\(\int (e x)^m (a+b x) (a c-b c x)^4 \, dx\) [68]

Optimal result

Integrand size = 22, antiderivative size = 145 \[ \int (e x)^m (a+b x) (a c-b c x)^4 \, dx=\frac {a^5 c^4 (e x)^{1+m}}{e (1+m)}-\frac {3 a^4 b c^4 (e x)^{2+m}}{e^2 (2+m)}+\frac {2 a^3 b^2 c^4 (e x)^{3+m}}{e^3 (3+m)}+\frac {2 a^2 b^3 c^4 (e x)^{4+m}}{e^4 (4+m)}-\frac {3 a b^4 c^4 (e x)^{5+m}}{e^5 (5+m)}+\frac {b^5 c^4 (e x)^{6+m}}{e^6 (6+m)} \]

[Out]

a^5*c^4*(e*x)^(1+m)/e/(1+m)-3*a^4*b*c^4*(e*x)^(2+m)/e^2/(2+m)+2*a^3*b^2*c^4*(e*x)^(3+m)/e^3/(3+m)+2*a^2*b^3*c^
4*(e*x)^(4+m)/e^4/(4+m)-3*a*b^4*c^4*(e*x)^(5+m)/e^5/(5+m)+b^5*c^4*(e*x)^(6+m)/e^6/(6+m)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {76} \[ \int (e x)^m (a+b x) (a c-b c x)^4 \, dx=\frac {a^5 c^4 (e x)^{m+1}}{e (m+1)}-\frac {3 a^4 b c^4 (e x)^{m+2}}{e^2 (m+2)}+\frac {2 a^3 b^2 c^4 (e x)^{m+3}}{e^3 (m+3)}+\frac {2 a^2 b^3 c^4 (e x)^{m+4}}{e^4 (m+4)}-\frac {3 a b^4 c^4 (e x)^{m+5}}{e^5 (m+5)}+\frac {b^5 c^4 (e x)^{m+6}}{e^6 (m+6)} \]

[In]

Int[(e*x)^m*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

(a^5*c^4*(e*x)^(1 + m))/(e*(1 + m)) - (3*a^4*b*c^4*(e*x)^(2 + m))/(e^2*(2 + m)) + (2*a^3*b^2*c^4*(e*x)^(3 + m)
)/(e^3*(3 + m)) + (2*a^2*b^3*c^4*(e*x)^(4 + m))/(e^4*(4 + m)) - (3*a*b^4*c^4*(e*x)^(5 + m))/(e^5*(5 + m)) + (b
^5*c^4*(e*x)^(6 + m))/(e^6*(6 + m))

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (a^5 c^4 (e x)^m-\frac {3 a^4 b c^4 (e x)^{1+m}}{e}+\frac {2 a^3 b^2 c^4 (e x)^{2+m}}{e^2}+\frac {2 a^2 b^3 c^4 (e x)^{3+m}}{e^3}-\frac {3 a b^4 c^4 (e x)^{4+m}}{e^4}+\frac {b^5 c^4 (e x)^{5+m}}{e^5}\right ) \, dx \\ & = \frac {a^5 c^4 (e x)^{1+m}}{e (1+m)}-\frac {3 a^4 b c^4 (e x)^{2+m}}{e^2 (2+m)}+\frac {2 a^3 b^2 c^4 (e x)^{3+m}}{e^3 (3+m)}+\frac {2 a^2 b^3 c^4 (e x)^{4+m}}{e^4 (4+m)}-\frac {3 a b^4 c^4 (e x)^{5+m}}{e^5 (5+m)}+\frac {b^5 c^4 (e x)^{6+m}}{e^6 (6+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.66 \[ \int (e x)^m (a+b x) (a c-b c x)^4 \, dx=\frac {c^4 x (e x)^m \left ((-a+b x)^5+a (7+2 m) \left (\frac {a^4}{1+m}-\frac {4 a^3 b x}{2+m}+\frac {6 a^2 b^2 x^2}{3+m}-\frac {4 a b^3 x^3}{4+m}+\frac {b^4 x^4}{5+m}\right )\right )}{6+m} \]

[In]

Integrate[(e*x)^m*(a + b*x)*(a*c - b*c*x)^4,x]

[Out]

(c^4*x*(e*x)^m*((-a + b*x)^5 + a*(7 + 2*m)*(a^4/(1 + m) - (4*a^3*b*x)/(2 + m) + (6*a^2*b^2*x^2)/(3 + m) - (4*a
*b^3*x^3)/(4 + m) + (b^4*x^4)/(5 + m))))/(6 + m)

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {a^{5} c^{4} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {b^{5} c^{4} x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}-\frac {3 a \,b^{4} c^{4} x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {2 a^{2} c^{4} b^{3} x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}+\frac {2 a^{3} c^{4} b^{2} x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}-\frac {3 a^{4} c^{4} b \,x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}\)	\(144\)
gosper	\(\frac {c^{4} \left (e x \right )^{m} \left (b^{5} m^{5} x^{5}-3 a \,b^{4} m^{5} x^{4}+15 b^{5} m^{4} x^{5}+2 a^{2} b^{3} m^{5} x^{3}-48 a \,b^{4} m^{4} x^{4}+85 b^{5} m^{3} x^{5}+2 a^{3} b^{2} m^{5} x^{2}+34 a^{2} b^{3} m^{4} x^{3}-285 a \,b^{4} m^{3} x^{4}+225 b^{5} m^{2} x^{5}-3 a^{4} b \,m^{5} x +36 a^{3} b^{2} m^{4} x^{2}+214 a^{2} b^{3} m^{3} x^{3}-780 a \,b^{4} m^{2} x^{4}+274 m \,x^{5} b^{5}+a^{5} m^{5}-57 a^{4} b \,m^{4} x +242 a^{3} b^{2} m^{3} x^{2}+614 a^{2} b^{3} m^{2} x^{3}-972 a \,b^{4} m \,x^{4}+120 b^{5} x^{5}+20 a^{5} m^{4}-411 a^{4} b \,m^{3} x +744 a^{3} b^{2} m^{2} x^{2}+792 a^{2} b^{3} m \,x^{3}-432 a \,b^{4} x^{4}+155 a^{5} m^{3}-1383 a^{4} b \,m^{2} x +1016 a^{3} b^{2} m \,x^{2}+360 a^{2} b^{3} x^{3}+580 a^{5} m^{2}-2106 a^{4} b m x +480 a^{3} b^{2} x^{2}+1044 a^{5} m -1080 a^{4} b x +720 a^{5}\right ) x}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(424\)
risch	\(\frac {c^{4} \left (e x \right )^{m} \left (b^{5} m^{5} x^{5}-3 a \,b^{4} m^{5} x^{4}+15 b^{5} m^{4} x^{5}+2 a^{2} b^{3} m^{5} x^{3}-48 a \,b^{4} m^{4} x^{4}+85 b^{5} m^{3} x^{5}+2 a^{3} b^{2} m^{5} x^{2}+34 a^{2} b^{3} m^{4} x^{3}-285 a \,b^{4} m^{3} x^{4}+225 b^{5} m^{2} x^{5}-3 a^{4} b \,m^{5} x +36 a^{3} b^{2} m^{4} x^{2}+214 a^{2} b^{3} m^{3} x^{3}-780 a \,b^{4} m^{2} x^{4}+274 m \,x^{5} b^{5}+a^{5} m^{5}-57 a^{4} b \,m^{4} x +242 a^{3} b^{2} m^{3} x^{2}+614 a^{2} b^{3} m^{2} x^{3}-972 a \,b^{4} m \,x^{4}+120 b^{5} x^{5}+20 a^{5} m^{4}-411 a^{4} b \,m^{3} x +744 a^{3} b^{2} m^{2} x^{2}+792 a^{2} b^{3} m \,x^{3}-432 a \,b^{4} x^{4}+155 a^{5} m^{3}-1383 a^{4} b \,m^{2} x +1016 a^{3} b^{2} m \,x^{2}+360 a^{2} b^{3} x^{3}+580 a^{5} m^{2}-2106 a^{4} b m x +480 a^{3} b^{2} x^{2}+1044 a^{5} m -1080 a^{4} b x +720 a^{5}\right ) x}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(424\)
parallelrisch	\(\frac {-3 x^{5} \left (e x \right )^{m} a \,b^{4} c^{4} m^{5}-48 x^{5} \left (e x \right )^{m} a \,b^{4} c^{4} m^{4}+2 x^{4} \left (e x \right )^{m} a^{2} b^{3} c^{4} m^{5}-285 x^{5} \left (e x \right )^{m} a \,b^{4} c^{4} m^{3}+34 x^{4} \left (e x \right )^{m} a^{2} b^{3} c^{4} m^{4}+2 x^{3} \left (e x \right )^{m} a^{3} b^{2} c^{4} m^{5}-780 x^{5} \left (e x \right )^{m} a \,b^{4} c^{4} m^{2}+214 x^{4} \left (e x \right )^{m} a^{2} b^{3} c^{4} m^{3}+36 x^{3} \left (e x \right )^{m} a^{3} b^{2} c^{4} m^{4}-3 x^{2} \left (e x \right )^{m} a^{4} b \,c^{4} m^{5}-972 x^{5} \left (e x \right )^{m} a \,b^{4} c^{4} m +614 x^{4} \left (e x \right )^{m} a^{2} b^{3} c^{4} m^{2}+242 x^{3} \left (e x \right )^{m} a^{3} b^{2} c^{4} m^{3}-57 x^{2} \left (e x \right )^{m} a^{4} b \,c^{4} m^{4}+792 x^{4} \left (e x \right )^{m} a^{2} b^{3} c^{4} m +744 x^{3} \left (e x \right )^{m} a^{3} b^{2} c^{4} m^{2}-411 x^{2} \left (e x \right )^{m} a^{4} b \,c^{4} m^{3}+1016 x^{3} \left (e x \right )^{m} a^{3} b^{2} c^{4} m -1383 x^{2} \left (e x \right )^{m} a^{4} b \,c^{4} m^{2}-2106 x^{2} \left (e x \right )^{m} a^{4} b \,c^{4} m +120 x^{6} \left (e x \right )^{m} b^{5} c^{4}+720 x \left (e x \right )^{m} a^{5} c^{4}+x^{6} \left (e x \right )^{m} b^{5} c^{4} m^{5}+15 x^{6} \left (e x \right )^{m} b^{5} c^{4} m^{4}+85 x^{6} \left (e x \right )^{m} b^{5} c^{4} m^{3}+225 x^{6} \left (e x \right )^{m} b^{5} c^{4} m^{2}+274 x^{6} \left (e x \right )^{m} b^{5} c^{4} m +x \left (e x \right )^{m} a^{5} c^{4} m^{5}-432 x^{5} \left (e x \right )^{m} a \,b^{4} c^{4}+20 x \left (e x \right )^{m} a^{5} c^{4} m^{4}+360 x^{4} \left (e x \right )^{m} a^{2} b^{3} c^{4}+155 x \left (e x \right )^{m} a^{5} c^{4} m^{3}+480 x^{3} \left (e x \right )^{m} a^{3} b^{2} c^{4}+580 x \left (e x \right )^{m} a^{5} c^{4} m^{2}-1080 x^{2} \left (e x \right )^{m} a^{4} b \,c^{4}+1044 x \left (e x \right )^{m} a^{5} c^{4} m}{\left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(721\)

[In]

int((e*x)^m*(b*x+a)*(-b*c*x+a*c)^4,x,method=_RETURNVERBOSE)

[Out]

a^5*c^4/(1+m)*x*exp(m*ln(e*x))+b^5*c^4/(6+m)*x^6*exp(m*ln(e*x))-3*a*b^4*c^4/(5+m)*x^5*exp(m*ln(e*x))+2*a^2*c^4
*b^3/(4+m)*x^4*exp(m*ln(e*x))+2*a^3*c^4*b^2/(3+m)*x^3*exp(m*ln(e*x))-3*a^4*c^4*b/(2+m)*x^2*exp(m*ln(e*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (145) = 290\).

Time = 0.24 (sec) , antiderivative size = 477, normalized size of antiderivative = 3.29 \[ \int (e x)^m (a+b x) (a c-b c x)^4 \, dx=\frac {{\left ({\left (b^{5} c^{4} m^{5} + 15 \, b^{5} c^{4} m^{4} + 85 \, b^{5} c^{4} m^{3} + 225 \, b^{5} c^{4} m^{2} + 274 \, b^{5} c^{4} m + 120 \, b^{5} c^{4}\right )} x^{6} - 3 \, {\left (a b^{4} c^{4} m^{5} + 16 \, a b^{4} c^{4} m^{4} + 95 \, a b^{4} c^{4} m^{3} + 260 \, a b^{4} c^{4} m^{2} + 324 \, a b^{4} c^{4} m + 144 \, a b^{4} c^{4}\right )} x^{5} + 2 \, {\left (a^{2} b^{3} c^{4} m^{5} + 17 \, a^{2} b^{3} c^{4} m^{4} + 107 \, a^{2} b^{3} c^{4} m^{3} + 307 \, a^{2} b^{3} c^{4} m^{2} + 396 \, a^{2} b^{3} c^{4} m + 180 \, a^{2} b^{3} c^{4}\right )} x^{4} + 2 \, {\left (a^{3} b^{2} c^{4} m^{5} + 18 \, a^{3} b^{2} c^{4} m^{4} + 121 \, a^{3} b^{2} c^{4} m^{3} + 372 \, a^{3} b^{2} c^{4} m^{2} + 508 \, a^{3} b^{2} c^{4} m + 240 \, a^{3} b^{2} c^{4}\right )} x^{3} - 3 \, {\left (a^{4} b c^{4} m^{5} + 19 \, a^{4} b c^{4} m^{4} + 137 \, a^{4} b c^{4} m^{3} + 461 \, a^{4} b c^{4} m^{2} + 702 \, a^{4} b c^{4} m + 360 \, a^{4} b c^{4}\right )} x^{2} + {\left (a^{5} c^{4} m^{5} + 20 \, a^{5} c^{4} m^{4} + 155 \, a^{5} c^{4} m^{3} + 580 \, a^{5} c^{4} m^{2} + 1044 \, a^{5} c^{4} m + 720 \, a^{5} c^{4}\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]

[In]

integrate((e*x)^m*(b*x+a)*(-b*c*x+a*c)^4,x, algorithm="fricas")

[Out]

((b^5*c^4*m^5 + 15*b^5*c^4*m^4 + 85*b^5*c^4*m^3 + 225*b^5*c^4*m^2 + 274*b^5*c^4*m + 120*b^5*c^4)*x^6 - 3*(a*b^
4*c^4*m^5 + 16*a*b^4*c^4*m^4 + 95*a*b^4*c^4*m^3 + 260*a*b^4*c^4*m^2 + 324*a*b^4*c^4*m + 144*a*b^4*c^4)*x^5 + 2
*(a^2*b^3*c^4*m^5 + 17*a^2*b^3*c^4*m^4 + 107*a^2*b^3*c^4*m^3 + 307*a^2*b^3*c^4*m^2 + 396*a^2*b^3*c^4*m + 180*a
^2*b^3*c^4)*x^4 + 2*(a^3*b^2*c^4*m^5 + 18*a^3*b^2*c^4*m^4 + 121*a^3*b^2*c^4*m^3 + 372*a^3*b^2*c^4*m^2 + 508*a^
3*b^2*c^4*m + 240*a^3*b^2*c^4)*x^3 - 3*(a^4*b*c^4*m^5 + 19*a^4*b*c^4*m^4 + 137*a^4*b*c^4*m^3 + 461*a^4*b*c^4*m
^2 + 702*a^4*b*c^4*m + 360*a^4*b*c^4)*x^2 + (a^5*c^4*m^5 + 20*a^5*c^4*m^4 + 155*a^5*c^4*m^3 + 580*a^5*c^4*m^2
+ 1044*a^5*c^4*m + 720*a^5*c^4)*x)*(e*x)^m/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2276 vs. \(2 (136) = 272\).

Time = 0.52 (sec) , antiderivative size = 2276, normalized size of antiderivative = 15.70 \[ \int (e x)^m (a+b x) (a c-b c x)^4 \, dx=\text {Too large to display} \]

[In]

integrate((e*x)**m*(b*x+a)*(-b*c*x+a*c)**4,x)

[Out]

Piecewise(((-a**5*c**4/(5*x**5) + 3*a**4*b*c**4/(4*x**4) - 2*a**3*b**2*c**4/(3*x**3) - a**2*b**3*c**4/x**2 + 3
*a*b**4*c**4/x + b**5*c**4*log(x))/e**6, Eq(m, -6)), ((-a**5*c**4/(4*x**4) + a**4*b*c**4/x**3 - a**3*b**2*c**4
/x**2 - 2*a**2*b**3*c**4/x - 3*a*b**4*c**4*log(x) + b**5*c**4*x)/e**5, Eq(m, -5)), ((-a**5*c**4/(3*x**3) + 3*a
**4*b*c**4/(2*x**2) - 2*a**3*b**2*c**4/x + 2*a**2*b**3*c**4*log(x) - 3*a*b**4*c**4*x + b**5*c**4*x**2/2)/e**4,
Eq(m, -4)), ((-a**5*c**4/(2*x**2) + 3*a**4*b*c**4/x + 2*a**3*b**2*c**4*log(x) + 2*a**2*b**3*c**4*x - 3*a*b**4
*c**4*x**2/2 + b**5*c**4*x**3/3)/e**3, Eq(m, -3)), ((-a**5*c**4/x - 3*a**4*b*c**4*log(x) + 2*a**3*b**2*c**4*x
+ a**2*b**3*c**4*x**2 - a*b**4*c**4*x**3 + b**5*c**4*x**4/4)/e**2, Eq(m, -2)), ((a**5*c**4*log(x) - 3*a**4*b*c
**4*x + a**3*b**2*c**4*x**2 + 2*a**2*b**3*c**4*x**3/3 - 3*a*b**4*c**4*x**4/4 + b**5*c**4*x**5/5)/e, Eq(m, -1))
, (a**5*c**4*m**5*x*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 20*a**5*c**4*
m**4*x*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 155*a**5*c**4*m**3*x*(e*x)
**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 580*a**5*c**4*m**2*x*(e*x)**m/(m**6 +
21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 1044*a**5*c**4*m*x*(e*x)**m/(m**6 + 21*m**5 + 175*
m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 720*a**5*c**4*x*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 +
1624*m**2 + 1764*m + 720) - 3*a**4*b*c**4*m**5*x**2*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**
2 + 1764*m + 720) - 57*a**4*b*c**4*m**4*x**2*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764
*m + 720) - 411*a**4*b*c**4*m**3*x**2*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 72
0) - 1383*a**4*b*c**4*m**2*x**2*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 2
106*a**4*b*c**4*m*x**2*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 1080*a**4*
b*c**4*x**2*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 2*a**3*b**2*c**4*m**5
*x**3*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 36*a**3*b**2*c**4*m**4*x**3
*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 242*a**3*b**2*c**4*m**3*x**3*(e*
x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 744*a**3*b**2*c**4*m**2*x**3*(e*x)**
m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 1016*a**3*b**2*c**4*m*x**3*(e*x)**m/(m**
6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 480*a**3*b**2*c**4*x**3*(e*x)**m/(m**6 + 21*m*
*5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 2*a**2*b**3*c**4*m**5*x**4*(e*x)**m/(m**6 + 21*m**5 + 1
75*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 34*a**2*b**3*c**4*m**4*x**4*(e*x)**m/(m**6 + 21*m**5 + 175*m*
*4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 214*a**2*b**3*c**4*m**3*x**4*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 +
735*m**3 + 1624*m**2 + 1764*m + 720) + 614*a**2*b**3*c**4*m**2*x**4*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735
*m**3 + 1624*m**2 + 1764*m + 720) + 792*a**2*b**3*c**4*m*x**4*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 +
1624*m**2 + 1764*m + 720) + 360*a**2*b**3*c**4*x**4*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**
2 + 1764*m + 720) - 3*a*b**4*c**4*m**5*x**5*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*
m + 720) - 48*a*b**4*c**4*m**4*x**5*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720)
- 285*a*b**4*c**4*m**3*x**5*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 780*
a*b**4*c**4*m**2*x**5*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 972*a*b**4*
c**4*m*x**5*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 432*a*b**4*c**4*x**5*
(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + b**5*c**4*m**5*x**6*(e*x)**m/(m**
6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 15*b**5*c**4*m**4*x**6*(e*x)**m/(m**6 + 21*m**
5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 85*b**5*c**4*m**3*x**6*(e*x)**m/(m**6 + 21*m**5 + 175*m*
*4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 225*b**5*c**4*m**2*x**6*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*
m**3 + 1624*m**2 + 1764*m + 720) + 274*b**5*c**4*m*x**6*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*
m**2 + 1764*m + 720) + 120*b**5*c**4*x**6*(e*x)**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m
+ 720), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.97 \[ \int (e x)^m (a+b x) (a c-b c x)^4 \, dx=\frac {b^{5} c^{4} e^{m} x^{6} x^{m}}{m + 6} - \frac {3 \, a b^{4} c^{4} e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a^{2} b^{3} c^{4} e^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, a^{3} b^{2} c^{4} e^{m} x^{3} x^{m}}{m + 3} - \frac {3 \, a^{4} b c^{4} e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} a^{5} c^{4}}{e {\left (m + 1\right )}} \]

[In]

integrate((e*x)^m*(b*x+a)*(-b*c*x+a*c)^4,x, algorithm="maxima")

[Out]

b^5*c^4*e^m*x^6*x^m/(m + 6) - 3*a*b^4*c^4*e^m*x^5*x^m/(m + 5) + 2*a^2*b^3*c^4*e^m*x^4*x^m/(m + 4) + 2*a^3*b^2*
c^4*e^m*x^3*x^m/(m + 3) - 3*a^4*b*c^4*e^m*x^2*x^m/(m + 2) + (e*x)^(m + 1)*a^5*c^4/(e*(m + 1))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 720 vs. \(2 (145) = 290\).

Time = 0.33 (sec) , antiderivative size = 720, normalized size of antiderivative = 4.97 \[ \int (e x)^m (a+b x) (a c-b c x)^4 \, dx=\frac {\left (e x\right )^{m} b^{5} c^{4} m^{5} x^{6} - 3 \, \left (e x\right )^{m} a b^{4} c^{4} m^{5} x^{5} + 15 \, \left (e x\right )^{m} b^{5} c^{4} m^{4} x^{6} + 2 \, \left (e x\right )^{m} a^{2} b^{3} c^{4} m^{5} x^{4} - 48 \, \left (e x\right )^{m} a b^{4} c^{4} m^{4} x^{5} + 85 \, \left (e x\right )^{m} b^{5} c^{4} m^{3} x^{6} + 2 \, \left (e x\right )^{m} a^{3} b^{2} c^{4} m^{5} x^{3} + 34 \, \left (e x\right )^{m} a^{2} b^{3} c^{4} m^{4} x^{4} - 285 \, \left (e x\right )^{m} a b^{4} c^{4} m^{3} x^{5} + 225 \, \left (e x\right )^{m} b^{5} c^{4} m^{2} x^{6} - 3 \, \left (e x\right )^{m} a^{4} b c^{4} m^{5} x^{2} + 36 \, \left (e x\right )^{m} a^{3} b^{2} c^{4} m^{4} x^{3} + 214 \, \left (e x\right )^{m} a^{2} b^{3} c^{4} m^{3} x^{4} - 780 \, \left (e x\right )^{m} a b^{4} c^{4} m^{2} x^{5} + 274 \, \left (e x\right )^{m} b^{5} c^{4} m x^{6} + \left (e x\right )^{m} a^{5} c^{4} m^{5} x - 57 \, \left (e x\right )^{m} a^{4} b c^{4} m^{4} x^{2} + 242 \, \left (e x\right )^{m} a^{3} b^{2} c^{4} m^{3} x^{3} + 614 \, \left (e x\right )^{m} a^{2} b^{3} c^{4} m^{2} x^{4} - 972 \, \left (e x\right )^{m} a b^{4} c^{4} m x^{5} + 120 \, \left (e x\right )^{m} b^{5} c^{4} x^{6} + 20 \, \left (e x\right )^{m} a^{5} c^{4} m^{4} x - 411 \, \left (e x\right )^{m} a^{4} b c^{4} m^{3} x^{2} + 744 \, \left (e x\right )^{m} a^{3} b^{2} c^{4} m^{2} x^{3} + 792 \, \left (e x\right )^{m} a^{2} b^{3} c^{4} m x^{4} - 432 \, \left (e x\right )^{m} a b^{4} c^{4} x^{5} + 155 \, \left (e x\right )^{m} a^{5} c^{4} m^{3} x - 1383 \, \left (e x\right )^{m} a^{4} b c^{4} m^{2} x^{2} + 1016 \, \left (e x\right )^{m} a^{3} b^{2} c^{4} m x^{3} + 360 \, \left (e x\right )^{m} a^{2} b^{3} c^{4} x^{4} + 580 \, \left (e x\right )^{m} a^{5} c^{4} m^{2} x - 2106 \, \left (e x\right )^{m} a^{4} b c^{4} m x^{2} + 480 \, \left (e x\right )^{m} a^{3} b^{2} c^{4} x^{3} + 1044 \, \left (e x\right )^{m} a^{5} c^{4} m x - 1080 \, \left (e x\right )^{m} a^{4} b c^{4} x^{2} + 720 \, \left (e x\right )^{m} a^{5} c^{4} x}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \]

[In]

integrate((e*x)^m*(b*x+a)*(-b*c*x+a*c)^4,x, algorithm="giac")

[Out]

((e*x)^m*b^5*c^4*m^5*x^6 - 3*(e*x)^m*a*b^4*c^4*m^5*x^5 + 15*(e*x)^m*b^5*c^4*m^4*x^6 + 2*(e*x)^m*a^2*b^3*c^4*m^
5*x^4 - 48*(e*x)^m*a*b^4*c^4*m^4*x^5 + 85*(e*x)^m*b^5*c^4*m^3*x^6 + 2*(e*x)^m*a^3*b^2*c^4*m^5*x^3 + 34*(e*x)^m
*a^2*b^3*c^4*m^4*x^4 - 285*(e*x)^m*a*b^4*c^4*m^3*x^5 + 225*(e*x)^m*b^5*c^4*m^2*x^6 - 3*(e*x)^m*a^4*b*c^4*m^5*x
^2 + 36*(e*x)^m*a^3*b^2*c^4*m^4*x^3 + 214*(e*x)^m*a^2*b^3*c^4*m^3*x^4 - 780*(e*x)^m*a*b^4*c^4*m^2*x^5 + 274*(e
*x)^m*b^5*c^4*m*x^6 + (e*x)^m*a^5*c^4*m^5*x - 57*(e*x)^m*a^4*b*c^4*m^4*x^2 + 242*(e*x)^m*a^3*b^2*c^4*m^3*x^3 +
614*(e*x)^m*a^2*b^3*c^4*m^2*x^4 - 972*(e*x)^m*a*b^4*c^4*m*x^5 + 120*(e*x)^m*b^5*c^4*x^6 + 20*(e*x)^m*a^5*c^4*
m^4*x - 411*(e*x)^m*a^4*b*c^4*m^3*x^2 + 744*(e*x)^m*a^3*b^2*c^4*m^2*x^3 + 792*(e*x)^m*a^2*b^3*c^4*m*x^4 - 432*
(e*x)^m*a*b^4*c^4*x^5 + 155*(e*x)^m*a^5*c^4*m^3*x - 1383*(e*x)^m*a^4*b*c^4*m^2*x^2 + 1016*(e*x)^m*a^3*b^2*c^4*
m*x^3 + 360*(e*x)^m*a^2*b^3*c^4*x^4 + 580*(e*x)^m*a^5*c^4*m^2*x - 2106*(e*x)^m*a^4*b*c^4*m*x^2 + 480*(e*x)^m*a
^3*b^2*c^4*x^3 + 1044*(e*x)^m*a^5*c^4*m*x - 1080*(e*x)^m*a^4*b*c^4*x^2 + 720*(e*x)^m*a^5*c^4*x)/(m^6 + 21*m^5
+ 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)

Mupad [B] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.72 \[ \int (e x)^m (a+b x) (a c-b c x)^4 \, dx={\left (e\,x\right )}^m\,\left (\frac {b^5\,c^4\,x^6\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {a^5\,c^4\,x\,\left (m^5+20\,m^4+155\,m^3+580\,m^2+1044\,m+720\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {3\,a\,b^4\,c^4\,x^5\,\left (m^5+16\,m^4+95\,m^3+260\,m^2+324\,m+144\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}-\frac {3\,a^4\,b\,c^4\,x^2\,\left (m^5+19\,m^4+137\,m^3+461\,m^2+702\,m+360\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {2\,a^2\,b^3\,c^4\,x^4\,\left (m^5+17\,m^4+107\,m^3+307\,m^2+396\,m+180\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}+\frac {2\,a^3\,b^2\,c^4\,x^3\,\left (m^5+18\,m^4+121\,m^3+372\,m^2+508\,m+240\right )}{m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720}\right ) \]

[In]

int((a*c - b*c*x)^4*(e*x)^m*(a + b*x),x)

[Out]

(e*x)^m*((b^5*c^4*x^6*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4
+ 21*m^5 + m^6 + 720) + (a^5*c^4*x*(1044*m + 580*m^2 + 155*m^3 + 20*m^4 + m^5 + 720))/(1764*m + 1624*m^2 + 735
*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) - (3*a*b^4*c^4*x^5*(324*m + 260*m^2 + 95*m^3 + 16*m^4 + m^5 + 144))/(1764
*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) - (3*a^4*b*c^4*x^2*(702*m + 461*m^2 + 137*m^3 + 19*m^4
+ m^5 + 360))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (2*a^2*b^3*c^4*x^4*(396*m + 307*
m^2 + 107*m^3 + 17*m^4 + m^5 + 180))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720) + (2*a^3*b^2
*c^4*x^3*(508*m + 372*m^2 + 121*m^3 + 18*m^4 + m^5 + 240))/(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m
^6 + 720))