Optimal. Leaf size=89 \[ \frac{b x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a (b c-a d)}-\frac{d x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right )}{2 c (b c-a d)} \]
[Out]
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Rubi [A] time = 0.111108, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{b x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a (b c-a d)}-\frac{d x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right )}{2 c (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x/((a + b*x^n)*(c + d*x^n)),x]
[Out]
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Rubi in Sympy [A] time = 15.6535, size = 60, normalized size = 0.67 \[ \frac{d x^{2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{2 c \left (a d - b c\right )} - \frac{b x^{2}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(a+b*x**n)/(c+d*x**n),x)
[Out]
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Mathematica [A] time = 0.0802597, size = 78, normalized size = 0.88 \[ \frac{b c x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )-a d x^2 \, _2F_1\left (1,\frac{2}{n};\frac{n+2}{n};-\frac{d x^n}{c}\right )}{2 a b c^2-2 a^2 c d} \]
Antiderivative was successfully verified.
[In] Integrate[x/((a + b*x^n)*(c + d*x^n)),x]
[Out]
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Maple [F] time = 0.098, size = 0, normalized size = 0. \[ \int{\frac{x}{ \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(a+b*x^n)/(c+d*x^n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^n + a)*(d*x^n + c)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{b d x^{2 \, n} + a c +{\left (b c + a d\right )} x^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^n + a)*(d*x^n + c)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (a + b x^{n}\right ) \left (c + d x^{n}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(a+b*x**n)/(c+d*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((b*x^n + a)*(d*x^n + c)),x, algorithm="giac")
[Out]