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

Optimal. Leaf size=115 \[ \frac{x (b c-a d) (a d (1-n)-b c (n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 d^2 n}-\frac{b x (a d-b c (n+1))}{c d^2 n}-\frac{x (b c-a d) \left (a+b x^n\right )}{c d n \left (c+d x^n\right )} \]

[Out]

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

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Rubi [A]  time = 0.244695, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{x (b c-a d) (a d (1-n)-b c (n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{c^2 d^2 n}-\frac{b x (a d-b c (n+1))}{c d^2 n}-\frac{x (b c-a d) \left (a+b x^n\right )}{c d n \left (c+d x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.0974, size = 95, normalized size = 0.83 \[ - \frac{b x \left (a d - b c \left (n + 1\right )\right )}{c d^{2} n} + \frac{x \left (a + b x^{n}\right ) \left (a d - b c\right )}{c d n \left (c + d x^{n}\right )} - \frac{x \left (a d - b c\right ) \left (- a d n + a d - b c n - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c^{2} d^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.190573, size = 95, normalized size = 0.83 \[ \frac{x \left (\frac{c \left (a^2 d^2-2 a b c d+b^2 c \left (c n+c+d n x^n\right )\right )}{c+d x^n}-(b c-a d) (a d (n-1)+b c (n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )\right )}{c^2 d^2 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \[ \int{\frac{ \left ( a+b{x}^{n} \right ) ^{2}}{ \left ( c+d{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^2/(c+d*x^n)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x**n)**2/(c + d*x**n)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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