3.285 \(\int \frac{\left (c+d x^{2 n}\right )^p}{\left (a-b x^n\right ) \left (a+b x^n\right )} \, dx\)

Optimal. Leaf size=76 \[ \frac{x \left (c+d x^{2 n}\right )^p \left (\frac{d x^{2 n}}{c}+1\right )^{-p} F_1\left (\frac{1}{2 n};1,-p;\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2},-\frac{d x^{2 n}}{c}\right )}{a^2} \]

[Out]

(x*(c + d*x^(2*n))^p*AppellF1[1/(2*n), 1, -p, (2 + n^(-1))/2, (b^2*x^(2*n))/a^2,
-((d*x^(2*n))/c)])/(a^2*(1 + (d*x^(2*n))/c)^p)

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Rubi [A]  time = 0.168164, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{x \left (c+d x^{2 n}\right )^p \left (\frac{d x^{2 n}}{c}+1\right )^{-p} F_1\left (\frac{1}{2 n};1,-p;\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2},-\frac{d x^{2 n}}{c}\right )}{a^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^(2*n))^p/((a - b*x^n)*(a + b*x^n)),x]

[Out]

(x*(c + d*x^(2*n))^p*AppellF1[1/(2*n), 1, -p, (2 + n^(-1))/2, (b^2*x^(2*n))/a^2,
-((d*x^(2*n))/c)])/(a^2*(1 + (d*x^(2*n))/c)^p)

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Rubi in Sympy [A]  time = 35.3518, size = 58, normalized size = 0.76 \[ \frac{x \left (1 + \frac{d x^{2 n}}{c}\right )^{- p} \left (c + d x^{2 n}\right )^{p} \operatorname{appellf_{1}}{\left (\frac{1}{2 n},1,- p,\frac{n + \frac{1}{2}}{n},\frac{b^{2} x^{2 n}}{a^{2}},- \frac{d x^{2 n}}{c} \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c+d*x**(2*n))**p/(a-b*x**n)/(a+b*x**n),x)

[Out]

x*(1 + d*x**(2*n)/c)**(-p)*(c + d*x**(2*n))**p*appellf1(1/(2*n), 1, -p, (n + 1/2
)/n, b**2*x**(2*n)/a**2, -d*x**(2*n)/c)/a**2

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Mathematica [B]  time = 0.492794, size = 258, normalized size = 3.39 \[ \frac{a^2 c (2 n+1) x \left (c+d x^{2 n}\right )^p F_1\left (\frac{1}{2 n};-p,1;1+\frac{1}{2 n};-\frac{d x^{2 n}}{c},\frac{b^2 x^{2 n}}{a^2}\right )}{\left (a^2-b^2 x^{2 n}\right ) \left (2 a^2 d n p x^{2 n} F_1\left (1+\frac{1}{2 n};1-p,1;2+\frac{1}{2 n};-\frac{d x^{2 n}}{c},\frac{b^2 x^{2 n}}{a^2}\right )+2 b^2 c n x^{2 n} F_1\left (1+\frac{1}{2 n};-p,2;2+\frac{1}{2 n};-\frac{d x^{2 n}}{c},\frac{b^2 x^{2 n}}{a^2}\right )+a^2 c (2 n+1) F_1\left (\frac{1}{2 n};-p,1;1+\frac{1}{2 n};-\frac{d x^{2 n}}{c},\frac{b^2 x^{2 n}}{a^2}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(c + d*x^(2*n))^p/((a - b*x^n)*(a + b*x^n)),x]

[Out]

(a^2*c*(1 + 2*n)*x*(c + d*x^(2*n))^p*AppellF1[1/(2*n), -p, 1, 1 + 1/(2*n), -((d*
x^(2*n))/c), (b^2*x^(2*n))/a^2])/((a^2 - b^2*x^(2*n))*(2*a^2*d*n*p*x^(2*n)*Appel
lF1[1 + 1/(2*n), 1 - p, 1, 2 + 1/(2*n), -((d*x^(2*n))/c), (b^2*x^(2*n))/a^2] + 2
*b^2*c*n*x^(2*n)*AppellF1[1 + 1/(2*n), -p, 2, 2 + 1/(2*n), -((d*x^(2*n))/c), (b^
2*x^(2*n))/a^2] + a^2*c*(1 + 2*n)*AppellF1[1/(2*n), -p, 1, 1 + 1/(2*n), -((d*x^(
2*n))/c), (b^2*x^(2*n))/a^2]))

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Maple [F]  time = 0.139, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c+d{x}^{2\,n} \right ) ^{p}}{ \left ( a-b{x}^{n} \right ) \left ( a+b{x}^{n} \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*x^(2*n))^p/(a-b*x^n)/(a+b*x^n),x)

[Out]

int((c+d*x^(2*n))^p/(a-b*x^n)/(a+b*x^n),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate((d*x^(2*n) + c)^p/((b*x^n + a)*(b*x^n - a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(d*x^(2*n) + c)^p/(b^2*x^(2*n) - a^2), 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((c+d*x**(2*n))**p/(a-b*x**n)/(a+b*x**n),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(-(d*x^(2*n) + c)^p/((b*x^n + a)*(b*x^n - a)), x)