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

Optimal. Leaf size=44 \[ \frac{c x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2}-\frac{d}{b n \left (a+b x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.0596695, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{c x \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2}-\frac{d}{b n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.99551, size = 32, normalized size = 0.73 \[ - \frac{d}{b n \left (a + b x^{n}\right )} + \frac{c x{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0879153, size = 56, normalized size = 1.27 \[ \frac{\frac{a (b c x-a d)}{b \left (a+b x^n\right )}+c (n-1) x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \[ \int{\frac{c+d{x}^{-1+n}}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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