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

Optimal. Leaf size=54 \[ \frac{c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^2} \]

[Out]

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

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Rubi [A]  time = 0.0502847, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.1553, size = 42, normalized size = 0.78 \[ \frac{c x \left (c + d x^{n}\right )^{- \frac{1}{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, 2 \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{x^{n} \left (- a d + b c\right )}{a \left (c + d x^{n}\right )}} \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.134758, size = 53, normalized size = 0.98 \[ \frac{c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};\frac{(a d-b c) x^n}{a \left (d x^n+c\right )}\right )}{a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{x}^{n} \right ) ^{2}} \left ( c+d{x}^{n} \right ) ^{1-{n}^{-1}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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