3.370 \(\int (e x)^m (a+b x) (a c-b c x) \, dx\)

Optimal. Leaf size=42 \[ \frac{a^2 c (e x)^{m+1}}{e (m+1)}-\frac{b^2 c (e x)^{m+3}}{e^3 (m+3)} \]

[Out]

(a^2*c*(e*x)^(1 + m))/(e*(1 + m)) - (b^2*c*(e*x)^(3 + m))/(e^3*(3 + m))

_______________________________________________________________________________________

Rubi [A]  time = 0.0565567, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{a^2 c (e x)^{m+1}}{e (m+1)}-\frac{b^2 c (e x)^{m+3}}{e^3 (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*c*(e*x)^(1 + m))/(e*(1 + m)) - (b^2*c*(e*x)^(3 + m))/(e^3*(3 + m))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.6794, size = 34, normalized size = 0.81 \[ \frac{a^{2} c \left (e x\right )^{m + 1}}{e \left (m + 1\right )} - \frac{b^{2} c \left (e x\right )^{m + 3}}{e^{3} \left (m + 3\right )} \]

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]

a**2*c*(e*x)**(m + 1)/(e*(m + 1)) - b**2*c*(e*x)**(m + 3)/(e**3*(m + 3))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0307721, size = 31, normalized size = 0.74 \[ c (e x)^m \left (\frac{a^2 x}{m+1}-\frac{b^2 x^3}{m+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

c*(e*x)^m*((a^2*x)/(1 + m) - (b^2*x^3)/(3 + m))

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 47, normalized size = 1.1 \[{\frac{c \left ( ex \right ) ^{m} \left ( -{b}^{2}m{x}^{2}-{b}^{2}{x}^{2}+{a}^{2}m+3\,{a}^{2} \right ) x}{ \left ( 3+m \right ) \left ( 1+m \right ) }} \]

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]

c*(e*x)^m*(-b^2*m*x^2-b^2*x^2+a^2*m+3*a^2)*x/(3+m)/(1+m)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.21958, size = 68, normalized size = 1.62 \[ -\frac{{\left ({\left (b^{2} c m + b^{2} c\right )} x^{3} -{\left (a^{2} c m + 3 \, a^{2} c\right )} x\right )} \left (e x\right )^{m}}{m^{2} + 4 \, m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-((b^2*c*m + b^2*c)*x^3 - (a^2*c*m + 3*a^2*c)*x)*(e*x)^m/(m^2 + 4*m + 3)

_______________________________________________________________________________________

Sympy [A]  time = 1.30074, size = 141, normalized size = 3.36 \[ \begin{cases} \frac{- \frac{a^{2} c}{2 x^{2}} - b^{2} c \log{\left (x \right )}}{e^{3}} & \text{for}\: m = -3 \\\frac{a^{2} c \log{\left (x \right )} - \frac{b^{2} c x^{2}}{2}}{e} & \text{for}\: m = -1 \\\frac{a^{2} c e^{m} m x x^{m}}{m^{2} + 4 m + 3} + \frac{3 a^{2} c e^{m} x x^{m}}{m^{2} + 4 m + 3} - \frac{b^{2} c e^{m} m x^{3} x^{m}}{m^{2} + 4 m + 3} - \frac{b^{2} c e^{m} x^{3} x^{m}}{m^{2} + 4 m + 3} & \text{otherwise} \end{cases} \]

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]

Piecewise(((-a**2*c/(2*x**2) - b**2*c*log(x))/e**3, Eq(m, -3)), ((a**2*c*log(x)
- b**2*c*x**2/2)/e, Eq(m, -1)), (a**2*c*e**m*m*x*x**m/(m**2 + 4*m + 3) + 3*a**2*
c*e**m*x*x**m/(m**2 + 4*m + 3) - b**2*c*e**m*m*x**3*x**m/(m**2 + 4*m + 3) - b**2
*c*e**m*x**3*x**m/(m**2 + 4*m + 3), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211266, size = 99, normalized size = 2.36 \[ -\frac{b^{2} c m x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} + b^{2} c x^{3} e^{\left (m{\rm ln}\left (x\right ) + m\right )} - a^{2} c m x e^{\left (m{\rm ln}\left (x\right ) + m\right )} - 3 \, a^{2} c x e^{\left (m{\rm ln}\left (x\right ) + m\right )}}{m^{2} + 4 \, m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b^2*c*m*x^3*e^(m*ln(x) + m) + b^2*c*x^3*e^(m*ln(x) + m) - a^2*c*m*x*e^(m*ln(x)
+ m) - 3*a^2*c*x*e^(m*ln(x) + m))/(m^2 + 4*m + 3)