Optimal. Leaf size=90 \[ \frac{d \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{d x^n}{c}\right )}{c x (b c-a d)}-\frac{b \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x (b c-a d)} \]
[Out]
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Rubi [A] time = 0.136407, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{d \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{d x^n}{c}\right )}{c x (b c-a d)}-\frac{b \, _2F_1\left (1,-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*x^n)*(c + d*x^n)),x]
[Out]
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Rubi in Sympy [A] time = 18.8438, size = 56, normalized size = 0.62 \[ - \frac{d{{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c x \left (a d - b c\right )} + \frac{b{{}_{2}F_{1}\left (\begin{matrix} 1, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a x \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(a+b*x**n)/(c+d*x**n),x)
[Out]
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Mathematica [A] time = 0.0839834, size = 74, normalized size = 0.82 \[ \frac{b c \, _2F_1\left (1,-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right )-a d \, _2F_1\left (1,-\frac{1}{n};\frac{n-1}{n};-\frac{d x^n}{c}\right )}{a c x (a d-b c)} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a + b*x^n)*(c + d*x^n)),x]
[Out]
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Maple [F] time = 0.102, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2} \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(a+b*x^n)/(c+d*x^n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)*(d*x^n + c)*x^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{b d x^{2} x^{2 \, n} +{\left (b c + a d\right )} x^{2} x^{n} + a c x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)*(d*x^n + c)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (a + b x^{n}\right ) \left (c + d x^{n}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(a+b*x**n)/(c+d*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)*(d*x^n + c)*x^2),x, algorithm="giac")
[Out]