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

Optimal. Leaf size=72 \[ \frac{x (a d-b c (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b n}+\frac{x (b c-a d)}{a b n \left (a+b x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.0844173, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x (a d-b c (1-n)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b n}+\frac{x (b c-a d)}{a b n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.0002, size = 53, normalized size = 0.74 \[ - \frac{x \left (a d - b c\right )}{a b n \left (a + b x^{n}\right )} + \frac{x \left (a d - b c \left (- n + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0768954, size = 68, normalized size = 0.94 \[ \frac{x \left (\left (a+b x^n\right ) (a d+b c (n-1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )+a (b c-a d)\right )}{a^2 b n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{c+d{x}^{n}}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*x^n)/(a+b*x^n)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((d*x^n + 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**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. \[ \int \frac{d x^{n} + c}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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