3.372 \(\int \frac{(e x)^m (a+b x)}{(a c-b c x)^2} \, dx\)

Optimal. Leaf size=66 \[ \frac{2 (e x)^{m+1}}{c^2 e (a-b x)}-\frac{(2 m+1) (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )}{a c^2 e (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0843184, antiderivative size = 66, 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 \[ \frac{2 (e x)^{m+1}}{c^2 e (a-b x)}-\frac{(2 m+1) (e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b x}{a}\right )}{a c^2 e (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.3447, size = 49, normalized size = 0.74 \[ \frac{2 \left (e x\right )^{m + 1}}{c^{2} e \left (a - b x\right )} - \frac{\left (e x\right )^{m + 1} \left (2 m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x}{a}} \right )}}{a c^{2} e \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(b*x+a)/(-b*c*x+a*c)**2,x)

[Out]

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

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Mathematica [A]  time = 0.119589, size = 103, normalized size = 1.56 \[ \frac{(e x)^m \left (\frac{b x}{b x-a}\right )^{1-m} \left (2 a m \, _2F_1\left (1-m,-m;2-m;\frac{a}{a-b x}\right )-(m-1) (a-b x) \, _2F_1\left (-m,-m;1-m;\frac{a}{a-b x}\right )\right )}{b^2 c^2 (m-1) m x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex \right ) ^{m} \left ( bx+a \right ) }{ \left ( -bcx+ac \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int((e*x)^m*(b*x+a)/(-b*c*x+a*c)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x)^m/(b*c*x - a*c)^2,x, algorithm="maxima")

[Out]

integrate((b*x + a)*(e*x)^m/(b*c*x - a*c)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )} \left (e x\right )^{m}}{b^{2} c^{2} x^{2} - 2 \, a b c^{2} x + a^{2} c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x)^m/(b*c*x - a*c)^2,x, algorithm="fricas")

[Out]

integral((b*x + a)*(e*x)^m/(b^2*c^2*x^2 - 2*a*b*c^2*x + a^2*c^2), x)

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Sympy [A]  time = 13.1543, size = 799, normalized size = 12.11 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(b*x+a)/(-b*c*x+a*c)**2,x)

[Out]

a*(a*e**m*m**2*x*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(
-a**3*c**2*gamma(m + 2) + a**2*b*c**2*x*gamma(m + 2)) + a*e**m*m*x*x**m*lerchphi
(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2
*b*c**2*x*gamma(m + 2)) - a*e**m*m*x*x**m*gamma(m + 1)/(-a**3*c**2*gamma(m + 2)
+ a**2*b*c**2*x*gamma(m + 2)) - a*e**m*x*x**m*gamma(m + 1)/(-a**3*c**2*gamma(m +
2) + a**2*b*c**2*x*gamma(m + 2)) - b*e**m*m**2*x**2*x**m*lerchphi(b*x*exp_polar
(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2*b*c**2*x*gamm
a(m + 2)) - b*e**m*m*x**2*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma
(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2*b*c**2*x*gamma(m + 2))) + b*(a*e**m*m**2
*x**2*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*
gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) + 3*a*e**m*m*x**2*x**m*lerchphi(b*x*e
xp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**
2*x*gamma(m + 3)) - a*e**m*m*x**2*x**m*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a
**2*b*c**2*x*gamma(m + 3)) + 2*a*e**m*x**2*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a
, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3))
- 2*a*e**m*x**2*x**m*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma
(m + 3)) - b*e**m*m**2*x**3*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 2)*gam
ma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) - 3*b*e**m*m*x*
*3*x**m*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*gam
ma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) - 2*b*e**m*x**3*x**m*lerchphi(b*x*exp_po
lar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*g
amma(m + 3)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*(e*x)^m/(b*c*x - a*c)^2,x, algorithm="giac")

[Out]

integrate((b*x + a)*(e*x)^m/(b*c*x - a*c)^2, x)