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

Optimal. Leaf size=58 \[ \frac{x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac{1}{n}-1}}{c (n+1)}+\frac{a n x \left (c+d x^n\right )^{-1/n}}{c^2 (n+1)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0483245, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac{1}{n}-1}}{c (n+1)}+\frac{a n x \left (c+d x^n\right )^{-1/n}}{c^2 (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.74866, size = 48, normalized size = 0.83 \[ \frac{a n x \left (c + d x^{n}\right )^{- \frac{1}{n}}}{c^{2} \left (n + 1\right )} + \frac{x \left (a + b x^{n}\right ) \left (c + d x^{n}\right )^{-1 - \frac{1}{n}}}{c \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.169288, size = 82, normalized size = 1.41 \[ \frac{x \left (c+d x^n\right )^{-\frac{n+1}{n}} \left (a (n+1) \left (c+d x^n\right ) \left (\frac{d x^n}{c}+1\right )^{\frac{1}{n}} \, _2F_1\left (2+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+b c x^n\right )}{c^2 (n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.165, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) ^{-2-{n}^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.267403, size = 115, normalized size = 1.98 \[ \frac{{\left (a d^{2} n + b c d\right )} x x^{2 \, n} +{\left (2 \, a c d n + b c^{2} + a c d\right )} x x^{n} +{\left (a c^{2} n + a c^{2}\right )} x}{{\left (c^{2} n + c^{2}\right )}{\left (d x^{n} + c\right )}^{\frac{2 \, n + 1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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)**(-2-1/n),x)

[Out]

Timed out

_______________________________________________________________________________________

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

[Out]

Exception raised: TypeError