )\right )}{a d (n+1) \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x^n)*(a + c*x^(2*n))),x]

[Out]

(x*(c*d^2*(1 + n)*Hypergeometric2F1[1, 1/(2*n), 1 + 1/(2*n), -((c*x^(2*n))/a)] +
e*(a*e*(1 + n)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((e*x^n)/d)] - c*d*x^n
*Hypergeometric2F1[1, (1 + n)/(2*n), (3 + n^(-1))/2, -((c*x^(2*n))/a)])))/(a*d*(
c*d^2 + a*e^2)*(1 + n))

_______________________________________________________________________________________

Maple [F]  time = 0.102, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( d+e{x}^{n} \right ) \left ( a+c{x}^{2\,n} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2 \, n} + a\right )}{\left (e x^{n} + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^(2*n) + a)*(e*x^n + d)),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{a e x^{n} + a d +{\left (c e x^{n} + c d\right )} x^{2 \, n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^(2*n) + a)*(e*x^n + d)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

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+c*x**(2*n)),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2 \, n} + a\right )}{\left (e x^{n} + d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^(2*n) + a)*(e*x^n + d)),x, algorithm="giac")

[Out]

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