Optimal. Leaf size=127 \[ \frac{(e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)}+\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c e (m+1) (b c-a d)} \]
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Rubi [A] time = 0.378685, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{(e x)^{m+1} (A b-a B) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a e (m+1) (b c-a d)}+\frac{(e x)^{m+1} (B c-A d) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )}{c e (m+1) (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^m*(A + B*x^n))/((a + b*x^n)*(c + d*x^n)),x]
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Rubi in Sympy [A] time = 90.2279, size = 199, normalized size = 1.57 \[ \frac{A d \left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c e \left (m + 1\right ) \left (a d - b c\right )} - \frac{A b \left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + 1}{n} \\ \frac{m + n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a e \left (m + 1\right ) \left (a d - b c\right )} + \frac{B d x^{n} \left (e x\right )^{- n} \left (e x\right )^{m + n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + n + 1}{n} \\ \frac{m + 2 n + 1}{n} \end{matrix}\middle |{- \frac{d x^{n}}{c}} \right )}}{c e \left (a d - b c\right ) \left (m + n + 1\right )} - \frac{B b x^{n} \left (e x\right )^{- n} \left (e x\right )^{m + n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{m + n + 1}{n} \\ \frac{m + 2 n + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a e \left (a d - b c\right ) \left (m + n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(A+B*x**n)/(a+b*x**n)/(c+d*x**n),x)
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Mathematica [A] time = 0.179007, size = 102, normalized size = 0.8 \[ \frac{x (e x)^m \left ((a B c-A b c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )+a (A d-B c) \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{d x^n}{c}\right )\right )}{a c (m+1) (a d-b c)} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^m*(A + B*x^n))/((a + b*x^n)*(c + d*x^n)),x]
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Maple [F] time = 0.097, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( A+B{x}^{n} \right ) }{ \left ( a+b{x}^{n} \right ) \left ( c+d{x}^{n} \right ) }}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(A+B*x^n)/(a+b*x^n)/(c+d*x^n),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(e*x)^m/((b*x^n + a)*(d*x^n + c)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{b d x^{2 \, n} + a c +{\left (b c + a d\right )} x^{n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(e*x)^m/((b*x^n + a)*(d*x^n + c)),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((e*x)**m*(A+B*x**n)/(a+b*x**n)/(c+d*x**n),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{n} + A\right )} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}{\left (d x^{n} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(e*x)^m/((b*x^n + a)*(d*x^n + c)),x, algorithm="giac")
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