Optimal. Leaf size=54 \[ \frac{x (b c-a d) \, _2F_1\left (1,\frac{1}{m-n};1+\frac{1}{m-n};-\frac{c x^{m-n}}{d}\right )}{c d}+\frac{a x}{c} \]
[Out]
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Rubi [A] time = 0.0939208, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{x (b c-a d) \, _2F_1\left (1,\frac{1}{m-n};1+\frac{1}{m-n};-\frac{c x^{m-n}}{d}\right )}{c d}+\frac{a x}{c} \]
Antiderivative was successfully verified.
[In] Int[(a*x^m + b*x^n)/(c*x^m + d*x^n),x]
[Out]
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Rubi in Sympy [A] time = 10.4547, size = 36, normalized size = 0.67 \[ \frac{a x}{c} - \frac{x \left (a d - b c\right ){{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{m - n} \\ 1 + \frac{1}{m - n} \end{matrix}\middle |{- \frac{c x^{m - n}}{d}} \right )}}{c d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*x**m+b*x**n)/(c*x**m+d*x**n),x)
[Out]
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Mathematica [A] time = 0.259487, size = 0, normalized size = 0. \[ \int \frac{a x^m+b x^n}{c x^m+d x^n} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(a*x^m + b*x^n)/(c*x^m + d*x^n),x]
[Out]
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Maple [F] time = 0.104, size = 0, normalized size = 0. \[ \int{\frac{a{x}^{m}+b{x}^{n}}{c{x}^{m}+d{x}^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*x^m+b*x^n)/(c*x^m+d*x^n),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -{\left (b c - a d\right )} \int \frac{x^{m}}{c d x^{m} + d^{2} x^{n}}\,{d x} + \frac{b x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x^m + b*x^n)/(c*x^m + d*x^n),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{a x^{m} + b x^{n}}{c x^{m} + d x^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x^m + b*x^n)/(c*x^m + d*x^n),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{a x^{m} + b x^{n}}{c x^{m} + d x^{n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x**m+b*x**n)/(c*x**m+d*x**n),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{a x^{m} + b x^{n}}{c x^{m} + d x^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*x^m + b*x^n)/(c*x^m + d*x^n),x, algorithm="giac")
[Out]