Optimal. Leaf size=341 \[ -\frac{x (b c-a d)^3 (b c (1-n)-a d (3 n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b^4 n}-\frac{d x \left (c+d x^n\right ) \left (a^2 d^2 \left (6 n^2+5 n+1\right )-2 a b c d \left (5 n^2+4 n+1\right )+b^2 c^2 \left (2 n^2+3 n+1\right )\right )}{a b^3 n (n+1) (2 n+1)}-\frac{d x \left (-a^3 d^3 \left (6 n^3+11 n^2+6 n+1\right )+a^2 b c d^2 \left (16 n^3+26 n^2+15 n+3\right )-a b^2 c^2 d \left (12 n^3+17 n^2+12 n+3\right )+b^3 c^3 \left (2 n^2+3 n+1\right )\right )}{a b^4 n (n+1) (2 n+1)}+\frac{d x \left (c+d x^n\right )^2 (a d (3 n+1)-b (2 c n+c))}{a b^2 n (2 n+1)}+\frac{x (b c-a d) \left (c+d x^n\right )^3}{a b n \left (a+b x^n\right )} \]
[Out]
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Rubi [A] time = 1.2872, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{x (b c-a d)^3 (b c (1-n)-a d (3 n+1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 b^4 n}-\frac{d x \left (c+d x^n\right ) \left (a^2 d^2 \left (6 n^2+5 n+1\right )-2 a b c d \left (5 n^2+4 n+1\right )+b^2 c^2 \left (2 n^2+3 n+1\right )\right )}{a b^3 n (n+1) (2 n+1)}-\frac{d x \left (-a^3 d^3 \left (6 n^3+11 n^2+6 n+1\right )+a^2 b c d^2 \left (16 n^3+26 n^2+15 n+3\right )-a b^2 c^2 d \left (12 n^3+17 n^2+12 n+3\right )+b^3 c^3 \left (2 n^2+3 n+1\right )\right )}{a b^4 n (n+1) (2 n+1)}+\frac{d x \left (c+d x^n\right )^2 (a d (3 n+1)-b (2 c n+c))}{a b^2 n (2 n+1)}+\frac{x (b c-a d) \left (c+d x^n\right )^3}{a b n \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^n)^4/(a + b*x^n)^2,x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c+d*x**n)**4/(a+b*x**n)**2,x)
[Out]
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Mathematica [A] time = 0.337356, size = 217, normalized size = 0.64 \[ \frac{x \left (\frac{(b c-a d)^3 (a d (3 n+1)+b c (n-1)) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^2 n}+\frac{(a d-b c)^3 (a d (3 n+1)+b c (n-1))}{a^2 n}+\frac{-a^4 d^4+4 a^3 b c d^3-6 a^2 b^2 c^2 d^2+4 a b^3 c^3 d+b^4 c^4 (n-1)}{a^2 n}+\frac{2 b d^3 x^n (2 b c-a d)}{n+1}+\frac{(b c-a d)^4}{a n \left (a+b x^n\right )}+\frac{b^2 d^4 x^{2 n}}{2 n+1}\right )}{b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^n)^4/(a + b*x^n)^2,x]
[Out]
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Maple [F] time = 0.078, size = 0, normalized size = 0. \[ \int{\frac{ \left ( c+d{x}^{n} \right ) ^{4}}{ \left ( a+b{x}^{n} \right ) ^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c+d*x^n)^4/(a+b*x^n)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -{\left (a^{4} d^{4}{\left (3 \, n + 1\right )} - 4 \, a^{3} b c d^{3}{\left (2 \, n + 1\right )} + 6 \, a^{2} b^{2} c^{2} d^{2}{\left (n + 1\right )} - b^{4} c^{4}{\left (n - 1\right )} - 4 \, a b^{3} c^{3} d\right )} \int \frac{1}{a b^{5} n x^{n} + a^{2} b^{4} n}\,{d x} + \frac{{\left (n^{2} + n\right )} a b^{3} d^{4} x x^{3 \, n} +{\left (4 \,{\left (2 \, n^{2} + n\right )} a b^{3} c d^{3} -{\left (3 \, n^{2} + n\right )} a^{2} b^{2} d^{4}\right )} x x^{2 \, n} +{\left (6 \,{\left (2 \, n^{3} + 3 \, n^{2} + n\right )} a b^{3} c^{2} d^{2} - 4 \,{\left (4 \, n^{3} + 4 \, n^{2} + n\right )} a^{2} b^{2} c d^{3} +{\left (6 \, n^{3} + 5 \, n^{2} + n\right )} a^{3} b d^{4}\right )} x x^{n} +{\left ({\left (2 \, n^{2} + 3 \, n + 1\right )} b^{4} c^{4} - 4 \,{\left (2 \, n^{2} + 3 \, n + 1\right )} a b^{3} c^{3} d + 6 \,{\left (2 \, n^{3} + 5 \, n^{2} + 4 \, n + 1\right )} a^{2} b^{2} c^{2} d^{2} - 4 \,{\left (4 \, n^{3} + 8 \, n^{2} + 5 \, n + 1\right )} a^{3} b c d^{3} +{\left (6 \, n^{3} + 11 \, n^{2} + 6 \, n + 1\right )} a^{4} d^{4}\right )} x}{{\left (2 \, n^{3} + 3 \, n^{2} + n\right )} a b^{5} x^{n} +{\left (2 \, n^{3} + 3 \, n^{2} + n\right )} a^{2} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + c)^4/(b*x^n + a)^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{d^{4} x^{4 \, n} + 4 \, c d^{3} x^{3 \, n} + 6 \, c^{2} d^{2} x^{2 \, n} + 4 \, c^{3} d x^{n} + c^{4}}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + c)^4/(b*x^n + a)^2,x, algorithm="fricas")
[Out]
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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((c+d*x**n)**4/(a+b*x**n)**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{n} + c\right )}^{4}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^n + c)^4/(b*x^n + a)^2,x, algorithm="giac")
[Out]