\(\int (e x)^m (a+b x)^4 (a d-b d x)^3 \, dx\) [62]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {90} \[ \int (e x)^m (a+b x)^4 (a d-b d x)^3 \, dx=\frac {a^7 d^3 (e x)^{m+1}}{e (m+1)}+\frac {a^6 b d^3 (e x)^{m+2}}{e^2 (m+2)}-\frac {3 a^5 b^2 d^3 (e x)^{m+3}}{e^3 (m+3)}-\frac {3 a^4 b^3 d^3 (e x)^{m+4}}{e^4 (m+4)}+\frac {3 a^3 b^4 d^3 (e x)^{m+5}}{e^5 (m+5)}+\frac {3 a^2 b^5 d^3 (e x)^{m+6}}{e^6 (m+6)}-\frac {a b^6 d^3 (e x)^{m+7}}{e^7 (m+7)}-\frac {b^7 d^3 (e x)^{m+8}}{e^8 (m+8)} \]

[In]

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

[Out]

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

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(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]))

Rubi steps \begin{align*} \text {integral}& = \int \left (a^7 d^3 (e x)^m+\frac {a^6 b d^3 (e x)^{1+m}}{e}-\frac {3 a^5 b^2 d^3 (e x)^{2+m}}{e^2}-\frac {3 a^4 b^3 d^3 (e x)^{3+m}}{e^3}+\frac {3 a^3 b^4 d^3 (e x)^{4+m}}{e^4}+\frac {3 a^2 b^5 d^3 (e x)^{5+m}}{e^5}-\frac {a b^6 d^3 (e x)^{6+m}}{e^6}-\frac {b^7 d^3 (e x)^{7+m}}{e^7}\right ) \, dx \\ & = \frac {a^7 d^3 (e x)^{1+m}}{e (1+m)}+\frac {a^6 b d^3 (e x)^{2+m}}{e^2 (2+m)}-\frac {3 a^5 b^2 d^3 (e x)^{3+m}}{e^3 (3+m)}-\frac {3 a^4 b^3 d^3 (e x)^{4+m}}{e^4 (4+m)}+\frac {3 a^3 b^4 d^3 (e x)^{5+m}}{e^5 (5+m)}+\frac {3 a^2 b^5 d^3 (e x)^{6+m}}{e^6 (6+m)}-\frac {a b^6 d^3 (e x)^{7+m}}{e^7 (7+m)}-\frac {b^7 d^3 (e x)^{8+m}}{e^8 (8+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.62 \[ \int (e x)^m (a+b x)^4 (a d-b d x)^3 \, dx=d^3 x (e x)^m \left (\frac {a^7}{1+m}+\frac {a^6 b x}{2+m}-\frac {3 a^5 b^2 x^2}{3+m}-\frac {3 a^4 b^3 x^3}{4+m}+\frac {3 a^3 b^4 x^4}{5+m}+\frac {3 a^2 b^5 x^5}{6+m}-\frac {a b^6 x^6}{7+m}-\frac {b^7 x^7}{8+m}\right ) \]

[In]

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

[Out]

d^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))

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.99

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

[In]

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

[Out]

a^7*d^3/(1+m)*x*exp(m*ln(e*x))+a^6*b*d^3/(2+m)*x^2*exp(m*ln(e*x))-b^7*d^3/(8+m)*x^8*exp(m*ln(e*x))-a*b^6*d^3/(
7+m)*x^7*exp(m*ln(e*x))+3*a^2*b^5*d^3/(6+m)*x^6*exp(m*ln(e*x))+3*a^3*b^4*d^3/(5+m)*x^5*exp(m*ln(e*x))-3*a^4*b^
3*d^3/(4+m)*x^4*exp(m*ln(e*x))-3*a^5*b^2*d^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 = 0.24 (sec) , antiderivative size = 860, normalized size of antiderivative = 4.37 \[ \int (e x)^m (a+b x)^4 (a d-b d x)^3 \, dx=-\frac {{\left ({\left (b^{7} d^{3} m^{7} + 28 \, b^{7} d^{3} m^{6} + 322 \, b^{7} d^{3} m^{5} + 1960 \, b^{7} d^{3} m^{4} + 6769 \, b^{7} d^{3} m^{3} + 13132 \, b^{7} d^{3} m^{2} + 13068 \, b^{7} d^{3} m + 5040 \, b^{7} d^{3}\right )} x^{8} + {\left (a b^{6} d^{3} m^{7} + 29 \, a b^{6} d^{3} m^{6} + 343 \, a b^{6} d^{3} m^{5} + 2135 \, a b^{6} d^{3} m^{4} + 7504 \, a b^{6} d^{3} m^{3} + 14756 \, a b^{6} d^{3} m^{2} + 14832 \, a b^{6} d^{3} m + 5760 \, a b^{6} d^{3}\right )} x^{7} - 3 \, {\left (a^{2} b^{5} d^{3} m^{7} + 30 \, a^{2} b^{5} d^{3} m^{6} + 366 \, a^{2} b^{5} d^{3} m^{5} + 2340 \, a^{2} b^{5} d^{3} m^{4} + 8409 \, a^{2} b^{5} d^{3} m^{3} + 16830 \, a^{2} b^{5} d^{3} m^{2} + 17144 \, a^{2} b^{5} d^{3} m + 6720 \, a^{2} b^{5} d^{3}\right )} x^{6} - 3 \, {\left (a^{3} b^{4} d^{3} m^{7} + 31 \, a^{3} b^{4} d^{3} m^{6} + 391 \, a^{3} b^{4} d^{3} m^{5} + 2581 \, a^{3} b^{4} d^{3} m^{4} + 9544 \, a^{3} b^{4} d^{3} m^{3} + 19564 \, a^{3} b^{4} d^{3} m^{2} + 20304 \, a^{3} b^{4} d^{3} m + 8064 \, a^{3} b^{4} d^{3}\right )} x^{5} + 3 \, {\left (a^{4} b^{3} d^{3} m^{7} + 32 \, a^{4} b^{3} d^{3} m^{6} + 418 \, a^{4} b^{3} d^{3} m^{5} + 2864 \, a^{4} b^{3} d^{3} m^{4} + 10993 \, a^{4} b^{3} d^{3} m^{3} + 23312 \, a^{4} b^{3} d^{3} m^{2} + 24876 \, a^{4} b^{3} d^{3} m + 10080 \, a^{4} b^{3} d^{3}\right )} x^{4} + 3 \, {\left (a^{5} b^{2} d^{3} m^{7} + 33 \, a^{5} b^{2} d^{3} m^{6} + 447 \, a^{5} b^{2} d^{3} m^{5} + 3195 \, a^{5} b^{2} d^{3} m^{4} + 12864 \, a^{5} b^{2} d^{3} m^{3} + 28692 \, a^{5} b^{2} d^{3} m^{2} + 32048 \, a^{5} b^{2} d^{3} m + 13440 \, a^{5} b^{2} d^{3}\right )} x^{3} - {\left (a^{6} b d^{3} m^{7} + 34 \, a^{6} b d^{3} m^{6} + 478 \, a^{6} b d^{3} m^{5} + 3580 \, a^{6} b d^{3} m^{4} + 15289 \, a^{6} b d^{3} m^{3} + 36706 \, a^{6} b d^{3} m^{2} + 44712 \, a^{6} b d^{3} m + 20160 \, a^{6} b d^{3}\right )} x^{2} - {\left (a^{7} d^{3} m^{7} + 35 \, a^{7} d^{3} m^{6} + 511 \, a^{7} d^{3} m^{5} + 4025 \, a^{7} d^{3} m^{4} + 18424 \, a^{7} d^{3} m^{3} + 48860 \, a^{7} d^{3} m^{2} + 69264 \, a^{7} d^{3} m + 40320 \, a^{7} d^{3}\right )} x\right )} \left (e x\right )^{m}}{m^{8} + 36 \, m^{7} + 546 \, m^{6} + 4536 \, m^{5} + 22449 \, m^{4} + 67284 \, m^{3} + 118124 \, m^{2} + 109584 \, m + 40320} \]

[In]

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

[Out]

-((b^7*d^3*m^7 + 28*b^7*d^3*m^6 + 322*b^7*d^3*m^5 + 1960*b^7*d^3*m^4 + 6769*b^7*d^3*m^3 + 13132*b^7*d^3*m^2 +
13068*b^7*d^3*m + 5040*b^7*d^3)*x^8 + (a*b^6*d^3*m^7 + 29*a*b^6*d^3*m^6 + 343*a*b^6*d^3*m^5 + 2135*a*b^6*d^3*m
^4 + 7504*a*b^6*d^3*m^3 + 14756*a*b^6*d^3*m^2 + 14832*a*b^6*d^3*m + 5760*a*b^6*d^3)*x^7 - 3*(a^2*b^5*d^3*m^7 +
30*a^2*b^5*d^3*m^6 + 366*a^2*b^5*d^3*m^5 + 2340*a^2*b^5*d^3*m^4 + 8409*a^2*b^5*d^3*m^3 + 16830*a^2*b^5*d^3*m^
2 + 17144*a^2*b^5*d^3*m + 6720*a^2*b^5*d^3)*x^6 - 3*(a^3*b^4*d^3*m^7 + 31*a^3*b^4*d^3*m^6 + 391*a^3*b^4*d^3*m^
5 + 2581*a^3*b^4*d^3*m^4 + 9544*a^3*b^4*d^3*m^3 + 19564*a^3*b^4*d^3*m^2 + 20304*a^3*b^4*d^3*m + 8064*a^3*b^4*d
^3)*x^5 + 3*(a^4*b^3*d^3*m^7 + 32*a^4*b^3*d^3*m^6 + 418*a^4*b^3*d^3*m^5 + 2864*a^4*b^3*d^3*m^4 + 10993*a^4*b^3
*d^3*m^3 + 23312*a^4*b^3*d^3*m^2 + 24876*a^4*b^3*d^3*m + 10080*a^4*b^3*d^3)*x^4 + 3*(a^5*b^2*d^3*m^7 + 33*a^5*
b^2*d^3*m^6 + 447*a^5*b^2*d^3*m^5 + 3195*a^5*b^2*d^3*m^4 + 12864*a^5*b^2*d^3*m^3 + 28692*a^5*b^2*d^3*m^2 + 320
48*a^5*b^2*d^3*m + 13440*a^5*b^2*d^3)*x^3 - (a^6*b*d^3*m^7 + 34*a^6*b*d^3*m^6 + 478*a^6*b*d^3*m^5 + 3580*a^6*b
*d^3*m^4 + 15289*a^6*b*d^3*m^3 + 36706*a^6*b*d^3*m^2 + 44712*a^6*b*d^3*m + 20160*a^6*b*d^3)*x^2 - (a^7*d^3*m^7
+ 35*a^7*d^3*m^6 + 511*a^7*d^3*m^5 + 4025*a^7*d^3*m^4 + 18424*a^7*d^3*m^3 + 48860*a^7*d^3*m^2 + 69264*a^7*d^3
*m + 40320*a^7*d^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 + 10958
4*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 = 0.72 (sec) , antiderivative size = 4779, normalized size of antiderivative = 24.26 \[ \int (e x)^m (a+b x)^4 (a d-b d x)^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise(((-a**7*d**3/(7*x**7) - a**6*b*d**3/(6*x**6) + 3*a**5*b**2*d**3/(5*x**5) + 3*a**4*b**3*d**3/(4*x**4)
- a**3*b**4*d**3/x**3 - 3*a**2*b**5*d**3/(2*x**2) + a*b**6*d**3/x - b**7*d**3*log(x))/e**8, Eq(m, -8)), ((-a*
*7*d**3/(6*x**6) - a**6*b*d**3/(5*x**5) + 3*a**5*b**2*d**3/(4*x**4) + a**4*b**3*d**3/x**3 - 3*a**3*b**4*d**3/(
2*x**2) - 3*a**2*b**5*d**3/x - a*b**6*d**3*log(x) - b**7*d**3*x)/e**7, Eq(m, -7)), ((-a**7*d**3/(5*x**5) - a**
6*b*d**3/(4*x**4) + a**5*b**2*d**3/x**3 + 3*a**4*b**3*d**3/(2*x**2) - 3*a**3*b**4*d**3/x + 3*a**2*b**5*d**3*lo
g(x) - a*b**6*d**3*x - b**7*d**3*x**2/2)/e**6, Eq(m, -6)), ((-a**7*d**3/(4*x**4) - a**6*b*d**3/(3*x**3) + 3*a*
*5*b**2*d**3/(2*x**2) + 3*a**4*b**3*d**3/x + 3*a**3*b**4*d**3*log(x) + 3*a**2*b**5*d**3*x - a*b**6*d**3*x**2/2
- b**7*d**3*x**3/3)/e**5, Eq(m, -5)), ((-a**7*d**3/(3*x**3) - a**6*b*d**3/(2*x**2) + 3*a**5*b**2*d**3/x - 3*a
**4*b**3*d**3*log(x) + 3*a**3*b**4*d**3*x + 3*a**2*b**5*d**3*x**2/2 - a*b**6*d**3*x**3/3 - b**7*d**3*x**4/4)/e
**4, Eq(m, -4)), ((-a**7*d**3/(2*x**2) - a**6*b*d**3/x - 3*a**5*b**2*d**3*log(x) - 3*a**4*b**3*d**3*x + 3*a**3
*b**4*d**3*x**2/2 + a**2*b**5*d**3*x**3 - a*b**6*d**3*x**4/4 - b**7*d**3*x**5/5)/e**3, Eq(m, -3)), ((-a**7*d**
3/x + a**6*b*d**3*log(x) - 3*a**5*b**2*d**3*x - 3*a**4*b**3*d**3*x**2/2 + a**3*b**4*d**3*x**3 + 3*a**2*b**5*d*
*3*x**4/4 - a*b**6*d**3*x**5/5 - b**7*d**3*x**6/6)/e**2, Eq(m, -2)), ((a**7*d**3*log(x) + a**6*b*d**3*x - 3*a*
*5*b**2*d**3*x**2/2 - a**4*b**3*d**3*x**3 + 3*a**3*b**4*d**3*x**4/4 + 3*a**2*b**5*d**3*x**5/5 - a*b**6*d**3*x*
*6/6 - b**7*d**3*x**7/7)/e, Eq(m, -1)), (a**7*d**3*m**7*x*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22
449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 35*a**7*d**3*m**6*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) + 511*a**7*d**3*m**5*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) + 4025*a**7*d
**3*m**4*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) + 18424*a**7*d**3*m**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) + 48860*a**7*d**3*m**2*x*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 224
49*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 69264*a**7*d**3*m*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) + 40320*a**7*d**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) + a**6*b*d**3*m**
7*x**2*(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 + 40
320) + 34*a**6*b*d**3*m**6*x**2*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 11
8124*m**2 + 109584*m + 40320) + 478*a**6*b*d**3*m**5*x**2*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22
449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 3580*a**6*b*d**3*m**4*x**2*(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) + 15289*a**6*b*d**3*m**3*x*
*2*(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)
+ 36706*a**6*b*d**3*m**2*x**2*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118
124*m**2 + 109584*m + 40320) + 44712*a**6*b*d**3*m*x**2*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 2244
9*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 20160*a**6*b*d**3*x**2*(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) - 3*a**5*b**2*d**3*m**7*x**3*(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) - 99*a
**5*b**2*d**3*m**6*x**3*(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) - 1341*a**5*b**2*d**3*m**5*x**3*(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) - 9585*a**5*b**2*d**3*m**4*x**3*(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) - 38592*a**5*b**2*d**3*m**3*
x**3*(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 + 4032
0) - 86076*a**5*b**2*d**3*m**2*x**3*(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) - 96144*a**5*b**2*d**3*m*x**3*(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) - 40320*a**5*b**2*d**3*x**3*(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) - 3*a**4*b**3*d**3*m**
7*x**4*(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 + 40
320) - 96*a**4*b**3*d**3*m**6*x**4*(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) - 1254*a**4*b**3*d**3*m**5*x**4*(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) - 8592*a**4*b**3*d**3*m**4*x**4*(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) - 32979*a**4*b**3
*d**3*m**3*x**4*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 1095
84*m + 40320) - 69936*a**4*b**3*d**3*m**2*x**4*(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) - 74628*a**4*b**3*d**3*m*x**4*(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) - 30240*a**4*b**3*d**3*x**4*(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) + 3*a**3*b*
*4*d**3*m**7*x**5*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 10
9584*m + 40320) + 93*a**3*b**4*d**3*m**6*x**5*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 6
7284*m**3 + 118124*m**2 + 109584*m + 40320) + 1173*a**3*b**4*d**3*m**5*x**5*(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) + 7743*a**3*b**4*d**3*m**4*x**5*(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) + 2863
2*a**3*b**4*d**3*m**3*x**5*(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) + 58692*a**3*b**4*d**3*m**2*x**5*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22
449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 60912*a**3*b**4*d**3*m*x**5*(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) + 24192*a**3*b**4*d**3*x**
5*(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)
+ 3*a**2*b**5*d**3*m**7*x**6*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 11812
4*m**2 + 109584*m + 40320) + 90*a**2*b**5*d**3*m**6*x**6*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 224
49*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 1098*a**2*b**5*d**3*m**5*x**6*(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) + 7020*a**2*b**5*d**3*m**
4*x**6*(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 + 40
320) + 25227*a**2*b**5*d**3*m**3*x**6*(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) + 50490*a**2*b**5*d**3*m**2*x**6*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 453
6*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 51432*a**2*b**5*d**3*m*x**6*(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) + 20160*a**2*b*
*5*d**3*x**6*(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) - a*b**6*d**3*m**7*x**7*(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) - 29*a*b**6*d**3*m**6*x**7*(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) - 343*a*b**6*d**3*m**5*x**7*(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) - 2135*a*b**6*d**3*m**4*x*
*7*(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)
- 7504*a*b**6*d**3*m**3*x**7*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 1181
24*m**2 + 109584*m + 40320) - 14756*a*b**6*d**3*m**2*x**7*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22
449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) - 14832*a*b**6*d**3*m*x**7*(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) - 5760*a*b**6*d**3*x**7*(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) - b**7*
d**3*m**7*x**8*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 10958
4*m + 40320) - 28*b**7*d**3*m**6*x**8*(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) - 322*b**7*d**3*m**5*x**8*(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) - 1960*b**7*d**3*m**4*x**8*(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) - 6769*b**7*d**3*m**3*x**
8*(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)
- 13132*b**7*d**3*m**2*x**8*(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) - 13068*b**7*d**3*m*x**8*(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) - 5040*b**7*d**3*x**8*(e*x)**m/(m**8 + 36*m**7 + 546*m**6 + 4
536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.96 \[ \int (e x)^m (a+b x)^4 (a d-b d x)^3 \, dx=-\frac {b^{7} d^{3} e^{m} x^{8} x^{m}}{m + 8} - \frac {a b^{6} d^{3} e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, a^{2} b^{5} d^{3} e^{m} x^{6} x^{m}}{m + 6} + \frac {3 \, a^{3} b^{4} d^{3} e^{m} x^{5} x^{m}}{m + 5} - \frac {3 \, a^{4} b^{3} d^{3} e^{m} x^{4} x^{m}}{m + 4} - \frac {3 \, a^{5} b^{2} d^{3} e^{m} x^{3} x^{m}}{m + 3} + \frac {a^{6} b d^{3} e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} a^{7} d^{3}}{e {\left (m + 1\right )}} \]

[In]

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

[Out]

-b^7*d^3*e^m*x^8*x^m/(m + 8) - a*b^6*d^3*e^m*x^7*x^m/(m + 7) + 3*a^2*b^5*d^3*e^m*x^6*x^m/(m + 6) + 3*a^3*b^4*d
^3*e^m*x^5*x^m/(m + 5) - 3*a^4*b^3*d^3*e^m*x^4*x^m/(m + 4) - 3*a^5*b^2*d^3*e^m*x^3*x^m/(m + 3) + a^6*b*d^3*e^m
*x^2*x^m/(m + 2) + (e*x)^(m + 1)*a^7*d^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 = 0.30 (sec) , antiderivative size = 1313, normalized size of antiderivative = 6.66 \[ \int (e x)^m (a+b x)^4 (a d-b d x)^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

-((e*x)^m*b^7*d^3*m^7*x^8 + (e*x)^m*a*b^6*d^3*m^7*x^7 + 28*(e*x)^m*b^7*d^3*m^6*x^8 - 3*(e*x)^m*a^2*b^5*d^3*m^7
*x^6 + 29*(e*x)^m*a*b^6*d^3*m^6*x^7 + 322*(e*x)^m*b^7*d^3*m^5*x^8 - 3*(e*x)^m*a^3*b^4*d^3*m^7*x^5 - 90*(e*x)^m
*a^2*b^5*d^3*m^6*x^6 + 343*(e*x)^m*a*b^6*d^3*m^5*x^7 + 1960*(e*x)^m*b^7*d^3*m^4*x^8 + 3*(e*x)^m*a^4*b^3*d^3*m^
7*x^4 - 93*(e*x)^m*a^3*b^4*d^3*m^6*x^5 - 1098*(e*x)^m*a^2*b^5*d^3*m^5*x^6 + 2135*(e*x)^m*a*b^6*d^3*m^4*x^7 + 6
769*(e*x)^m*b^7*d^3*m^3*x^8 + 3*(e*x)^m*a^5*b^2*d^3*m^7*x^3 + 96*(e*x)^m*a^4*b^3*d^3*m^6*x^4 - 1173*(e*x)^m*a^
3*b^4*d^3*m^5*x^5 - 7020*(e*x)^m*a^2*b^5*d^3*m^4*x^6 + 7504*(e*x)^m*a*b^6*d^3*m^3*x^7 + 13132*(e*x)^m*b^7*d^3*
m^2*x^8 - (e*x)^m*a^6*b*d^3*m^7*x^2 + 99*(e*x)^m*a^5*b^2*d^3*m^6*x^3 + 1254*(e*x)^m*a^4*b^3*d^3*m^5*x^4 - 7743
*(e*x)^m*a^3*b^4*d^3*m^4*x^5 - 25227*(e*x)^m*a^2*b^5*d^3*m^3*x^6 + 14756*(e*x)^m*a*b^6*d^3*m^2*x^7 + 13068*(e*
x)^m*b^7*d^3*m*x^8 - (e*x)^m*a^7*d^3*m^7*x - 34*(e*x)^m*a^6*b*d^3*m^6*x^2 + 1341*(e*x)^m*a^5*b^2*d^3*m^5*x^3 +
8592*(e*x)^m*a^4*b^3*d^3*m^4*x^4 - 28632*(e*x)^m*a^3*b^4*d^3*m^3*x^5 - 50490*(e*x)^m*a^2*b^5*d^3*m^2*x^6 + 14
832*(e*x)^m*a*b^6*d^3*m*x^7 + 5040*(e*x)^m*b^7*d^3*x^8 - 35*(e*x)^m*a^7*d^3*m^6*x - 478*(e*x)^m*a^6*b*d^3*m^5*
x^2 + 9585*(e*x)^m*a^5*b^2*d^3*m^4*x^3 + 32979*(e*x)^m*a^4*b^3*d^3*m^3*x^4 - 58692*(e*x)^m*a^3*b^4*d^3*m^2*x^5
- 51432*(e*x)^m*a^2*b^5*d^3*m*x^6 + 5760*(e*x)^m*a*b^6*d^3*x^7 - 511*(e*x)^m*a^7*d^3*m^5*x - 3580*(e*x)^m*a^6
*b*d^3*m^4*x^2 + 38592*(e*x)^m*a^5*b^2*d^3*m^3*x^3 + 69936*(e*x)^m*a^4*b^3*d^3*m^2*x^4 - 60912*(e*x)^m*a^3*b^4
*d^3*m*x^5 - 20160*(e*x)^m*a^2*b^5*d^3*x^6 - 4025*(e*x)^m*a^7*d^3*m^4*x - 15289*(e*x)^m*a^6*b*d^3*m^3*x^2 + 86
076*(e*x)^m*a^5*b^2*d^3*m^2*x^3 + 74628*(e*x)^m*a^4*b^3*d^3*m*x^4 - 24192*(e*x)^m*a^3*b^4*d^3*x^5 - 18424*(e*x
)^m*a^7*d^3*m^3*x - 36706*(e*x)^m*a^6*b*d^3*m^2*x^2 + 96144*(e*x)^m*a^5*b^2*d^3*m*x^3 + 30240*(e*x)^m*a^4*b^3*
d^3*x^4 - 48860*(e*x)^m*a^7*d^3*m^2*x - 44712*(e*x)^m*a^6*b*d^3*m*x^2 + 40320*(e*x)^m*a^5*b^2*d^3*x^3 - 69264*
(e*x)^m*a^7*d^3*m*x - 20160*(e*x)^m*a^6*b*d^3*x^2 - 40320*(e*x)^m*a^7*d^3*x)/(m^8 + 36*m^7 + 546*m^6 + 4536*m^
5 + 22449*m^4 + 67284*m^3 + 118124*m^2 + 109584*m + 40320)

Mupad [B] (verification not implemented)

Time = 0.82 (sec) , antiderivative size = 723, normalized size of antiderivative = 3.67 \[ \int (e x)^m (a+b x)^4 (a d-b d x)^3 \, dx=\frac {a^7\,d^3\,x\,{\left (e\,x\right )}^m\,\left (m^7+35\,m^6+511\,m^5+4025\,m^4+18424\,m^3+48860\,m^2+69264\,m+40320\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}-\frac {b^7\,d^3\,x^8\,{\left (e\,x\right )}^m\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}-\frac {a\,b^6\,d^3\,x^7\,{\left (e\,x\right )}^m\,\left (m^7+29\,m^6+343\,m^5+2135\,m^4+7504\,m^3+14756\,m^2+14832\,m+5760\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {a^6\,b\,d^3\,x^2\,{\left (e\,x\right )}^m\,\left (m^7+34\,m^6+478\,m^5+3580\,m^4+15289\,m^3+36706\,m^2+44712\,m+20160\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,a^2\,b^5\,d^3\,x^6\,{\left (e\,x\right )}^m\,\left (m^7+30\,m^6+366\,m^5+2340\,m^4+8409\,m^3+16830\,m^2+17144\,m+6720\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,a^3\,b^4\,d^3\,x^5\,{\left (e\,x\right )}^m\,\left (m^7+31\,m^6+391\,m^5+2581\,m^4+9544\,m^3+19564\,m^2+20304\,m+8064\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}-\frac {3\,a^4\,b^3\,d^3\,x^4\,{\left (e\,x\right )}^m\,\left (m^7+32\,m^6+418\,m^5+2864\,m^4+10993\,m^3+23312\,m^2+24876\,m+10080\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}-\frac {3\,a^5\,b^2\,d^3\,x^3\,{\left (e\,x\right )}^m\,\left (m^7+33\,m^6+447\,m^5+3195\,m^4+12864\,m^3+28692\,m^2+32048\,m+13440\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320} \]

[In]

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

[Out]

(a^7*d^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 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) - (b^7*d^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))/(109584*m + 118124*m^2 + 67284*m^3 + 224
49*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) - (a*b^6*d^3*x^7*(e*x)^m*(14832*m + 14756*m^2 + 7504*m^3 +
2135*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*d^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*d^3*x^6*(e*x)^m*(17144*m + 16830*m^2 + 8409*m^3 + 2340*m^4 + 366*m^5 + 30*m^6 + m^7 + 6720))/(1
09584*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*d^3*x^5
*(e*x)^m*(20304*m + 19564*m^2 + 9544*m^3 + 2581*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*d^3*x^4*(e*x)^m*(24876*m + 23
312*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*d^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))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*
m^6 + 36*m^7 + m^8 + 40320)