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

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

[Out]

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

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Rubi [A]  time = 0.236724, 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 \[ -\frac{x (b c-a d) (b c (1-n)-a d (n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b^2 n}-\frac{d x (b c-a d (n+1))}{a b^2 n}+\frac{x (b c-a d) \left (c+d x^n\right )}{a b n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.5983, size = 97, normalized size = 0.84 \[ - \frac{x \left (c + d x^{n}\right ) \left (a d - b c\right )}{a b n \left (a + b x^{n}\right )} - \frac{d x \left (- a d \left (n + 1\right ) + b c\right )}{a b^{2} n} - \frac{x \left (a d - b c\right ) \left (a d n + a d + b c n - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.199505, size = 96, normalized size = 0.83 \[ \frac{x \left (\frac{a \left (a^2 d^2 (n+1)+a b d \left (d n x^n-2 c\right )+b^2 c^2\right )}{a+b x^n}+(b c-a d) (a d (n+1)+b c (n-1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )\right )}{a^2 b^2 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c+d{x}^{n} \right ) ^{2}}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c+d*x^n)^2/(a+b*x^n)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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