3.223 \(\int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac{1}{n}} \, dx\)

Optimal. Leaf size=116 \[ \frac{2 a^2 n^2 x \left (c+d x^n\right )^{-1/n}}{c^3 (n+1) (2 n+1)}+\frac{2 a n x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac{1}{n}-1}}{c^2 (n+1) (2 n+1)}+\frac{x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac{1}{n}-2}}{c (2 n+1)} \]

[Out]

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

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Rubi [A]  time = 0.112751, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{2 a^2 n^2 x \left (c+d x^n\right )^{-1/n}}{c^3 (n+1) (2 n+1)}+\frac{2 a n x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac{1}{n}-1}}{c^2 (n+1) (2 n+1)}+\frac{x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac{1}{n}-2}}{c (2 n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.1007, size = 100, normalized size = 0.86 \[ \frac{2 a^{2} n^{2} x \left (c + d x^{n}\right )^{- \frac{1}{n}}}{c^{3} \left (n + 1\right ) \left (2 n + 1\right )} + \frac{2 a n x \left (a + b x^{n}\right ) \left (c + d x^{n}\right )^{-1 - \frac{1}{n}}}{c^{2} \left (n + 1\right ) \left (2 n + 1\right )} + \frac{x \left (a + b x^{n}\right )^{2} \left (c + d x^{n}\right )^{-2 - \frac{1}{n}}}{c \left (2 n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.313694, size = 139, normalized size = 1.2 \[ \frac{x \left (c+d x^n\right )^{-1/n} \left (a^2 \left (\frac{d x^n}{c}+1\right )^{\frac{1}{n}} \, _2F_1\left (3+\frac{1}{n},\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+\frac{2 a b x^n \left (\frac{d x^n}{c}+1\right )^{\frac{1}{n}} \, _2F_1\left (1+\frac{1}{n},3+\frac{1}{n};2+\frac{1}{n};-\frac{d x^n}{c}\right )}{n+1}+\frac{b^2 c^2 x^{2 n}}{(2 n+1) \left (c+d x^n\right )^2}\right )}{c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.177, size = 0, normalized size = 0. \[ \int \left ( a+b{x}^{n} \right ) ^{2} \left ( c+d{x}^{n} \right ) ^{-3-{n}^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*(d*x^n + c)^(-1/n - 3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.256654, size = 312, normalized size = 2.69 \[ \frac{{\left (2 \, a^{2} d^{3} n^{2} + b^{2} c^{2} d +{\left (b^{2} c^{2} d + 2 \, a b c d^{2}\right )} n\right )} x x^{3 \, n} +{\left (6 \, a^{2} c d^{2} n^{2} + b^{2} c^{3} + 2 \, a b c^{2} d +{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} n\right )} x x^{2 \, n} +{\left (6 \, a^{2} c^{2} d n^{2} + 2 \, a b c^{3} + a^{2} c^{2} d +{\left (4 \, a b c^{3} + 5 \, a^{2} c^{2} d\right )} n\right )} x x^{n} +{\left (2 \, a^{2} c^{3} n^{2} + 3 \, a^{2} c^{3} n + a^{2} c^{3}\right )} x}{{\left (2 \, c^{3} n^{2} + 3 \, c^{3} n + c^{3}\right )}{\left (d x^{n} + c\right )}^{\frac{3 \, n + 1}{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^2*(d*x^n + c)^(-1/n - 3),x, algorithm="fricas")

[Out]

((2*a^2*d^3*n^2 + b^2*c^2*d + (b^2*c^2*d + 2*a*b*c*d^2)*n)*x*x^(3*n) + (6*a^2*c*
d^2*n^2 + b^2*c^3 + 2*a*b*c^2*d + (b^2*c^3 + 6*a*b*c^2*d + 2*a^2*c*d^2)*n)*x*x^(
2*n) + (6*a^2*c^2*d*n^2 + 2*a*b*c^3 + a^2*c^2*d + (4*a*b*c^3 + 5*a^2*c^2*d)*n)*x
*x^n + (2*a^2*c^3*n^2 + 3*a^2*c^3*n + a^2*c^3)*x)/((2*c^3*n^2 + 3*c^3*n + c^3)*(
d*x^n + c)^((3*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)**(-3-1/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 - 3),x, algorithm="giac")

[Out]

Exception raised: TypeError