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