\(\int (e x)^m (a+b x)^4 (a c-b c x)^3 \, dx\) [301]

Optimal result

Integrand size = 24, antiderivative size = 197 \[ \int (e x)^m (a+b x)^4 (a c-b c x)^3 \, dx=\frac {a^7 c^3 (e x)^{1+m}}{e (1+m)}+\frac {a^6 b c^3 (e x)^{2+m}}{e^2 (2+m)}-\frac {3 a^5 b^2 c^3 (e x)^{3+m}}{e^3 (3+m)}-\frac {3 a^4 b^3 c^3 (e x)^{4+m}}{e^4 (4+m)}+\frac {3 a^3 b^4 c^3 (e x)^{5+m}}{e^5 (5+m)}+\frac {3 a^2 b^5 c^3 (e x)^{6+m}}{e^6 (6+m)}-\frac {a b^6 c^3 (e x)^{7+m}}{e^7 (7+m)}-\frac {b^7 c^3 (e x)^{8+m}}{e^8 (8+m)} \] Output:

a^7*c^3*(e*x)^(1+m)/e/(1+m)+a^6*b*c^3*(e*x)^(2+m)/e^2/(2+m)-3*a^5*b^2*c^3* 
(e*x)^(3+m)/e^3/(3+m)-3*a^4*b^3*c^3*(e*x)^(4+m)/e^4/(4+m)+3*a^3*b^4*c^3*(e 
*x)^(5+m)/e^5/(5+m)+3*a^2*b^5*c^3*(e*x)^(6+m)/e^6/(6+m)-a*b^6*c^3*(e*x)^(7 
+m)/e^7/(7+m)-b^7*c^3*(e*x)^(8+m)/e^8/(8+m)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.62 \[ \int (e x)^m (a+b x)^4 (a c-b c x)^3 \, dx=c^3 x (e x)^m \left (\frac {a^7}{1+m}+\frac {a^6 b x}{2+m}-\frac {3 a^5 b^2 x^2}{3+m}-\frac {3 a^4 b^3 x^3}{4+m}+\frac {3 a^3 b^4 x^4}{5+m}+\frac {3 a^2 b^5 x^5}{6+m}-\frac {a b^6 x^6}{7+m}-\frac {b^7 x^7}{8+m}\right ) \] Input:

Integrate[(e*x)^m*(a + b*x)^4*(a*c - b*c*x)^3,x]
 

Output:

c^3*x*(e*x)^m*(a^7/(1 + m) + (a^6*b*x)/(2 + m) - (3*a^5*b^2*x^2)/(3 + m) - 
 (3*a^4*b^3*x^3)/(4 + m) + (3*a^3*b^4*x^4)/(5 + m) + (3*a^2*b^5*x^5)/(6 + 
m) - (a*b^6*x^6)/(7 + m) - (b^7*x^7)/(8 + m))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {99, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (a+b x)^4 (e x)^m (a c-b c x)^3 \, dx\)

\(\Big \downarrow \) 99

\(\displaystyle \int \left (a^7 c^3 (e x)^m+\frac {a^6 b c^3 (e x)^{m+1}}{e}-\frac {3 a^5 b^2 c^3 (e x)^{m+2}}{e^2}-\frac {3 a^4 b^3 c^3 (e x)^{m+3}}{e^3}+\frac {3 a^3 b^4 c^3 (e x)^{m+4}}{e^4}+\frac {3 a^2 b^5 c^3 (e x)^{m+5}}{e^5}-\frac {a b^6 c^3 (e x)^{m+6}}{e^6}-\frac {b^7 c^3 (e x)^{m+7}}{e^7}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^7 c^3 (e x)^{m+1}}{e (m+1)}+\frac {a^6 b c^3 (e x)^{m+2}}{e^2 (m+2)}-\frac {3 a^5 b^2 c^3 (e x)^{m+3}}{e^3 (m+3)}-\frac {3 a^4 b^3 c^3 (e x)^{m+4}}{e^4 (m+4)}+\frac {3 a^3 b^4 c^3 (e x)^{m+5}}{e^5 (m+5)}+\frac {3 a^2 b^5 c^3 (e x)^{m+6}}{e^6 (m+6)}-\frac {a b^6 c^3 (e x)^{m+7}}{e^7 (m+7)}-\frac {b^7 c^3 (e x)^{m+8}}{e^8 (m+8)}\)

Input:

Int[(e*x)^m*(a + b*x)^4*(a*c - b*c*x)^3,x]
 

Output:

(a^7*c^3*(e*x)^(1 + m))/(e*(1 + m)) + (a^6*b*c^3*(e*x)^(2 + m))/(e^2*(2 + 
m)) - (3*a^5*b^2*c^3*(e*x)^(3 + m))/(e^3*(3 + m)) - (3*a^4*b^3*c^3*(e*x)^( 
4 + m))/(e^4*(4 + m)) + (3*a^3*b^4*c^3*(e*x)^(5 + m))/(e^5*(5 + m)) + (3*a 
^2*b^5*c^3*(e*x)^(6 + m))/(e^6*(6 + m)) - (a*b^6*c^3*(e*x)^(7 + m))/(e^7*( 
7 + m)) - (b^7*c^3*(e*x)^(8 + m))/(e^8*(8 + m))
 

Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {a^{7} c^{3} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {a^{6} b \,c^{3} x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}-\frac {b^{7} c^{3} x^{8} {\mathrm e}^{m \ln \left (e x \right )}}{8+m}-\frac {a \,b^{6} c^{3} x^{7} {\mathrm e}^{m \ln \left (e x \right )}}{7+m}+\frac {3 a^{2} b^{5} c^{3} x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}+\frac {3 a^{3} b^{4} c^{3} x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}-\frac {3 a^{4} b^{3} c^{3} x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}-\frac {3 a^{5} b^{2} c^{3} x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}\)	\(196\)
gosper	\(\frac {c^{3} \left (e x \right )^{m} \left (-b^{7} m^{7} x^{7}-a \,b^{6} m^{7} x^{6}-28 b^{7} m^{6} x^{7}+3 a^{2} b^{5} m^{7} x^{5}-29 a \,b^{6} m^{6} x^{6}-322 b^{7} m^{5} x^{7}+3 a^{3} b^{4} m^{7} x^{4}+90 a^{2} b^{5} m^{6} x^{5}-343 a \,b^{6} m^{5} x^{6}-1960 b^{7} m^{4} x^{7}-3 a^{4} b^{3} m^{7} x^{3}+93 a^{3} b^{4} m^{6} x^{4}+1098 a^{2} b^{5} m^{5} x^{5}-2135 a \,b^{6} m^{4} x^{6}-6769 b^{7} m^{3} x^{7}-3 a^{5} b^{2} m^{7} x^{2}-96 a^{4} b^{3} m^{6} x^{3}+1173 a^{3} b^{4} m^{5} x^{4}+7020 a^{2} b^{5} m^{4} x^{5}-7504 a \,b^{6} m^{3} x^{6}-13132 b^{7} m^{2} x^{7}+a^{6} b \,m^{7} x -99 a^{5} b^{2} m^{6} x^{2}-1254 a^{4} b^{3} m^{5} x^{3}+7743 a^{3} b^{4} m^{4} x^{4}+25227 a^{2} b^{5} m^{3} x^{5}-14756 a \,b^{6} m^{2} x^{6}-13068 m \,x^{7} b^{7}+a^{7} m^{7}+34 a^{6} b \,m^{6} x -1341 a^{5} b^{2} m^{5} x^{2}-8592 a^{4} b^{3} m^{4} x^{3}+28632 a^{3} b^{4} m^{3} x^{4}+50490 a^{2} b^{5} m^{2} x^{5}-14832 a \,b^{6} m \,x^{6}-5040 b^{7} x^{7}+35 a^{7} m^{6}+478 a^{6} b \,m^{5} x -9585 a^{5} b^{2} m^{4} x^{2}-32979 a^{4} b^{3} m^{3} x^{3}+58692 a^{3} b^{4} m^{2} x^{4}+51432 a^{2} b^{5} m \,x^{5}-5760 a \,b^{6} x^{6}+511 a^{7} m^{5}+3580 a^{6} b \,m^{4} x -38592 a^{5} b^{2} m^{3} x^{2}-69936 a^{4} b^{3} m^{2} x^{3}+60912 a^{3} b^{4} m \,x^{4}+20160 a^{2} b^{5} x^{5}+4025 a^{7} m^{4}+15289 a^{6} b \,m^{3} x -86076 a^{5} b^{2} m^{2} x^{2}-74628 a^{4} b^{3} m \,x^{3}+24192 a^{3} b^{4} x^{4}+18424 a^{7} m^{3}+36706 a^{6} b \,m^{2} x -96144 a^{5} b^{2} m \,x^{2}-30240 a^{4} b^{3} x^{3}+48860 a^{7} m^{2}+44712 a^{6} b m x -40320 a^{5} b^{2} x^{2}+69264 a^{7} m +20160 a^{6} b x +40320 a^{7}\right ) x}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(786\)
risch	\(\frac {c^{3} \left (e x \right )^{m} \left (-b^{7} m^{7} x^{7}-a \,b^{6} m^{7} x^{6}-28 b^{7} m^{6} x^{7}+3 a^{2} b^{5} m^{7} x^{5}-29 a \,b^{6} m^{6} x^{6}-322 b^{7} m^{5} x^{7}+3 a^{3} b^{4} m^{7} x^{4}+90 a^{2} b^{5} m^{6} x^{5}-343 a \,b^{6} m^{5} x^{6}-1960 b^{7} m^{4} x^{7}-3 a^{4} b^{3} m^{7} x^{3}+93 a^{3} b^{4} m^{6} x^{4}+1098 a^{2} b^{5} m^{5} x^{5}-2135 a \,b^{6} m^{4} x^{6}-6769 b^{7} m^{3} x^{7}-3 a^{5} b^{2} m^{7} x^{2}-96 a^{4} b^{3} m^{6} x^{3}+1173 a^{3} b^{4} m^{5} x^{4}+7020 a^{2} b^{5} m^{4} x^{5}-7504 a \,b^{6} m^{3} x^{6}-13132 b^{7} m^{2} x^{7}+a^{6} b \,m^{7} x -99 a^{5} b^{2} m^{6} x^{2}-1254 a^{4} b^{3} m^{5} x^{3}+7743 a^{3} b^{4} m^{4} x^{4}+25227 a^{2} b^{5} m^{3} x^{5}-14756 a \,b^{6} m^{2} x^{6}-13068 m \,x^{7} b^{7}+a^{7} m^{7}+34 a^{6} b \,m^{6} x -1341 a^{5} b^{2} m^{5} x^{2}-8592 a^{4} b^{3} m^{4} x^{3}+28632 a^{3} b^{4} m^{3} x^{4}+50490 a^{2} b^{5} m^{2} x^{5}-14832 a \,b^{6} m \,x^{6}-5040 b^{7} x^{7}+35 a^{7} m^{6}+478 a^{6} b \,m^{5} x -9585 a^{5} b^{2} m^{4} x^{2}-32979 a^{4} b^{3} m^{3} x^{3}+58692 a^{3} b^{4} m^{2} x^{4}+51432 a^{2} b^{5} m \,x^{5}-5760 a \,b^{6} x^{6}+511 a^{7} m^{5}+3580 a^{6} b \,m^{4} x -38592 a^{5} b^{2} m^{3} x^{2}-69936 a^{4} b^{3} m^{2} x^{3}+60912 a^{3} b^{4} m \,x^{4}+20160 a^{2} b^{5} x^{5}+4025 a^{7} m^{4}+15289 a^{6} b \,m^{3} x -86076 a^{5} b^{2} m^{2} x^{2}-74628 a^{4} b^{3} m \,x^{3}+24192 a^{3} b^{4} x^{4}+18424 a^{7} m^{3}+36706 a^{6} b \,m^{2} x -96144 a^{5} b^{2} m \,x^{2}-30240 a^{4} b^{3} x^{3}+48860 a^{7} m^{2}+44712 a^{6} b m x -40320 a^{5} b^{2} x^{2}+69264 a^{7} m +20160 a^{6} b x +40320 a^{7}\right ) x}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(786\)
orering	\(\frac {\left (-b^{7} m^{7} x^{7}-a \,b^{6} m^{7} x^{6}-28 b^{7} m^{6} x^{7}+3 a^{2} b^{5} m^{7} x^{5}-29 a \,b^{6} m^{6} x^{6}-322 b^{7} m^{5} x^{7}+3 a^{3} b^{4} m^{7} x^{4}+90 a^{2} b^{5} m^{6} x^{5}-343 a \,b^{6} m^{5} x^{6}-1960 b^{7} m^{4} x^{7}-3 a^{4} b^{3} m^{7} x^{3}+93 a^{3} b^{4} m^{6} x^{4}+1098 a^{2} b^{5} m^{5} x^{5}-2135 a \,b^{6} m^{4} x^{6}-6769 b^{7} m^{3} x^{7}-3 a^{5} b^{2} m^{7} x^{2}-96 a^{4} b^{3} m^{6} x^{3}+1173 a^{3} b^{4} m^{5} x^{4}+7020 a^{2} b^{5} m^{4} x^{5}-7504 a \,b^{6} m^{3} x^{6}-13132 b^{7} m^{2} x^{7}+a^{6} b \,m^{7} x -99 a^{5} b^{2} m^{6} x^{2}-1254 a^{4} b^{3} m^{5} x^{3}+7743 a^{3} b^{4} m^{4} x^{4}+25227 a^{2} b^{5} m^{3} x^{5}-14756 a \,b^{6} m^{2} x^{6}-13068 m \,x^{7} b^{7}+a^{7} m^{7}+34 a^{6} b \,m^{6} x -1341 a^{5} b^{2} m^{5} x^{2}-8592 a^{4} b^{3} m^{4} x^{3}+28632 a^{3} b^{4} m^{3} x^{4}+50490 a^{2} b^{5} m^{2} x^{5}-14832 a \,b^{6} m \,x^{6}-5040 b^{7} x^{7}+35 a^{7} m^{6}+478 a^{6} b \,m^{5} x -9585 a^{5} b^{2} m^{4} x^{2}-32979 a^{4} b^{3} m^{3} x^{3}+58692 a^{3} b^{4} m^{2} x^{4}+51432 a^{2} b^{5} m \,x^{5}-5760 a \,b^{6} x^{6}+511 a^{7} m^{5}+3580 a^{6} b \,m^{4} x -38592 a^{5} b^{2} m^{3} x^{2}-69936 a^{4} b^{3} m^{2} x^{3}+60912 a^{3} b^{4} m \,x^{4}+20160 a^{2} b^{5} x^{5}+4025 a^{7} m^{4}+15289 a^{6} b \,m^{3} x -86076 a^{5} b^{2} m^{2} x^{2}-74628 a^{4} b^{3} m \,x^{3}+24192 a^{3} b^{4} x^{4}+18424 a^{7} m^{3}+36706 a^{6} b \,m^{2} x -96144 a^{5} b^{2} m \,x^{2}-30240 a^{4} b^{3} x^{3}+48860 a^{7} m^{2}+44712 a^{6} b m x -40320 a^{5} b^{2} x^{2}+69264 a^{7} m +20160 a^{6} b x +40320 a^{7}\right ) x \left (e x \right )^{m} \left (-b c x +a c \right )^{3}}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) \left (-b x +a \right )^{3}}\)	\(802\)
parallelrisch	\(\text {Expression too large to display}\)	\(1314\)

Input:

int((e*x)^m*(b*x+a)^4*(-b*c*x+a*c)^3,x,method=_RETURNVERBOSE)
 

Output:

a^7*c^3/(1+m)*x*exp(m*ln(e*x))+a^6*b*c^3/(2+m)*x^2*exp(m*ln(e*x))-b^7*c^3/ 
(8+m)*x^8*exp(m*ln(e*x))-a*b^6*c^3/(7+m)*x^7*exp(m*ln(e*x))+3*a^2*b^5*c^3/ 
(6+m)*x^6*exp(m*ln(e*x))+3*a^3*b^4*c^3/(5+m)*x^5*exp(m*ln(e*x))-3*a^4*b^3* 
c^3/(4+m)*x^4*exp(m*ln(e*x))-3*a^5*b^2*c^3/(3+m)*x^3*exp(m*ln(e*x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 860 vs. \(2 (197) = 394\).

Time = 0.08 (sec) , antiderivative size = 860, normalized size of antiderivative = 4.37 \[ \int (e x)^m (a+b x)^4 (a c-b c x)^3 \, dx =\text {Too large to display} \] Input:

integrate((e*x)^m*(b*x+a)^4*(-b*c*x+a*c)^3,x, algorithm="fricas")
 

Output:

-((b^7*c^3*m^7 + 28*b^7*c^3*m^6 + 322*b^7*c^3*m^5 + 1960*b^7*c^3*m^4 + 676 
9*b^7*c^3*m^3 + 13132*b^7*c^3*m^2 + 13068*b^7*c^3*m + 5040*b^7*c^3)*x^8 + 
(a*b^6*c^3*m^7 + 29*a*b^6*c^3*m^6 + 343*a*b^6*c^3*m^5 + 2135*a*b^6*c^3*m^4 
 + 7504*a*b^6*c^3*m^3 + 14756*a*b^6*c^3*m^2 + 14832*a*b^6*c^3*m + 5760*a*b 
^6*c^3)*x^7 - 3*(a^2*b^5*c^3*m^7 + 30*a^2*b^5*c^3*m^6 + 366*a^2*b^5*c^3*m^ 
5 + 2340*a^2*b^5*c^3*m^4 + 8409*a^2*b^5*c^3*m^3 + 16830*a^2*b^5*c^3*m^2 + 
17144*a^2*b^5*c^3*m + 6720*a^2*b^5*c^3)*x^6 - 3*(a^3*b^4*c^3*m^7 + 31*a^3* 
b^4*c^3*m^6 + 391*a^3*b^4*c^3*m^5 + 2581*a^3*b^4*c^3*m^4 + 9544*a^3*b^4*c^ 
3*m^3 + 19564*a^3*b^4*c^3*m^2 + 20304*a^3*b^4*c^3*m + 8064*a^3*b^4*c^3)*x^ 
5 + 3*(a^4*b^3*c^3*m^7 + 32*a^4*b^3*c^3*m^6 + 418*a^4*b^3*c^3*m^5 + 2864*a 
^4*b^3*c^3*m^4 + 10993*a^4*b^3*c^3*m^3 + 23312*a^4*b^3*c^3*m^2 + 24876*a^4 
*b^3*c^3*m + 10080*a^4*b^3*c^3)*x^4 + 3*(a^5*b^2*c^3*m^7 + 33*a^5*b^2*c^3* 
m^6 + 447*a^5*b^2*c^3*m^5 + 3195*a^5*b^2*c^3*m^4 + 12864*a^5*b^2*c^3*m^3 + 
 28692*a^5*b^2*c^3*m^2 + 32048*a^5*b^2*c^3*m + 13440*a^5*b^2*c^3)*x^3 - (a 
^6*b*c^3*m^7 + 34*a^6*b*c^3*m^6 + 478*a^6*b*c^3*m^5 + 3580*a^6*b*c^3*m^4 + 
 15289*a^6*b*c^3*m^3 + 36706*a^6*b*c^3*m^2 + 44712*a^6*b*c^3*m + 20160*a^6 
*b*c^3)*x^2 - (a^7*c^3*m^7 + 35*a^7*c^3*m^6 + 511*a^7*c^3*m^5 + 4025*a^7*c 
^3*m^4 + 18424*a^7*c^3*m^3 + 48860*a^7*c^3*m^2 + 69264*a^7*c^3*m + 40320*a 
^7*c^3)*x)*(e*x)^m/(m^8 + 36*m^7 + 546*m^6 + 4536*m^5 + 22449*m^4 + 67284* 
m^3 + 118124*m^2 + 109584*m + 40320)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4779 vs. \(2 (184) = 368\).

Time = 0.71 (sec) , antiderivative size = 4779, normalized size of antiderivative = 24.26 \[ \int (e x)^m (a+b x)^4 (a c-b c x)^3 \, dx=\text {Too large to display} \] Input:

integrate((e*x)**m*(b*x+a)**4*(-b*c*x+a*c)**3,x)
 

Output:

Piecewise(((-a**7*c**3/(7*x**7) - a**6*b*c**3/(6*x**6) + 3*a**5*b**2*c**3/ 
(5*x**5) + 3*a**4*b**3*c**3/(4*x**4) - a**3*b**4*c**3/x**3 - 3*a**2*b**5*c 
**3/(2*x**2) + a*b**6*c**3/x - b**7*c**3*log(x))/e**8, Eq(m, -8)), ((-a**7 
*c**3/(6*x**6) - a**6*b*c**3/(5*x**5) + 3*a**5*b**2*c**3/(4*x**4) + a**4*b 
**3*c**3/x**3 - 3*a**3*b**4*c**3/(2*x**2) - 3*a**2*b**5*c**3/x - a*b**6*c* 
*3*log(x) - b**7*c**3*x)/e**7, Eq(m, -7)), ((-a**7*c**3/(5*x**5) - a**6*b* 
c**3/(4*x**4) + a**5*b**2*c**3/x**3 + 3*a**4*b**3*c**3/(2*x**2) - 3*a**3*b 
**4*c**3/x + 3*a**2*b**5*c**3*log(x) - a*b**6*c**3*x - b**7*c**3*x**2/2)/e 
**6, Eq(m, -6)), ((-a**7*c**3/(4*x**4) - a**6*b*c**3/(3*x**3) + 3*a**5*b** 
2*c**3/(2*x**2) + 3*a**4*b**3*c**3/x + 3*a**3*b**4*c**3*log(x) + 3*a**2*b* 
*5*c**3*x - a*b**6*c**3*x**2/2 - b**7*c**3*x**3/3)/e**5, Eq(m, -5)), ((-a* 
*7*c**3/(3*x**3) - a**6*b*c**3/(2*x**2) + 3*a**5*b**2*c**3/x - 3*a**4*b**3 
*c**3*log(x) + 3*a**3*b**4*c**3*x + 3*a**2*b**5*c**3*x**2/2 - a*b**6*c**3* 
x**3/3 - b**7*c**3*x**4/4)/e**4, Eq(m, -4)), ((-a**7*c**3/(2*x**2) - a**6* 
b*c**3/x - 3*a**5*b**2*c**3*log(x) - 3*a**4*b**3*c**3*x + 3*a**3*b**4*c**3 
*x**2/2 + a**2*b**5*c**3*x**3 - a*b**6*c**3*x**4/4 - b**7*c**3*x**5/5)/e** 
3, Eq(m, -3)), ((-a**7*c**3/x + a**6*b*c**3*log(x) - 3*a**5*b**2*c**3*x - 
3*a**4*b**3*c**3*x**2/2 + a**3*b**4*c**3*x**3 + 3*a**2*b**5*c**3*x**4/4 - 
a*b**6*c**3*x**5/5 - b**7*c**3*x**6/6)/e**2, Eq(m, -2)), ((a**7*c**3*log(x 
) + a**6*b*c**3*x - 3*a**5*b**2*c**3*x**2/2 - a**4*b**3*c**3*x**3 + 3*a...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.96 \[ \int (e x)^m (a+b x)^4 (a c-b c x)^3 \, dx=-\frac {b^{7} c^{3} e^{m} x^{8} x^{m}}{m + 8} - \frac {a b^{6} c^{3} e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a^{2} b^{5} c^{3} e^{m} x^{6} x^{m}}{m + 6} + \frac {3 \, a^{3} b^{4} c^{3} e^{m} x^{5} x^{m}}{m + 5} - \frac {3 \, a^{4} b^{3} c^{3} e^{m} x^{4} x^{m}}{m + 4} - \frac {3 \, a^{5} b^{2} c^{3} e^{m} x^{3} x^{m}}{m + 3} + \frac {a^{6} b c^{3} e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} a^{7} c^{3}}{e {\left (m + 1\right )}} \] Input:

integrate((e*x)^m*(b*x+a)^4*(-b*c*x+a*c)^3,x, algorithm="maxima")
 

Output:

-b^7*c^3*e^m*x^8*x^m/(m + 8) - a*b^6*c^3*e^m*x^7*x^m/(m + 7) + 3*a^2*b^5*c 
^3*e^m*x^6*x^m/(m + 6) + 3*a^3*b^4*c^3*e^m*x^5*x^m/(m + 5) - 3*a^4*b^3*c^3 
*e^m*x^4*x^m/(m + 4) - 3*a^5*b^2*c^3*e^m*x^3*x^m/(m + 3) + a^6*b*c^3*e^m*x 
^2*x^m/(m + 2) + (e*x)^(m + 1)*a^7*c^3/(e*(m + 1))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1313 vs. \(2 (197) = 394\).

Time = 0.13 (sec) , antiderivative size = 1313, normalized size of antiderivative = 6.66 \[ \int (e x)^m (a+b x)^4 (a c-b c x)^3 \, dx=\text {Too large to display} \] Input:

integrate((e*x)^m*(b*x+a)^4*(-b*c*x+a*c)^3,x, algorithm="giac")
 

Output:

-((e*x)^m*b^7*c^3*m^7*x^8 + (e*x)^m*a*b^6*c^3*m^7*x^7 + 28*(e*x)^m*b^7*c^3 
*m^6*x^8 - 3*(e*x)^m*a^2*b^5*c^3*m^7*x^6 + 29*(e*x)^m*a*b^6*c^3*m^6*x^7 + 
322*(e*x)^m*b^7*c^3*m^5*x^8 - 3*(e*x)^m*a^3*b^4*c^3*m^7*x^5 - 90*(e*x)^m*a 
^2*b^5*c^3*m^6*x^6 + 343*(e*x)^m*a*b^6*c^3*m^5*x^7 + 1960*(e*x)^m*b^7*c^3* 
m^4*x^8 + 3*(e*x)^m*a^4*b^3*c^3*m^7*x^4 - 93*(e*x)^m*a^3*b^4*c^3*m^6*x^5 - 
 1098*(e*x)^m*a^2*b^5*c^3*m^5*x^6 + 2135*(e*x)^m*a*b^6*c^3*m^4*x^7 + 6769* 
(e*x)^m*b^7*c^3*m^3*x^8 + 3*(e*x)^m*a^5*b^2*c^3*m^7*x^3 + 96*(e*x)^m*a^4*b 
^3*c^3*m^6*x^4 - 1173*(e*x)^m*a^3*b^4*c^3*m^5*x^5 - 7020*(e*x)^m*a^2*b^5*c 
^3*m^4*x^6 + 7504*(e*x)^m*a*b^6*c^3*m^3*x^7 + 13132*(e*x)^m*b^7*c^3*m^2*x^ 
8 - (e*x)^m*a^6*b*c^3*m^7*x^2 + 99*(e*x)^m*a^5*b^2*c^3*m^6*x^3 + 1254*(e*x 
)^m*a^4*b^3*c^3*m^5*x^4 - 7743*(e*x)^m*a^3*b^4*c^3*m^4*x^5 - 25227*(e*x)^m 
*a^2*b^5*c^3*m^3*x^6 + 14756*(e*x)^m*a*b^6*c^3*m^2*x^7 + 13068*(e*x)^m*b^7 
*c^3*m*x^8 - (e*x)^m*a^7*c^3*m^7*x - 34*(e*x)^m*a^6*b*c^3*m^6*x^2 + 1341*( 
e*x)^m*a^5*b^2*c^3*m^5*x^3 + 8592*(e*x)^m*a^4*b^3*c^3*m^4*x^4 - 28632*(e*x 
)^m*a^3*b^4*c^3*m^3*x^5 - 50490*(e*x)^m*a^2*b^5*c^3*m^2*x^6 + 14832*(e*x)^ 
m*a*b^6*c^3*m*x^7 + 5040*(e*x)^m*b^7*c^3*x^8 - 35*(e*x)^m*a^7*c^3*m^6*x - 
478*(e*x)^m*a^6*b*c^3*m^5*x^2 + 9585*(e*x)^m*a^5*b^2*c^3*m^4*x^3 + 32979*( 
e*x)^m*a^4*b^3*c^3*m^3*x^4 - 58692*(e*x)^m*a^3*b^4*c^3*m^2*x^5 - 51432*(e* 
x)^m*a^2*b^5*c^3*m*x^6 + 5760*(e*x)^m*a*b^6*c^3*x^7 - 511*(e*x)^m*a^7*c^3* 
m^5*x - 3580*(e*x)^m*a^6*b*c^3*m^4*x^2 + 38592*(e*x)^m*a^5*b^2*c^3*m^3*...
 

Mupad [B] (verification not implemented)

Time = 0.81 (sec) , antiderivative size = 723, normalized size of antiderivative = 3.67 \[ \int (e x)^m (a+b x)^4 (a c-b c x)^3 \, dx=\frac {a^7\,c^3\,x\,{\left (e\,x\right )}^m\,\left (m^7+35\,m^6+511\,m^5+4025\,m^4+18424\,m^3+48860\,m^2+69264\,m+40320\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}-\frac {b^7\,c^3\,x^8\,{\left (e\,x\right )}^m\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}-\frac {a\,b^6\,c^3\,x^7\,{\left (e\,x\right )}^m\,\left (m^7+29\,m^6+343\,m^5+2135\,m^4+7504\,m^3+14756\,m^2+14832\,m+5760\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {a^6\,b\,c^3\,x^2\,{\left (e\,x\right )}^m\,\left (m^7+34\,m^6+478\,m^5+3580\,m^4+15289\,m^3+36706\,m^2+44712\,m+20160\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,a^2\,b^5\,c^3\,x^6\,{\left (e\,x\right )}^m\,\left (m^7+30\,m^6+366\,m^5+2340\,m^4+8409\,m^3+16830\,m^2+17144\,m+6720\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,a^3\,b^4\,c^3\,x^5\,{\left (e\,x\right )}^m\,\left (m^7+31\,m^6+391\,m^5+2581\,m^4+9544\,m^3+19564\,m^2+20304\,m+8064\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}-\frac {3\,a^4\,b^3\,c^3\,x^4\,{\left (e\,x\right )}^m\,\left (m^7+32\,m^6+418\,m^5+2864\,m^4+10993\,m^3+23312\,m^2+24876\,m+10080\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}-\frac {3\,a^5\,b^2\,c^3\,x^3\,{\left (e\,x\right )}^m\,\left (m^7+33\,m^6+447\,m^5+3195\,m^4+12864\,m^3+28692\,m^2+32048\,m+13440\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320} \] Input:

int((a*c - b*c*x)^3*(e*x)^m*(a + b*x)^4,x)
 

Output:

(a^7*c^3*x*(e*x)^m*(69264*m + 48860*m^2 + 18424*m^3 + 4025*m^4 + 511*m^5 + 
 35*m^6 + m^7 + 40320))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4 
536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) - (b^7*c^3*x^8*(e*x)^m*(13068*m 
+ 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040))/(10958 
4*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m 
^8 + 40320) - (a*b^6*c^3*x^7*(e*x)^m*(14832*m + 14756*m^2 + 7504*m^3 + 213 
5*m^4 + 343*m^5 + 29*m^6 + m^7 + 5760))/(109584*m + 118124*m^2 + 67284*m^3 
 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (a^6*b*c^3*x^2 
*(e*x)^m*(44712*m + 36706*m^2 + 15289*m^3 + 3580*m^4 + 478*m^5 + 34*m^6 + 
m^7 + 20160))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 
546*m^6 + 36*m^7 + m^8 + 40320) + (3*a^2*b^5*c^3*x^6*(e*x)^m*(17144*m + 16 
830*m^2 + 8409*m^3 + 2340*m^4 + 366*m^5 + 30*m^6 + m^7 + 6720))/(109584*m 
+ 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 
 40320) + (3*a^3*b^4*c^3*x^5*(e*x)^m*(20304*m + 19564*m^2 + 9544*m^3 + 258 
1*m^4 + 391*m^5 + 31*m^6 + m^7 + 8064))/(109584*m + 118124*m^2 + 67284*m^3 
 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) - (3*a^4*b^3*c^3 
*x^4*(e*x)^m*(24876*m + 23312*m^2 + 10993*m^3 + 2864*m^4 + 418*m^5 + 32*m^ 
6 + m^7 + 10080))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^ 
5 + 546*m^6 + 36*m^7 + m^8 + 40320) - (3*a^5*b^2*c^3*x^3*(e*x)^m*(32048*m 
+ 28692*m^2 + 12864*m^3 + 3195*m^4 + 447*m^5 + 33*m^6 + m^7 + 13440))/(...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 786, normalized size of antiderivative = 3.99 \[ \int (e x)^m (a+b x)^4 (a c-b c x)^3 \, dx=\frac {x^{m} e^{m} c^{3} x \left (-b^{7} m^{7} x^{7}-a \,b^{6} m^{7} x^{6}-28 b^{7} m^{6} x^{7}+3 a^{2} b^{5} m^{7} x^{5}-29 a \,b^{6} m^{6} x^{6}-322 b^{7} m^{5} x^{7}+3 a^{3} b^{4} m^{7} x^{4}+90 a^{2} b^{5} m^{6} x^{5}-343 a \,b^{6} m^{5} x^{6}-1960 b^{7} m^{4} x^{7}-3 a^{4} b^{3} m^{7} x^{3}+93 a^{3} b^{4} m^{6} x^{4}+1098 a^{2} b^{5} m^{5} x^{5}-2135 a \,b^{6} m^{4} x^{6}-6769 b^{7} m^{3} x^{7}-3 a^{5} b^{2} m^{7} x^{2}-96 a^{4} b^{3} m^{6} x^{3}+1173 a^{3} b^{4} m^{5} x^{4}+7020 a^{2} b^{5} m^{4} x^{5}-7504 a \,b^{6} m^{3} x^{6}-13132 b^{7} m^{2} x^{7}+a^{6} b \,m^{7} x -99 a^{5} b^{2} m^{6} x^{2}-1254 a^{4} b^{3} m^{5} x^{3}+7743 a^{3} b^{4} m^{4} x^{4}+25227 a^{2} b^{5} m^{3} x^{5}-14756 a \,b^{6} m^{2} x^{6}-13068 b^{7} m \,x^{7}+a^{7} m^{7}+34 a^{6} b \,m^{6} x -1341 a^{5} b^{2} m^{5} x^{2}-8592 a^{4} b^{3} m^{4} x^{3}+28632 a^{3} b^{4} m^{3} x^{4}+50490 a^{2} b^{5} m^{2} x^{5}-14832 a \,b^{6} m \,x^{6}-5040 b^{7} x^{7}+35 a^{7} m^{6}+478 a^{6} b \,m^{5} x -9585 a^{5} b^{2} m^{4} x^{2}-32979 a^{4} b^{3} m^{3} x^{3}+58692 a^{3} b^{4} m^{2} x^{4}+51432 a^{2} b^{5} m \,x^{5}-5760 a \,b^{6} x^{6}+511 a^{7} m^{5}+3580 a^{6} b \,m^{4} x -38592 a^{5} b^{2} m^{3} x^{2}-69936 a^{4} b^{3} m^{2} x^{3}+60912 a^{3} b^{4} m \,x^{4}+20160 a^{2} b^{5} x^{5}+4025 a^{7} m^{4}+15289 a^{6} b \,m^{3} x -86076 a^{5} b^{2} m^{2} x^{2}-74628 a^{4} b^{3} m \,x^{3}+24192 a^{3} b^{4} x^{4}+18424 a^{7} m^{3}+36706 a^{6} b \,m^{2} x -96144 a^{5} b^{2} m \,x^{2}-30240 a^{4} b^{3} x^{3}+48860 a^{7} m^{2}+44712 a^{6} b m x -40320 a^{5} b^{2} x^{2}+69264 a^{7} m +20160 a^{6} b x +40320 a^{7}\right )}{m^{8}+36 m^{7}+546 m^{6}+4536 m^{5}+22449 m^{4}+67284 m^{3}+118124 m^{2}+109584 m +40320} \] Input:

int((e*x)^m*(b*x+a)^4*(-b*c*x+a*c)^3,x)
 

Output:

(x**m*e**m*c**3*x*(a**7*m**7 + 35*a**7*m**6 + 511*a**7*m**5 + 4025*a**7*m* 
*4 + 18424*a**7*m**3 + 48860*a**7*m**2 + 69264*a**7*m + 40320*a**7 + a**6* 
b*m**7*x + 34*a**6*b*m**6*x + 478*a**6*b*m**5*x + 3580*a**6*b*m**4*x + 152 
89*a**6*b*m**3*x + 36706*a**6*b*m**2*x + 44712*a**6*b*m*x + 20160*a**6*b*x 
 - 3*a**5*b**2*m**7*x**2 - 99*a**5*b**2*m**6*x**2 - 1341*a**5*b**2*m**5*x* 
*2 - 9585*a**5*b**2*m**4*x**2 - 38592*a**5*b**2*m**3*x**2 - 86076*a**5*b** 
2*m**2*x**2 - 96144*a**5*b**2*m*x**2 - 40320*a**5*b**2*x**2 - 3*a**4*b**3* 
m**7*x**3 - 96*a**4*b**3*m**6*x**3 - 1254*a**4*b**3*m**5*x**3 - 8592*a**4* 
b**3*m**4*x**3 - 32979*a**4*b**3*m**3*x**3 - 69936*a**4*b**3*m**2*x**3 - 7 
4628*a**4*b**3*m*x**3 - 30240*a**4*b**3*x**3 + 3*a**3*b**4*m**7*x**4 + 93* 
a**3*b**4*m**6*x**4 + 1173*a**3*b**4*m**5*x**4 + 7743*a**3*b**4*m**4*x**4 
+ 28632*a**3*b**4*m**3*x**4 + 58692*a**3*b**4*m**2*x**4 + 60912*a**3*b**4* 
m*x**4 + 24192*a**3*b**4*x**4 + 3*a**2*b**5*m**7*x**5 + 90*a**2*b**5*m**6* 
x**5 + 1098*a**2*b**5*m**5*x**5 + 7020*a**2*b**5*m**4*x**5 + 25227*a**2*b* 
*5*m**3*x**5 + 50490*a**2*b**5*m**2*x**5 + 51432*a**2*b**5*m*x**5 + 20160* 
a**2*b**5*x**5 - a*b**6*m**7*x**6 - 29*a*b**6*m**6*x**6 - 343*a*b**6*m**5* 
x**6 - 2135*a*b**6*m**4*x**6 - 7504*a*b**6*m**3*x**6 - 14756*a*b**6*m**2*x 
**6 - 14832*a*b**6*m*x**6 - 5760*a*b**6*x**6 - b**7*m**7*x**7 - 28*b**7*m* 
*6*x**7 - 322*b**7*m**5*x**7 - 1960*b**7*m**4*x**7 - 6769*b**7*m**3*x**7 - 
 13132*b**7*m**2*x**7 - 13068*b**7*m*x**7 - 5040*b**7*x**7))/(m**8 + 36...