3.371 \(\int \frac{(e x)^m (a+b x)}{a c-b c x} \, dx\)

Optimal. Leaf size=55 \[ \frac{2 (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )}{c e (m+1)}-\frac{(e x)^{m+1}}{c e (m+1)} \]

[Out]

-((e*x)^(1 + m)/(c*e*(1 + m))) + (2*(e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2
+ m, (b*x)/a])/(c*e*(1 + m))

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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]

[Out]

-((e*x)^(1 + m)/(c*e*(1 + m))) + (2*(e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2
+ m, (b*x)/a])/(c*e*(1 + m))

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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]

2*(e*x)**(m + 1)*hyper((1, m + 1), (m + 2,), b*x/a)/(c*e*(m + 1)) - (e*x)**(m +
1)/(c*e*(m + 1))

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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]

(x*(e*x)^m*(-1 + 2*Hypergeometric2F1[1, 1 + m, 2 + m, (b*x)/a]))/(c*(1 + m))

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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]

int((e*x)^m*(b*x+a)/(-b*c*x+a*c),x)

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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]

-integrate((b*x + a)*(e*x)^m/(b*c*x - a*c), x)

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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]

integral(-(b*x + a)*(e*x)^m/(b*c*x - a*c), x)

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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]

e**m*m*x*x**m*lerchphi(b*x/a, 1, m + 1)*gamma(m + 1)/(c*gamma(m + 2)) + e**m*x*x
**m*lerchphi(b*x/a, 1, m + 1)*gamma(m + 1)/(c*gamma(m + 2)) + b*e**m*m*x**2*x**m
*lerchphi(b*x/a, 1, m + 2)*gamma(m + 2)/(a*c*gamma(m + 3)) + 2*b*e**m*x**2*x**m*
lerchphi(b*x/a, 1, m + 2)*gamma(m + 2)/(a*c*gamma(m + 3))

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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]

integrate(-(b*x + a)*(e*x)^m/(b*c*x - a*c), x)