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

Optimal. Leaf size=193 \[ \frac{x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{-\frac{1}{n}-p-1} (n (p+1) (b c-a d)+b c) \left (\frac{c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p-1} \, _2F_1\left (\frac{1}{n},-p-1;1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a c n (p+1) (b c-a d)}-\frac{b x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{-\frac{1}{n}-p-1}}{a n (p+1) (b c-a d)} \]

[Out]

-((b*x*(a + b*x^n)^(1 + p)*(c + d*x^n)^(-1 - n^(-1) - p))/(a*(b*c - a*d)*n*(1 +
p))) + ((b*c + (b*c - a*d)*n*(1 + p))*x*(a + b*x^n)^(1 + p)*((c*(a + b*x^n))/(a*
(c + d*x^n)))^(-1 - p)*(c + d*x^n)^(-1 - n^(-1) - p)*Hypergeometric2F1[n^(-1), -
1 - p, 1 + n^(-1), -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(a*c*(b*c - a*d)*n*(1
+ p))

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Rubi [A]  time = 0.206221, antiderivative size = 179, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{-\frac{1}{n}-p-1} \left (\frac{b}{n (p+1) (b c-a d)}+\frac{1}{c}\right ) \left (\frac{c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p-1} \, _2F_1\left (\frac{1}{n},-p-1;1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a}-\frac{b x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{-\frac{1}{n}-p-1}}{a n (p+1) (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-((b*x*(a + b*x^n)^(1 + p)*(c + d*x^n)^(-1 - n^(-1) - p))/(a*(b*c - a*d)*n*(1 +
p))) + ((c^(-1) + b/((b*c - a*d)*n*(1 + p)))*x*(a + b*x^n)^(1 + p)*((c*(a + b*x^
n))/(a*(c + d*x^n)))^(-1 - p)*(c + d*x^n)^(-1 - n^(-1) - p)*Hypergeometric2F1[n^
(-1), -1 - p, 1 + n^(-1), -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/a

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Rubi in Sympy [A]  time = 25.2427, size = 155, normalized size = 0.8 \[ \frac{b x \left (a + b x^{n}\right )^{p + 1} \left (c + d x^{n}\right )^{- p - 1 - \frac{1}{n}}}{a n \left (p + 1\right ) \left (a d - b c\right )} - \frac{x \left (\frac{a \left (c + d x^{n}\right )}{c \left (a + b x^{n}\right )}\right )^{p + 2 + \frac{1}{n}} \left (a + b x^{n}\right )^{p + 2} \left (c + d x^{n}\right )^{- p - 2 - \frac{1}{n}} \left (b c - n \left (p + 1\right ) \left (a d - b c\right )\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, p + 2 + \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{x^{n} \left (a d - b c\right )}{c \left (a + b x^{n}\right )}} \right )}}{a^{2} n \left (p + 1\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 53.2575, size = 1414, normalized size = 7.33 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

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

[Out]

(c^4*(1 + n)*(1 + 2*n)*(1 + 3*n)*x*(a + b*x^n)^(3 + p)*(c + d*x^n)^(-2 - n^(-1)
- p)*(1 + (d*x^n)/c)*Gamma[2 + n^(-1)]*Gamma[-p]*(Hypergeometric2F1[1, -p, 1 + n
^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + (d*n*x^n*((c*Hypergeometric2F1[1, -p
, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])/(1 + n) + ((b*c - a*d)*x^n*Gam
ma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)
*x^n)/(c*(a + b*x^n))])/((1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p])))/c^
2))/(-(c*d*(1 + 3*n)*(1 + n + n*p)*x^n*(a + b*x^n)^2*(c^2*(1 + n)*(1 + 2*n)*(a +
b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 1 + n^(-1), ((b*c -
a*d)*x^n)/(c*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]
*Gamma[-p]*Hypergeometric2F1[1, -p, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n)
)] + (b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[2,
1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]))) + b*c*n*(1 + 3*n)*p*x^
n*(a + b*x^n)*(c + d*x^n)*(c^2*(1 + n)*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*G
amma[-p]*Hypergeometric2F1[1, -p, 1 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]
+ d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F
1[1, -p, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + (b*c - a*d)*(1 + n)*x^
n*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c -
a*d)*x^n)/(c*(a + b*x^n))])) + c*(1 + 3*n)*(a + b*x^n)^2*(c + d*x^n)*(c^2*(1 +
n)*(1 + 2*n)*(a + b*x^n)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 1
+ n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + d*n*x^n*(c*(1 + 2*n)*(a + b*x^n)*
Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 2 + n^(-1), ((b*c - a*d)*x^
n)/(c*(a + b*x^n))] + (b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hyp
ergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))])) + n^2*
x^n*(c + d*x^n)*(a*c^2*(-(b*c) + a*d)*(1 + 2*n)*(1 + 3*n)*p*(a + b*x^n)*Gamma[2
+ n^(-1)]*Gamma[-p]*Hypergeometric2F1[2, 1 - p, 2 + n^(-1), ((b*c - a*d)*x^n)/(c
*(a + b*x^n))] + c*d*(1 + 3*n)*(a + b*x^n)^2*(c*(1 + 2*n)*(a + b*x^n)*Gamma[2 +
n^(-1)]*Gamma[-p]*Hypergeometric2F1[1, -p, 2 + n^(-1), ((b*c - a*d)*x^n)/(c*(a +
b*x^n))] + (b*c - a*d)*(1 + n)*x^n*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometri
c2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))]) - d*(b*c - a*d)*x
^n*(b*c*(1 + n)*(1 + 3*n)*x^n*(a + b*x^n)*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hyperge
ometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] - c*(1 + n)*
(1 + 3*n)*(a + b*x^n)^2*Gamma[1 + n^(-1)]*Gamma[1 - p]*Hypergeometric2F1[2, 1 -
p, 3 + n^(-1), ((b*c - a*d)*x^n)/(c*(a + b*x^n))] + a*c*n*(1 + 3*n)*p*(a + b*x^n
)*Gamma[2 + n^(-1)]*Gamma[-p]*Hypergeometric2F1[2, 1 - p, 3 + n^(-1), ((b*c - a*
d)*x^n)/(c*(a + b*x^n))] - 2*a*(-(b*c) + a*d)*n*(1 + n)*(-1 + p)*x^n*Gamma[1 + n
^(-1)]*Gamma[1 - p]*Hypergeometric2F1[3, 2 - p, 4 + n^(-1), ((b*c - a*d)*x^n)/(c
*(a + b*x^n))])))

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Maple [F]  time = 0.264, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{n} \right ) ^{-2-{n}^{-1}-p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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