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

Optimal. Leaf size=93 \[ \frac{x \left (a+b x^n\right )^p \left (c+d x^n\right )^{-\frac{1}{n}-p} \left (\frac{c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{c} \]

[Out]

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

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Rubi [A]  time = 0.0726765, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{x \left (a+b x^n\right )^p \left (c+d x^n\right )^{-\frac{1}{n}-p} \left (\frac{c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.19064, size = 83, normalized size = 0.89 \[ \frac{x \left (\frac{a \left (c + d x^{n}\right )}{c \left (a + b x^{n}\right )}\right )^{p + 1 + \frac{1}{n}} \left (a + b x^{n}\right )^{p + 1} \left (c + d x^{n}\right )^{- p - 1 - \frac{1}{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, p + 1 + \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{x^{n} \left (a d - b c\right )}{c \left (a + b x^{n}\right )}} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.558127, size = 93, normalized size = 1. \[ \frac{x \left (a+b x^n\right )^p \left (c+d x^n\right )^{-\frac{n p+1}{n}} \left (\frac{c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \, _2F_1\left (\frac{1}{n},-p;1+\frac{1}{n};\frac{(a d-b c) x^n}{a \left (d x^n+c\right )}\right )}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.249, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{p} \left ( c+d{x}^{n} \right ) ^{-1-{n}^{-1}-p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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