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

Optimal. Leaf size=327 \[ -\frac{2 a^2 n^2 x \left (c+d x^n\right )^{-1/n} (3 a d n-b (3 c n+c))}{c^4 (n+1) (2 n+1) (3 n+1) (b c-a d)}-\frac{2 a n x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac{1}{n}-1} (3 a d n-b (3 c n+c))}{c^3 (n+1) (2 n+1) (3 n+1) (b c-a d)}-\frac{x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac{1}{n}-2} (3 a d n-b (3 c n+c))}{c^2 \left (6 n^2+5 n+1\right ) (b c-a d)}-\frac{x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac{1}{n}-3} (3 a d n-b (3 c n+c))}{3 a c n (3 n+1) (b c-a d)}-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac{1}{n}-3}}{3 a n (b c-a d)} \]

[Out]

-(b*x*(a + b*x^n)^3*(c + d*x^n)^(-3 - n^(-1)))/(3*a*(b*c - a*d)*n) - ((3*a*d*n -
b*(c + 3*c*n))*x*(a + b*x^n)^3*(c + d*x^n)^(-3 - n^(-1)))/(3*a*c*(b*c - a*d)*n*
(1 + 3*n)) - ((3*a*d*n - b*(c + 3*c*n))*x*(a + b*x^n)^2*(c + d*x^n)^(-2 - n^(-1)
))/(c^2*(b*c - a*d)*(1 + 5*n + 6*n^2)) - (2*a*n*(3*a*d*n - b*(c + 3*c*n))*x*(a +
b*x^n)*(c + d*x^n)^(-1 - n^(-1)))/(c^3*(b*c - a*d)*(1 + n)*(1 + 2*n)*(1 + 3*n))
- (2*a^2*n^2*(3*a*d*n - b*(c + 3*c*n))*x)/(c^4*(b*c - a*d)*(1 + n)*(1 + 2*n)*(1
+ 3*n)*(c + d*x^n)^n^(-1))

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Rubi [A]  time = 0.548162, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{2 a^2 n^2 x \left (c+d x^n\right )^{-1/n} (3 a d n-b (3 c n+c))}{c^4 (n+1) (2 n+1) (3 n+1) (b c-a d)}-\frac{2 a n x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac{1}{n}-1} (3 a d n-b (3 c n+c))}{c^3 (n+1) (2 n+1) (3 n+1) (b c-a d)}-\frac{x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac{1}{n}-2} (3 a d n-b (3 c n+c))}{c^2 \left (6 n^2+5 n+1\right ) (b c-a d)}-\frac{x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac{1}{n}-3} (3 a d n-b (3 c n+c))}{3 a c n (3 n+1) (b c-a d)}-\frac{b x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac{1}{n}-3}}{3 a n (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

-(b*x*(a + b*x^n)^3*(c + d*x^n)^(-3 - n^(-1)))/(3*a*(b*c - a*d)*n) - ((3*a*d*n -
b*(c + 3*c*n))*x*(a + b*x^n)^3*(c + d*x^n)^(-3 - n^(-1)))/(3*a*c*(b*c - a*d)*n*
(1 + 3*n)) - ((3*a*d*n - b*(c + 3*c*n))*x*(a + b*x^n)^2*(c + d*x^n)^(-2 - n^(-1)
))/(c^2*(b*c - a*d)*(1 + 5*n + 6*n^2)) - (2*a*n*(3*a*d*n - b*(c + 3*c*n))*x*(a +
b*x^n)*(c + d*x^n)^(-1 - n^(-1)))/(c^3*(b*c - a*d)*(1 + n)*(1 + 2*n)*(1 + 3*n))
- (2*a^2*n^2*(3*a*d*n - b*(c + 3*c*n))*x)/(c^4*(b*c - a*d)*(1 + n)*(1 + 2*n)*(1
+ 3*n)*(c + d*x^n)^n^(-1))

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Rubi in Sympy [A]  time = 81.8544, size = 286, normalized size = 0.87 \[ - \frac{2 a^{2} n^{2} x \left (c + d x^{n}\right )^{- \frac{1}{n}} \left (- 3 a d n + 3 b c n + b c\right )}{c^{4} \left (n + 1\right ) \left (2 n + 1\right ) \left (3 n + 1\right ) \left (a d - b c\right )} - \frac{2 a n x \left (a + b x^{n}\right ) \left (c + d x^{n}\right )^{-1 - \frac{1}{n}} \left (- 3 a d n + 3 b c n + b c\right )}{c^{3} \left (n + 1\right ) \left (2 n + 1\right ) \left (3 n + 1\right ) \left (a d - b c\right )} - \frac{x \left (a + b x^{n}\right )^{2} \left (c + d x^{n}\right )^{-2 - \frac{1}{n}} \left (- 3 a d n + 3 b c n + b c\right )}{c^{2} \left (2 n + 1\right ) \left (3 n + 1\right ) \left (a d - b c\right )} + \frac{b x \left (a + b x^{n}\right )^{3} \left (c + d x^{n}\right )^{-3 - \frac{1}{n}}}{3 a n \left (a d - b c\right )} - \frac{x \left (a + b x^{n}\right )^{3} \left (c + d x^{n}\right )^{-3 - \frac{1}{n}} \left (- 3 a d n + 3 b c n + b c\right )}{3 a c n \left (3 n + 1\right ) \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.304528, size = 153, normalized size = 0.47 \[ \frac{x \left (c+d x^n\right )^{-1/n} \left (\frac{d x^n}{c}+1\right )^{\frac{1}{n}} \left ((n+1) \left (a^2 (2 n+1) \, _2F_1\left (4+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+b^2 x^{2 n} \, _2F_1\left (2+\frac{1}{n},4+\frac{1}{n};3+\frac{1}{n};-\frac{d x^n}{c}\right )\right )+2 a b (2 n+1) x^n \, _2F_1\left (1+\frac{1}{n},4+\frac{1}{n};2+\frac{1}{n};-\frac{d x^n}{c}\right )\right )}{c^4 (n+1) (2 n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.21, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{2} \left ( c+d{x}^{n} \right ) ^{-4-{n}^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.260355, size = 540, normalized size = 1.65 \[ \frac{{\left (6 \, a^{2} d^{4} n^{3} + b^{2} c^{2} d^{2} n +{\left (b^{2} c^{2} d^{2} + 4 \, a b c d^{3}\right )} n^{2}\right )} x x^{4 \, n} +{\left (24 \, a^{2} c d^{3} n^{3} + b^{2} c^{3} d + 2 \,{\left (2 \, b^{2} c^{3} d + 8 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3}\right )} n^{2} +{\left (5 \, b^{2} c^{3} d + 4 \, a b c^{2} d^{2}\right )} n\right )} x x^{3 \, n} +{\left (36 \, a^{2} c^{2} d^{2} n^{3} + b^{2} c^{4} + 2 \, a b c^{3} d + 3 \,{\left (b^{2} c^{4} + 8 \, a b c^{3} d + 7 \, a^{2} c^{2} d^{2}\right )} n^{2} +{\left (4 \, b^{2} c^{4} + 14 \, a b c^{3} d + 3 \, a^{2} c^{2} d^{2}\right )} n\right )} x x^{2 \, n} +{\left (24 \, a^{2} c^{3} d n^{3} + 2 \, a b c^{4} + a^{2} c^{3} d + 2 \,{\left (6 \, a b c^{4} + 13 \, a^{2} c^{3} d\right )} n^{2} +{\left (10 \, a b c^{4} + 9 \, a^{2} c^{3} d\right )} n\right )} x x^{n} +{\left (6 \, a^{2} c^{4} n^{3} + 11 \, a^{2} c^{4} n^{2} + 6 \, a^{2} c^{4} n + a^{2} c^{4}\right )} x}{{\left (6 \, c^{4} n^{3} + 11 \, c^{4} n^{2} + 6 \, c^{4} n + c^{4}\right )}{\left (d x^{n} + c\right )}^{\frac{4 \, n + 1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((6*a^2*d^4*n^3 + b^2*c^2*d^2*n + (b^2*c^2*d^2 + 4*a*b*c*d^3)*n^2)*x*x^(4*n) + (
24*a^2*c*d^3*n^3 + b^2*c^3*d + 2*(2*b^2*c^3*d + 8*a*b*c^2*d^2 + 3*a^2*c*d^3)*n^2
+ (5*b^2*c^3*d + 4*a*b*c^2*d^2)*n)*x*x^(3*n) + (36*a^2*c^2*d^2*n^3 + b^2*c^4 +
2*a*b*c^3*d + 3*(b^2*c^4 + 8*a*b*c^3*d + 7*a^2*c^2*d^2)*n^2 + (4*b^2*c^4 + 14*a*
b*c^3*d + 3*a^2*c^2*d^2)*n)*x*x^(2*n) + (24*a^2*c^3*d*n^3 + 2*a*b*c^4 + a^2*c^3*
d + 2*(6*a*b*c^4 + 13*a^2*c^3*d)*n^2 + (10*a*b*c^4 + 9*a^2*c^3*d)*n)*x*x^n + (6*
a^2*c^4*n^3 + 11*a^2*c^4*n^2 + 6*a^2*c^4*n + a^2*c^4)*x)/((6*c^4*n^3 + 11*c^4*n^
2 + 6*c^4*n + c^4)*(d*x^n + c)^((4*n + 1)/n))

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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)**2*(c+d*x**n)**(-4-1/n),x)

[Out]

Timed out

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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)^2*(d*x^n + c)^(-1/n - 4),x, algorithm="giac")

[Out]

Exception raised: TypeError