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

Optimal. Leaf size=78 \[ \frac{x (b c-a d (1-2 n)) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 c^3 d n}-\frac{x (b c-a d)}{2 c d n \left (c+d x^n\right )^2} \]

[Out]

-((b*c - a*d)*x)/(2*c*d*n*(c + d*x^n)^2) + ((b*c - a*d*(1 - 2*n))*x*Hypergeometr
ic2F1[2, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/(2*c^3*d*n)

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Rubi [A]  time = 0.092494, antiderivative size = 78, 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 \[ \frac{x (b c-a d (1-2 n)) \, _2F_1\left (2,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 c^3 d n}-\frac{x (b c-a d)}{2 c d n \left (c+d x^n\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)*x)/(2*c*d*n*(c + d*x^n)^2) + ((b*c - a*d*(1 - 2*n))*x*Hypergeometr
ic2F1[2, n^(-1), 1 + n^(-1), -((d*x^n)/c)])/(2*c^3*d*n)

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Rubi in Sympy [A]  time = 10.1555, size = 60, normalized size = 0.77 \[ \frac{x \left (a d - b c\right )}{2 c d n \left (c + d x^{n}\right )^{2}} + \frac{x \left (- a d \left (- 2 n + 1\right ) + b c\right ){{}_{2}F_{1}\left (\begin{matrix} 2, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{2 c^{3} d n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.107669, size = 96, normalized size = 1.23 \[ \frac{x \left (-\frac{c^2 n (b c-a d)}{\left (c+d x^n\right )^2}+(n-1) (a d (2 n-1)+b c) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+\frac{c (a d (2 n-1)+b c)}{c+d x^n}\right )}{2 c^3 d n^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{a+b{x}^{n}}{ \left ( c+d{x}^{n} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)/(c+d*x^n)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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