Optimal. Leaf size=55 \[ \frac{2 (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )}{c e (m+1)}-\frac{(e x)^{m+1}}{c e (m+1)} \]
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Rubi [A] time = 0.0687365, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2 (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )}{c e (m+1)}-\frac{(e x)^{m+1}}{c e (m+1)} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^m*(a + b*x))/(a*c - b*c*x),x]
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Rubi in Sympy [A] time = 10.3751, size = 37, normalized size = 0.67 \[ \frac{2 \left (e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x}{a}} \right )}}{c e \left (m + 1\right )} - \frac{\left (e x\right )^{m + 1}}{c e \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(b*x+a)/(-b*c*x+a*c),x)
[Out]
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Mathematica [A] time = 0.0398232, size = 33, normalized size = 0.6 \[ \frac{x (e x)^m \left (2 \, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )-1\right )}{c (m+1)} \]
Antiderivative was successfully verified.
[In] Integrate[((e*x)^m*(a + b*x))/(a*c - b*c*x),x]
[Out]
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Maple [F] time = 0.045, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( bx+a \right ) }{-bcx+ac}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(b*x+a)/(-b*c*x+a*c),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (b x + a\right )} \left (e x\right )^{m}}{b c x - a c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*x + a)*(e*x)^m/(b*c*x - a*c),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (b x + a\right )} \left (e x\right )^{m}}{b c x - a c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*x + a)*(e*x)^m/(b*c*x - a*c),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.22665, size = 129, normalized size = 2.35 \[ \frac{e^{m} m x x^{m} \Phi \left (\frac{b x}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{c \Gamma \left (m + 2\right )} + \frac{e^{m} x x^{m} \Phi \left (\frac{b x}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{c \Gamma \left (m + 2\right )} + \frac{b e^{m} m x^{2} x^{m} \Phi \left (\frac{b x}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a c \Gamma \left (m + 3\right )} + \frac{2 b e^{m} x^{2} x^{m} \Phi \left (\frac{b x}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a c \Gamma \left (m + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**m*(b*x+a)/(-b*c*x+a*c),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (b x + a\right )} \left (e x\right )^{m}}{b c x - a c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*x + a)*(e*x)^m/(b*c*x - a*c),x, algorithm="giac")
[Out]