Optimal. Leaf size=210 \[ -\frac{d x \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (3 n^2-4 n+1\right )+b^2 c^2 \left (6 n^2-5 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 c^3 n^2 (b c-a d)^3}+\frac{b^3 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a (b c-a d)^3}-\frac{d x (a d (1-2 n)-b (c-4 c n))}{2 c^2 n^2 (b c-a d)^2 \left (c+d x^n\right )}-\frac{d x}{2 c n (b c-a d) \left (c+d x^n\right )^2} \]
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Rubi [A] time = 0.818794, antiderivative size = 210, 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 \[ -\frac{d x \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (3 n^2-4 n+1\right )+b^2 c^2 \left (6 n^2-5 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )}{2 c^3 n^2 (b c-a d)^3}+\frac{b^3 x \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a (b c-a d)^3}+\frac{d x (b c (1-4 n)-a d (1-2 n))}{2 c^2 n^2 (b c-a d)^2 \left (c+d x^n\right )}-\frac{d x}{2 c n (b c-a d) \left (c+d x^n\right )^2} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^n)*(c + d*x^n)^3),x]
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Rubi in Sympy [A] time = 112.827, size = 218, normalized size = 1.04 \[ \frac{d x}{2 c n \left (c + d x^{n}\right )^{2} \left (a d - b c\right )} - \frac{d x \left (- 2 a d n + a d + 4 b c n - b c\right )}{2 c^{2} n^{2} \left (c + d x^{n}\right ) \left (a d - b c\right )^{2}} + \frac{d x \left (a d \left (- 2 a d n + a d + 2 b c n - b c \left (- 2 n + 1\right )\right ) - b c \left (- n + 1\right ) \left (- 2 a d n + a d + 4 b c n - b c\right ) - n \left (a d - b c\right ) \left (- 2 a d n + a d + 2 b c n\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{2 c^{3} n^{2} \left (a d - b c\right )^{3}} - \frac{b^{3} x{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \left (a d - b c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*x**n)/(c+d*x**n)**3,x)
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Mathematica [A] time = 0.333671, size = 210, normalized size = 1. \[ \frac{x \left (-a d \left (c+d x^n\right )^2 \left (a^2 d^2 \left (2 n^2-3 n+1\right )-2 a b c d \left (3 n^2-4 n+1\right )+b^2 c^2 \left (6 n^2-5 n+1\right )\right ) \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{d x^n}{c}\right )+2 b^3 c^3 n^2 \left (c+d x^n\right )^2 \, _2F_1\left (1,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )-a c^2 d n (b c-a d)^2+a c d (b c-a d) \left (c+d x^n\right ) (a d (2 n-1)+b (c-4 c n))\right )}{2 a c^3 n^2 (b c-a d)^3 \left (c+d x^n\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^n)*(c + d*x^n)^3),x]
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Maple [F] time = 0.175, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) ^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*x^n)/(c+d*x^n)^3,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -b^{3} \int -\frac{1}{a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3} +{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{n}}\,{d x} +{\left ({\left (6 \, n^{2} - 5 \, n + 1\right )} b^{2} c^{2} d - 2 \,{\left (3 \, n^{2} - 4 \, n + 1\right )} a b c d^{2} +{\left (2 \, n^{2} - 3 \, n + 1\right )} a^{2} d^{3}\right )} \int -\frac{1}{2 \,{\left (b^{3} c^{6} n^{2} - 3 \, a b^{2} c^{5} d n^{2} + 3 \, a^{2} b c^{4} d^{2} n^{2} - a^{3} c^{3} d^{3} n^{2} +{\left (b^{3} c^{5} d n^{2} - 3 \, a b^{2} c^{4} d^{2} n^{2} + 3 \, a^{2} b c^{3} d^{3} n^{2} - a^{3} c^{2} d^{4} n^{2}\right )} x^{n}\right )}}\,{d x} - \frac{{\left (b c d^{2}{\left (4 \, n - 1\right )} - a d^{3}{\left (2 \, n - 1\right )}\right )} x x^{n} +{\left (b c^{2} d{\left (5 \, n - 1\right )} - a c d^{2}{\left (3 \, n - 1\right )}\right )} x}{2 \,{\left (b^{2} c^{6} n^{2} - 2 \, a b c^{5} d n^{2} + a^{2} c^{4} d^{2} n^{2} +{\left (b^{2} c^{4} d^{2} n^{2} - 2 \, a b c^{3} d^{3} n^{2} + a^{2} c^{2} d^{4} n^{2}\right )} x^{2 \, n} + 2 \,{\left (b^{2} c^{5} d n^{2} - 2 \, a b c^{4} d^{2} n^{2} + a^{2} c^{3} d^{3} n^{2}\right )} x^{n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)*(d*x^n + c)^3),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{b d^{3} x^{4 \, n} + a c^{3} +{\left (3 \, b c d^{2} + a d^{3}\right )} x^{3 \, n} + 3 \,{\left (b c^{2} d + a c d^{2}\right )} x^{2 \, n} +{\left (b c^{3} + 3 \, a c^{2} d\right )} x^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)*(d*x^n + c)^3),x, algorithm="fricas")
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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(1/(a+b*x**n)/(c+d*x**n)**3,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)*(d*x^n + c)^3),x, algorithm="giac")
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