3.571 \(\int \frac{c+d x^{-1+n}}{\left (a+b x^n\right )^3} \, dx\)

Optimal. Leaf size=46 \[ \frac{c x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3}-\frac{d}{2 b n \left (a+b x^n\right )^2} \]

[Out]

-d/(2*b*n*(a + b*x^n)^2) + (c*x*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((b*x^
n)/a)])/a^3

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Rubi [A]  time = 0.0598445, antiderivative size = 46, 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 \[ \frac{c x \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3}-\frac{d}{2 b n \left (a+b x^n\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^(-1 + n))/(a + b*x^n)^3,x]

[Out]

-d/(2*b*n*(a + b*x^n)^2) + (c*x*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((b*x^
n)/a)])/a^3

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Rubi in Sympy [A]  time = 9.02079, size = 36, normalized size = 0.78 \[ - \frac{d}{2 b n \left (a + b x^{n}\right )^{2}} + \frac{c x{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c+d*x**(-1+n))/(a+b*x**n)**3,x)

[Out]

-d/(2*b*n*(a + b*x**n)**2) + c*x*hyper((3, 1/n), (1 + 1/n,), -b*x**n/a)/a**3

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Mathematica [B]  time = 0.13708, size = 108, normalized size = 2.35 \[ \frac{x \left (c+d x^{n-1}\right ) \left (\frac{a^2 n (b c x-a d)}{b \left (a+b x^n\right )^2}+c \left (2 n^2-3 n+1\right ) x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )+\frac{a c (2 n-1) x}{a+b x^n}\right )}{2 a^3 n^2 \left (c x+d x^n\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^(-1 + n))/(a + b*x^n)^3,x]

[Out]

(x*(c + d*x^(-1 + n))*((a^2*n*(-(a*d) + b*c*x))/(b*(a + b*x^n)^2) + (a*c*(-1 + 2
*n)*x)/(a + b*x^n) + c*(1 - 3*n + 2*n^2)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-
1), -((b*x^n)/a)]))/(2*a^3*n^2*(c*x + d*x^n))

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \[ \int{\frac{c+d{x}^{-1+n}}{ \left ( a+b{x}^{n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*x^(-1+n))/(a+b*x^n)^3,x)

[Out]

int((c+d*x^(-1+n))/(a+b*x^n)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[{\left (2 \, n^{2} - 3 \, n + 1\right )} c \int \frac{1}{2 \,{\left (a^{2} b n^{2} x^{n} + a^{3} n^{2}\right )}}\,{d x} + \frac{b^{2} c{\left (2 \, n - 1\right )} x x^{n} + a b c{\left (3 \, n - 1\right )} x - a^{2} d n}{2 \,{\left (a^{2} b^{3} n^{2} x^{2 \, n} + 2 \, a^{3} b^{2} n^{2} x^{n} + a^{4} b n^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^(n - 1) + c)/(b*x^n + a)^3,x, algorithm="maxima")

[Out]

(2*n^2 - 3*n + 1)*c*integrate(1/2/(a^2*b*n^2*x^n + a^3*n^2), x) + 1/2*(b^2*c*(2*
n - 1)*x*x^n + a*b*c*(3*n - 1)*x - a^2*d*n)/(a^2*b^3*n^2*x^(2*n) + 2*a^3*b^2*n^2
*x^n + a^4*b*n^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d x^{n - 1} + c}{b^{3} x^{3 \, n} + 3 \, a b^{2} x^{2 \, n} + 3 \, a^{2} b x^{n} + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^(n - 1) + c)/(b*x^n + a)^3,x, algorithm="fricas")

[Out]

integral((d*x^(n - 1) + c)/(b^3*x^(3*n) + 3*a*b^2*x^(2*n) + 3*a^2*b*x^n + a^3),
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**(-1+n))/(a+b*x**n)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{d x^{n - 1} + c}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^(n - 1) + c)/(b*x^n + a)^3,x, algorithm="giac")

[Out]

integrate((d*x^(n - 1) + c)/(b*x^n + a)^3, x)