3.237 \(\int \frac{\left (c+d x^n\right )^{2-\frac{1}{n}}}{\left (a+b x^n\right )^4} \, dx\)

Optimal. Leaf size=133 \[ \frac{b x \left (c+d x^n\right )^{3-\frac{1}{n}}}{3 a n (b c-a d) \left (a+b x^n\right )^3}-\frac{c^2 x \left (c+d x^n\right )^{-1/n} (3 a d n+b c (1-3 n)) \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{3 a^4 n (b c-a d)} \]

[Out]

(b*x*(c + d*x^n)^(3 - n^(-1)))/(3*a*(b*c - a*d)*n*(a + b*x^n)^3) - (c^2*(b*c*(1
- 3*n) + 3*a*d*n)*x*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -(((b*c - a*d)*x^n)
/(a*(c + d*x^n)))])/(3*a^4*(b*c - a*d)*n*(c + d*x^n)^n^(-1))

_______________________________________________________________________________________

Rubi [A]  time = 0.171492, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{b x \left (c+d x^n\right )^{3-\frac{1}{n}}}{3 a n (b c-a d) \left (a+b x^n\right )^3}-\frac{c^2 x \left (c+d x^n\right )^{-1/n} (3 a d n+b (c-3 c n)) \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{3 a^4 n (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^n)^(2 - n^(-1))/(a + b*x^n)^4,x]

[Out]

(b*x*(c + d*x^n)^(3 - n^(-1)))/(3*a*(b*c - a*d)*n*(a + b*x^n)^3) - (c^2*(3*a*d*n
+ b*(c - 3*c*n))*x*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -(((b*c - a*d)*x^n)
/(a*(c + d*x^n)))])/(3*a^4*(b*c - a*d)*n*(c + d*x^n)^n^(-1))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 20.6569, size = 107, normalized size = 0.8 \[ - \frac{b x \left (c + d x^{n}\right )^{3 - \frac{1}{n}}}{3 a n \left (a + b x^{n}\right )^{3} \left (a d - b c\right )} + \frac{c^{2} x \left (c + d x^{n}\right )^{- \frac{1}{n}} \left (3 a d n - 3 b c n + b c\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{n}, 3 \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{x^{n} \left (- a d + b c\right )}{a \left (c + d x^{n}\right )}} \right )}}{3 a^{4} n \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c+d*x**n)**(2-1/n)/(a+b*x**n)**4,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [F]  time = 179.999, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]  Integrate[(c + d*x^n)^(2 - n^(-1))/(a + b*x^n)^4,x]

[Out]

$Aborted

_______________________________________________________________________________________

Maple [F]  time = 0.137, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{x}^{n} \right ) ^{4}} \left ( c+d{x}^{n} \right ) ^{2-{n}^{-1}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*x^n)^(2-1/n)/(a+b*x^n)^4,x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)^(-1/n + 2)/(b*x^n + a)^4,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (d x^{n} + c\right )}^{\frac{2 \, n - 1}{n}}}{b^{4} x^{4 \, n} + 4 \, a b^{3} x^{3 \, n} + 6 \, a^{2} b^{2} x^{2 \, n} + 4 \, a^{3} b x^{n} + a^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)^(-1/n + 2)/(b*x^n + a)^4,x, algorithm="fricas")

[Out]

integral((d*x^n + c)^((2*n - 1)/n)/(b^4*x^(4*n) + 4*a*b^3*x^(3*n) + 6*a^2*b^2*x^
(2*n) + 4*a^3*b*x^n + a^4), x)

_______________________________________________________________________________________

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)**(2-1/n)/(a+b*x**n)**4,x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^n + c)^(-1/n + 2)/(b*x^n + a)^4,x, algorithm="giac")

[Out]

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