\(\int (e x)^m (A+B x) (a+b x^2)^3 \, dx\) [114]

Optimal result

Integrand size = 20, antiderivative size = 169 \[ \int (e x)^m (A+B x) \left (a+b x^2\right )^3 \, dx=\frac {a^3 A (e x)^{1+m}}{e (1+m)}+\frac {a^3 B (e x)^{2+m}}{e^2 (2+m)}+\frac {3 a^2 A b (e x)^{3+m}}{e^3 (3+m)}+\frac {3 a^2 b B (e x)^{4+m}}{e^4 (4+m)}+\frac {3 a A b^2 (e x)^{5+m}}{e^5 (5+m)}+\frac {3 a b^2 B (e x)^{6+m}}{e^6 (6+m)}+\frac {A b^3 (e x)^{7+m}}{e^7 (7+m)}+\frac {b^3 B (e x)^{8+m}}{e^8 (8+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.60 \[ \int (e x)^m (A+B x) \left (a+b x^2\right )^3 \, dx=x (e x)^m \left (a^3 \left (\frac {A}{1+m}+\frac {B x}{2+m}\right )+3 a^2 b x^2 \left (\frac {A}{3+m}+\frac {B x}{4+m}\right )+3 a b^2 x^4 \left (\frac {A}{5+m}+\frac {B x}{6+m}\right )+b^3 x^6 \left (\frac {A}{7+m}+\frac {B x}{8+m}\right )\right ) \] Input:

Integrate[(e*x)^m*(A + B*x)*(a + b*x^2)^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {522, 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.

\(\displaystyle \int \left (a+b x^2\right )^3 (A+B x) (e x)^m \, dx\)

\(\Big \downarrow \) 522

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

\(\Big \downarrow \) 2009

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

Input:

Int[(e*x)^m*(A + B*x)*(a + b*x^2)^3,x]
 

Output:

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

Defintions of rubi rules used

rule 522

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], 
x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
 

rule 2009

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

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {A \,b^{3} x^{7} {\mathrm e}^{m \ln \left (e x \right )}}{7+m}+\frac {B \,a^{3} x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}+\frac {B \,b^{3} x^{8} {\mathrm e}^{m \ln \left (e x \right )}}{8+m}+\frac {a^{3} A x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {3 A a \,b^{2} x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {3 A \,a^{2} b \,x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}+\frac {3 B a \,b^{2} x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}+\frac {3 B \,a^{2} b \,x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}\)	\(168\)
gosper	\(\frac {x \left (B \,b^{3} m^{7} x^{7}+A \,b^{3} m^{7} x^{6}+28 B \,b^{3} m^{6} x^{7}+29 A \,b^{3} m^{6} x^{6}+3 B a \,b^{2} m^{7} x^{5}+322 B \,b^{3} m^{5} x^{7}+3 A a \,b^{2} m^{7} x^{4}+343 A \,b^{3} m^{5} x^{6}+90 B a \,b^{2} m^{6} x^{5}+1960 B \,b^{3} m^{4} x^{7}+93 A a \,b^{2} m^{6} x^{4}+2135 A \,b^{3} m^{4} x^{6}+3 B \,a^{2} b \,m^{7} x^{3}+1098 B a \,b^{2} m^{5} x^{5}+6769 B \,b^{3} m^{3} x^{7}+3 A \,a^{2} b \,m^{7} x^{2}+1173 A a \,b^{2} m^{5} x^{4}+7504 A \,b^{3} m^{3} x^{6}+96 B \,a^{2} b \,m^{6} x^{3}+7020 B a \,b^{2} m^{4} x^{5}+13132 B \,b^{3} m^{2} x^{7}+99 A \,a^{2} b \,m^{6} x^{2}+7743 A a \,b^{2} m^{4} x^{4}+14756 A \,b^{3} m^{2} x^{6}+B \,a^{3} m^{7} x +1254 B \,a^{2} b \,m^{5} x^{3}+25227 B a \,b^{2} m^{3} x^{5}+13068 B \,b^{3} m \,x^{7}+A \,a^{3} m^{7}+1341 A \,a^{2} b \,m^{5} x^{2}+28632 A a \,b^{2} m^{3} x^{4}+14832 A \,b^{3} m \,x^{6}+34 B \,a^{3} m^{6} x +8592 B \,a^{2} b \,m^{4} x^{3}+50490 B a \,b^{2} m^{2} x^{5}+5040 B \,b^{3} x^{7}+35 A \,a^{3} m^{6}+9585 A \,a^{2} b \,m^{4} x^{2}+58692 A a \,b^{2} m^{2} x^{4}+5760 A \,b^{3} x^{6}+478 B \,a^{3} m^{5} x +32979 B \,a^{2} b \,m^{3} x^{3}+51432 B a \,b^{2} m \,x^{5}+511 A \,a^{3} m^{5}+38592 A \,a^{2} b \,m^{3} x^{2}+60912 A a \,b^{2} m \,x^{4}+3580 B \,a^{3} m^{4} x +69936 B \,a^{2} b \,m^{2} x^{3}+20160 B a \,b^{2} x^{5}+4025 A \,a^{3} m^{4}+86076 A \,a^{2} b \,m^{2} x^{2}+24192 A a \,b^{2} x^{4}+15289 B \,a^{3} m^{3} x +74628 B \,a^{2} b m \,x^{3}+18424 A \,a^{3} m^{3}+96144 A \,a^{2} b m \,x^{2}+36706 B \,a^{3} m^{2} x +30240 a^{2} B b \,x^{3}+48860 A \,a^{3} m^{2}+40320 A \,a^{2} b \,x^{2}+44712 B \,a^{3} m x +69264 a^{3} A m +20160 B \,a^{3} x +40320 a^{3} A \right ) \left (e x \right )^{m}}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(765\)
risch	\(\frac {x \left (B \,b^{3} m^{7} x^{7}+A \,b^{3} m^{7} x^{6}+28 B \,b^{3} m^{6} x^{7}+29 A \,b^{3} m^{6} x^{6}+3 B a \,b^{2} m^{7} x^{5}+322 B \,b^{3} m^{5} x^{7}+3 A a \,b^{2} m^{7} x^{4}+343 A \,b^{3} m^{5} x^{6}+90 B a \,b^{2} m^{6} x^{5}+1960 B \,b^{3} m^{4} x^{7}+93 A a \,b^{2} m^{6} x^{4}+2135 A \,b^{3} m^{4} x^{6}+3 B \,a^{2} b \,m^{7} x^{3}+1098 B a \,b^{2} m^{5} x^{5}+6769 B \,b^{3} m^{3} x^{7}+3 A \,a^{2} b \,m^{7} x^{2}+1173 A a \,b^{2} m^{5} x^{4}+7504 A \,b^{3} m^{3} x^{6}+96 B \,a^{2} b \,m^{6} x^{3}+7020 B a \,b^{2} m^{4} x^{5}+13132 B \,b^{3} m^{2} x^{7}+99 A \,a^{2} b \,m^{6} x^{2}+7743 A a \,b^{2} m^{4} x^{4}+14756 A \,b^{3} m^{2} x^{6}+B \,a^{3} m^{7} x +1254 B \,a^{2} b \,m^{5} x^{3}+25227 B a \,b^{2} m^{3} x^{5}+13068 B \,b^{3} m \,x^{7}+A \,a^{3} m^{7}+1341 A \,a^{2} b \,m^{5} x^{2}+28632 A a \,b^{2} m^{3} x^{4}+14832 A \,b^{3} m \,x^{6}+34 B \,a^{3} m^{6} x +8592 B \,a^{2} b \,m^{4} x^{3}+50490 B a \,b^{2} m^{2} x^{5}+5040 B \,b^{3} x^{7}+35 A \,a^{3} m^{6}+9585 A \,a^{2} b \,m^{4} x^{2}+58692 A a \,b^{2} m^{2} x^{4}+5760 A \,b^{3} x^{6}+478 B \,a^{3} m^{5} x +32979 B \,a^{2} b \,m^{3} x^{3}+51432 B a \,b^{2} m \,x^{5}+511 A \,a^{3} m^{5}+38592 A \,a^{2} b \,m^{3} x^{2}+60912 A a \,b^{2} m \,x^{4}+3580 B \,a^{3} m^{4} x +69936 B \,a^{2} b \,m^{2} x^{3}+20160 B a \,b^{2} x^{5}+4025 A \,a^{3} m^{4}+86076 A \,a^{2} b \,m^{2} x^{2}+24192 A a \,b^{2} x^{4}+15289 B \,a^{3} m^{3} x +74628 B \,a^{2} b m \,x^{3}+18424 A \,a^{3} m^{3}+96144 A \,a^{2} b m \,x^{2}+36706 B \,a^{3} m^{2} x +30240 a^{2} B b \,x^{3}+48860 A \,a^{3} m^{2}+40320 A \,a^{2} b \,x^{2}+44712 B \,a^{3} m x +69264 a^{3} A m +20160 B \,a^{3} x +40320 a^{3} A \right ) \left (e x \right )^{m}}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(765\)
orering	\(\frac {x \left (B \,b^{3} m^{7} x^{7}+A \,b^{3} m^{7} x^{6}+28 B \,b^{3} m^{6} x^{7}+29 A \,b^{3} m^{6} x^{6}+3 B a \,b^{2} m^{7} x^{5}+322 B \,b^{3} m^{5} x^{7}+3 A a \,b^{2} m^{7} x^{4}+343 A \,b^{3} m^{5} x^{6}+90 B a \,b^{2} m^{6} x^{5}+1960 B \,b^{3} m^{4} x^{7}+93 A a \,b^{2} m^{6} x^{4}+2135 A \,b^{3} m^{4} x^{6}+3 B \,a^{2} b \,m^{7} x^{3}+1098 B a \,b^{2} m^{5} x^{5}+6769 B \,b^{3} m^{3} x^{7}+3 A \,a^{2} b \,m^{7} x^{2}+1173 A a \,b^{2} m^{5} x^{4}+7504 A \,b^{3} m^{3} x^{6}+96 B \,a^{2} b \,m^{6} x^{3}+7020 B a \,b^{2} m^{4} x^{5}+13132 B \,b^{3} m^{2} x^{7}+99 A \,a^{2} b \,m^{6} x^{2}+7743 A a \,b^{2} m^{4} x^{4}+14756 A \,b^{3} m^{2} x^{6}+B \,a^{3} m^{7} x +1254 B \,a^{2} b \,m^{5} x^{3}+25227 B a \,b^{2} m^{3} x^{5}+13068 B \,b^{3} m \,x^{7}+A \,a^{3} m^{7}+1341 A \,a^{2} b \,m^{5} x^{2}+28632 A a \,b^{2} m^{3} x^{4}+14832 A \,b^{3} m \,x^{6}+34 B \,a^{3} m^{6} x +8592 B \,a^{2} b \,m^{4} x^{3}+50490 B a \,b^{2} m^{2} x^{5}+5040 B \,b^{3} x^{7}+35 A \,a^{3} m^{6}+9585 A \,a^{2} b \,m^{4} x^{2}+58692 A a \,b^{2} m^{2} x^{4}+5760 A \,b^{3} x^{6}+478 B \,a^{3} m^{5} x +32979 B \,a^{2} b \,m^{3} x^{3}+51432 B a \,b^{2} m \,x^{5}+511 A \,a^{3} m^{5}+38592 A \,a^{2} b \,m^{3} x^{2}+60912 A a \,b^{2} m \,x^{4}+3580 B \,a^{3} m^{4} x +69936 B \,a^{2} b \,m^{2} x^{3}+20160 B a \,b^{2} x^{5}+4025 A \,a^{3} m^{4}+86076 A \,a^{2} b \,m^{2} x^{2}+24192 A a \,b^{2} x^{4}+15289 B \,a^{3} m^{3} x +74628 B \,a^{2} b m \,x^{3}+18424 A \,a^{3} m^{3}+96144 A \,a^{2} b m \,x^{2}+36706 B \,a^{3} m^{2} x +30240 a^{2} B b \,x^{3}+48860 A \,a^{3} m^{2}+40320 A \,a^{2} b \,x^{2}+44712 B \,a^{3} m x +69264 a^{3} A m +20160 B \,a^{3} x +40320 a^{3} A \right ) \left (e x \right )^{m}}{\left (8+m \right ) \left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(765\)
parallelrisch	\(\text {Expression too large to display}\)	\(1103\)

Input:

int((e*x)^m*(B*x+A)*(b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (169) = 338\).

Time = 0.09 (sec) , antiderivative size = 649, normalized size of antiderivative = 3.84 \[ \int (e x)^m (A+B x) \left (a+b x^2\right )^3 \, dx=\frac {{\left ({\left (B b^{3} m^{7} + 28 \, B b^{3} m^{6} + 322 \, B b^{3} m^{5} + 1960 \, B b^{3} m^{4} + 6769 \, B b^{3} m^{3} + 13132 \, B b^{3} m^{2} + 13068 \, B b^{3} m + 5040 \, B b^{3}\right )} x^{8} + {\left (A b^{3} m^{7} + 29 \, A b^{3} m^{6} + 343 \, A b^{3} m^{5} + 2135 \, A b^{3} m^{4} + 7504 \, A b^{3} m^{3} + 14756 \, A b^{3} m^{2} + 14832 \, A b^{3} m + 5760 \, A b^{3}\right )} x^{7} + 3 \, {\left (B a b^{2} m^{7} + 30 \, B a b^{2} m^{6} + 366 \, B a b^{2} m^{5} + 2340 \, B a b^{2} m^{4} + 8409 \, B a b^{2} m^{3} + 16830 \, B a b^{2} m^{2} + 17144 \, B a b^{2} m + 6720 \, B a b^{2}\right )} x^{6} + 3 \, {\left (A a b^{2} m^{7} + 31 \, A a b^{2} m^{6} + 391 \, A a b^{2} m^{5} + 2581 \, A a b^{2} m^{4} + 9544 \, A a b^{2} m^{3} + 19564 \, A a b^{2} m^{2} + 20304 \, A a b^{2} m + 8064 \, A a b^{2}\right )} x^{5} + 3 \, {\left (B a^{2} b m^{7} + 32 \, B a^{2} b m^{6} + 418 \, B a^{2} b m^{5} + 2864 \, B a^{2} b m^{4} + 10993 \, B a^{2} b m^{3} + 23312 \, B a^{2} b m^{2} + 24876 \, B a^{2} b m + 10080 \, B a^{2} b\right )} x^{4} + 3 \, {\left (A a^{2} b m^{7} + 33 \, A a^{2} b m^{6} + 447 \, A a^{2} b m^{5} + 3195 \, A a^{2} b m^{4} + 12864 \, A a^{2} b m^{3} + 28692 \, A a^{2} b m^{2} + 32048 \, A a^{2} b m + 13440 \, A a^{2} b\right )} x^{3} + {\left (B a^{3} m^{7} + 34 \, B a^{3} m^{6} + 478 \, B a^{3} m^{5} + 3580 \, B a^{3} m^{4} + 15289 \, B a^{3} m^{3} + 36706 \, B a^{3} m^{2} + 44712 \, B a^{3} m + 20160 \, B a^{3}\right )} x^{2} + {\left (A a^{3} m^{7} + 35 \, A a^{3} m^{6} + 511 \, A a^{3} m^{5} + 4025 \, A a^{3} m^{4} + 18424 \, A a^{3} m^{3} + 48860 \, A a^{3} m^{2} + 69264 \, A a^{3} m + 40320 \, A a^{3}\right )} x\right )} \left (e x\right )^{m}}{m^{8} + 36 \, m^{7} + 546 \, m^{6} + 4536 \, m^{5} + 22449 \, m^{4} + 67284 \, m^{3} + 118124 \, m^{2} + 109584 \, m + 40320} \] Input:

integrate((e*x)^m*(B*x+A)*(b*x^2+a)^3,x, algorithm="fricas")
 

Output:

((B*b^3*m^7 + 28*B*b^3*m^6 + 322*B*b^3*m^5 + 1960*B*b^3*m^4 + 6769*B*b^3*m 
^3 + 13132*B*b^3*m^2 + 13068*B*b^3*m + 5040*B*b^3)*x^8 + (A*b^3*m^7 + 29*A 
*b^3*m^6 + 343*A*b^3*m^5 + 2135*A*b^3*m^4 + 7504*A*b^3*m^3 + 14756*A*b^3*m 
^2 + 14832*A*b^3*m + 5760*A*b^3)*x^7 + 3*(B*a*b^2*m^7 + 30*B*a*b^2*m^6 + 3 
66*B*a*b^2*m^5 + 2340*B*a*b^2*m^4 + 8409*B*a*b^2*m^3 + 16830*B*a*b^2*m^2 + 
 17144*B*a*b^2*m + 6720*B*a*b^2)*x^6 + 3*(A*a*b^2*m^7 + 31*A*a*b^2*m^6 + 3 
91*A*a*b^2*m^5 + 2581*A*a*b^2*m^4 + 9544*A*a*b^2*m^3 + 19564*A*a*b^2*m^2 + 
 20304*A*a*b^2*m + 8064*A*a*b^2)*x^5 + 3*(B*a^2*b*m^7 + 32*B*a^2*b*m^6 + 4 
18*B*a^2*b*m^5 + 2864*B*a^2*b*m^4 + 10993*B*a^2*b*m^3 + 23312*B*a^2*b*m^2 
+ 24876*B*a^2*b*m + 10080*B*a^2*b)*x^4 + 3*(A*a^2*b*m^7 + 33*A*a^2*b*m^6 + 
 447*A*a^2*b*m^5 + 3195*A*a^2*b*m^4 + 12864*A*a^2*b*m^3 + 28692*A*a^2*b*m^ 
2 + 32048*A*a^2*b*m + 13440*A*a^2*b)*x^3 + (B*a^3*m^7 + 34*B*a^3*m^6 + 478 
*B*a^3*m^5 + 3580*B*a^3*m^4 + 15289*B*a^3*m^3 + 36706*B*a^3*m^2 + 44712*B* 
a^3*m + 20160*B*a^3)*x^2 + (A*a^3*m^7 + 35*A*a^3*m^6 + 511*A*a^3*m^5 + 402 
5*A*a^3*m^4 + 18424*A*a^3*m^3 + 48860*A*a^3*m^2 + 69264*A*a^3*m + 40320*A* 
a^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. 4398 vs. \(2 (160) = 320\).

Time = 0.62 (sec) , antiderivative size = 4398, normalized size of antiderivative = 26.02 \[ \int (e x)^m (A+B x) \left (a+b x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((e*x)**m*(B*x+A)*(b*x**2+a)**3,x)
 

Output:

Piecewise(((-A*a**3/(7*x**7) - 3*A*a**2*b/(5*x**5) - A*a*b**2/x**3 - A*b** 
3/x - B*a**3/(6*x**6) - 3*B*a**2*b/(4*x**4) - 3*B*a*b**2/(2*x**2) + B*b**3 
*log(x))/e**8, Eq(m, -8)), ((-A*a**3/(6*x**6) - 3*A*a**2*b/(4*x**4) - 3*A* 
a*b**2/(2*x**2) + A*b**3*log(x) - B*a**3/(5*x**5) - B*a**2*b/x**3 - 3*B*a* 
b**2/x + B*b**3*x)/e**7, Eq(m, -7)), ((-A*a**3/(5*x**5) - A*a**2*b/x**3 - 
3*A*a*b**2/x + A*b**3*x - B*a**3/(4*x**4) - 3*B*a**2*b/(2*x**2) + 3*B*a*b* 
*2*log(x) + B*b**3*x**2/2)/e**6, Eq(m, -6)), ((-A*a**3/(4*x**4) - 3*A*a**2 
*b/(2*x**2) + 3*A*a*b**2*log(x) + A*b**3*x**2/2 - B*a**3/(3*x**3) - 3*B*a* 
*2*b/x + 3*B*a*b**2*x + B*b**3*x**3/3)/e**5, Eq(m, -5)), ((-A*a**3/(3*x**3 
) - 3*A*a**2*b/x + 3*A*a*b**2*x + A*b**3*x**3/3 - B*a**3/(2*x**2) + 3*B*a* 
*2*b*log(x) + 3*B*a*b**2*x**2/2 + B*b**3*x**4/4)/e**4, Eq(m, -4)), ((-A*a* 
*3/(2*x**2) + 3*A*a**2*b*log(x) + 3*A*a*b**2*x**2/2 + A*b**3*x**4/4 - B*a* 
*3/x + 3*B*a**2*b*x + B*a*b**2*x**3 + B*b**3*x**5/5)/e**3, Eq(m, -3)), ((- 
A*a**3/x + 3*A*a**2*b*x + A*a*b**2*x**3 + A*b**3*x**5/5 + B*a**3*log(x) + 
3*B*a**2*b*x**2/2 + 3*B*a*b**2*x**4/4 + B*b**3*x**6/6)/e**2, Eq(m, -2)), ( 
(A*a**3*log(x) + 3*A*a**2*b*x**2/2 + 3*A*a*b**2*x**4/4 + A*b**3*x**6/6 + B 
*a**3*x + B*a**2*b*x**3 + 3*B*a*b**2*x**5/5 + B*b**3*x**7/7)/e, Eq(m, -1)) 
, (A*a**3*m**7*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) + 35*A*a**3*m**6*x*(e*x 
)**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 +...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.96 \[ \int (e x)^m (A+B x) \left (a+b x^2\right )^3 \, dx=\frac {B b^{3} e^{m} x^{8} x^{m}}{m + 8} + \frac {A b^{3} e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B a b^{2} e^{m} x^{6} x^{m}}{m + 6} + \frac {3 \, A a b^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, B a^{2} b e^{m} x^{4} x^{m}}{m + 4} + \frac {3 \, A a^{2} b e^{m} x^{3} x^{m}}{m + 3} + \frac {B a^{3} e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} A a^{3}}{e {\left (m + 1\right )}} \] Input:

integrate((e*x)^m*(B*x+A)*(b*x^2+a)^3,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1102 vs. \(2 (169) = 338\).

Time = 0.13 (sec) , antiderivative size = 1102, normalized size of antiderivative = 6.52 \[ \int (e x)^m (A+B x) \left (a+b x^2\right )^3 \, dx=\text {Too large to display} \] Input:

integrate((e*x)^m*(B*x+A)*(b*x^2+a)^3,x, algorithm="giac")
 

Output:

((e*x)^m*B*b^3*m^7*x^8 + (e*x)^m*A*b^3*m^7*x^7 + 28*(e*x)^m*B*b^3*m^6*x^8 
+ 3*(e*x)^m*B*a*b^2*m^7*x^6 + 29*(e*x)^m*A*b^3*m^6*x^7 + 322*(e*x)^m*B*b^3 
*m^5*x^8 + 3*(e*x)^m*A*a*b^2*m^7*x^5 + 90*(e*x)^m*B*a*b^2*m^6*x^6 + 343*(e 
*x)^m*A*b^3*m^5*x^7 + 1960*(e*x)^m*B*b^3*m^4*x^8 + 3*(e*x)^m*B*a^2*b*m^7*x 
^4 + 93*(e*x)^m*A*a*b^2*m^6*x^5 + 1098*(e*x)^m*B*a*b^2*m^5*x^6 + 2135*(e*x 
)^m*A*b^3*m^4*x^7 + 6769*(e*x)^m*B*b^3*m^3*x^8 + 3*(e*x)^m*A*a^2*b*m^7*x^3 
 + 96*(e*x)^m*B*a^2*b*m^6*x^4 + 1173*(e*x)^m*A*a*b^2*m^5*x^5 + 7020*(e*x)^ 
m*B*a*b^2*m^4*x^6 + 7504*(e*x)^m*A*b^3*m^3*x^7 + 13132*(e*x)^m*B*b^3*m^2*x 
^8 + (e*x)^m*B*a^3*m^7*x^2 + 99*(e*x)^m*A*a^2*b*m^6*x^3 + 1254*(e*x)^m*B*a 
^2*b*m^5*x^4 + 7743*(e*x)^m*A*a*b^2*m^4*x^5 + 25227*(e*x)^m*B*a*b^2*m^3*x^ 
6 + 14756*(e*x)^m*A*b^3*m^2*x^7 + 13068*(e*x)^m*B*b^3*m*x^8 + (e*x)^m*A*a^ 
3*m^7*x + 34*(e*x)^m*B*a^3*m^6*x^2 + 1341*(e*x)^m*A*a^2*b*m^5*x^3 + 8592*( 
e*x)^m*B*a^2*b*m^4*x^4 + 28632*(e*x)^m*A*a*b^2*m^3*x^5 + 50490*(e*x)^m*B*a 
*b^2*m^2*x^6 + 14832*(e*x)^m*A*b^3*m*x^7 + 5040*(e*x)^m*B*b^3*x^8 + 35*(e* 
x)^m*A*a^3*m^6*x + 478*(e*x)^m*B*a^3*m^5*x^2 + 9585*(e*x)^m*A*a^2*b*m^4*x^ 
3 + 32979*(e*x)^m*B*a^2*b*m^3*x^4 + 58692*(e*x)^m*A*a*b^2*m^2*x^5 + 51432* 
(e*x)^m*B*a*b^2*m*x^6 + 5760*(e*x)^m*A*b^3*x^7 + 511*(e*x)^m*A*a^3*m^5*x + 
 3580*(e*x)^m*B*a^3*m^4*x^2 + 38592*(e*x)^m*A*a^2*b*m^3*x^3 + 69936*(e*x)^ 
m*B*a^2*b*m^2*x^4 + 60912*(e*x)^m*A*a*b^2*m*x^5 + 20160*(e*x)^m*B*a*b^2*x^ 
6 + 4025*(e*x)^m*A*a^3*m^4*x + 15289*(e*x)^m*B*a^3*m^3*x^2 + 86076*(e*x...
 

Mupad [B] (verification not implemented)

Time = 9.05 (sec) , antiderivative size = 695, normalized size of antiderivative = 4.11 \[ \int (e x)^m (A+B x) \left (a+b x^2\right )^3 \, dx=\frac {A\,a^3\,x\,{\left (e\,x\right )}^m\,\left (m^7+35\,m^6+511\,m^5+4025\,m^4+18424\,m^3+48860\,m^2+69264\,m+40320\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {A\,b^3\,x^7\,{\left (e\,x\right )}^m\,\left (m^7+29\,m^6+343\,m^5+2135\,m^4+7504\,m^3+14756\,m^2+14832\,m+5760\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {B\,a^3\,x^2\,{\left (e\,x\right )}^m\,\left (m^7+34\,m^6+478\,m^5+3580\,m^4+15289\,m^3+36706\,m^2+44712\,m+20160\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {B\,b^3\,x^8\,{\left (e\,x\right )}^m\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,A\,a\,b^2\,x^5\,{\left (e\,x\right )}^m\,\left (m^7+31\,m^6+391\,m^5+2581\,m^4+9544\,m^3+19564\,m^2+20304\,m+8064\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,A\,a^2\,b\,x^3\,{\left (e\,x\right )}^m\,\left (m^7+33\,m^6+447\,m^5+3195\,m^4+12864\,m^3+28692\,m^2+32048\,m+13440\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,B\,a\,b^2\,x^6\,{\left (e\,x\right )}^m\,\left (m^7+30\,m^6+366\,m^5+2340\,m^4+8409\,m^3+16830\,m^2+17144\,m+6720\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,B\,a^2\,b\,x^4\,{\left (e\,x\right )}^m\,\left (m^7+32\,m^6+418\,m^5+2864\,m^4+10993\,m^3+23312\,m^2+24876\,m+10080\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320} \] Input:

int((e*x)^m*(a + b*x^2)^3*(A + B*x),x)
 

Output:

(A*a^3*x*(e*x)^m*(69264*m + 48860*m^2 + 18424*m^3 + 4025*m^4 + 511*m^5 + 3 
5*m^6 + m^7 + 40320))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 453 
6*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (A*b^3*x^7*(e*x)^m*(14832*m + 14 
756*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) + (B*a^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 + 224 
49*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (B*b^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 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36 
*m^7 + m^8 + 40320) + (3*A*a*b^2*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 + 6 
7284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (3*A*a 
^2*b*x^3*(e*x)^m*(32048*m + 28692*m^2 + 12864*m^3 + 3195*m^4 + 447*m^5 + 3 
3*m^6 + m^7 + 13440))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 453 
6*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (3*B*a*b^2*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))/(10958 
4*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m 
^8 + 40320) + (3*B*a^2*b*x^4*(e*x)^m*(24876*m + 23312*m^2 + 10993*m^3 + 28 
64*m^4 + 418*m^5 + 32*m^6 + m^7 + 10080))/(109584*m + 118124*m^2 + 6728...
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 749, normalized size of antiderivative = 4.43 \[ \int (e x)^m (A+B x) \left (a+b x^2\right )^3 \, dx=\frac {x^{m} e^{m} x \left (b^{4} m^{7} x^{7}+a \,b^{3} m^{7} x^{6}+28 b^{4} m^{6} x^{7}+3 a \,b^{3} m^{7} x^{5}+29 a \,b^{3} m^{6} x^{6}+322 b^{4} m^{5} x^{7}+3 a^{2} b^{2} m^{7} x^{4}+90 a \,b^{3} m^{6} x^{5}+343 a \,b^{3} m^{5} x^{6}+1960 b^{4} m^{4} x^{7}+3 a^{2} b^{2} m^{7} x^{3}+93 a^{2} b^{2} m^{6} x^{4}+1098 a \,b^{3} m^{5} x^{5}+2135 a \,b^{3} m^{4} x^{6}+6769 b^{4} m^{3} x^{7}+3 a^{3} b \,m^{7} x^{2}+96 a^{2} b^{2} m^{6} x^{3}+1173 a^{2} b^{2} m^{5} x^{4}+7020 a \,b^{3} m^{4} x^{5}+7504 a \,b^{3} m^{3} x^{6}+13132 b^{4} m^{2} x^{7}+a^{3} b \,m^{7} x +99 a^{3} b \,m^{6} x^{2}+1254 a^{2} b^{2} m^{5} x^{3}+7743 a^{2} b^{2} m^{4} x^{4}+25227 a \,b^{3} m^{3} x^{5}+14756 a \,b^{3} m^{2} x^{6}+13068 b^{4} m \,x^{7}+a^{4} m^{7}+34 a^{3} b \,m^{6} x +1341 a^{3} b \,m^{5} x^{2}+8592 a^{2} b^{2} m^{4} x^{3}+28632 a^{2} b^{2} m^{3} x^{4}+50490 a \,b^{3} m^{2} x^{5}+14832 a \,b^{3} m \,x^{6}+5040 b^{4} x^{7}+35 a^{4} m^{6}+478 a^{3} b \,m^{5} x +9585 a^{3} b \,m^{4} x^{2}+32979 a^{2} b^{2} m^{3} x^{3}+58692 a^{2} b^{2} m^{2} x^{4}+51432 a \,b^{3} m \,x^{5}+5760 a \,b^{3} x^{6}+511 a^{4} m^{5}+3580 a^{3} b \,m^{4} x +38592 a^{3} b \,m^{3} x^{2}+69936 a^{2} b^{2} m^{2} x^{3}+60912 a^{2} b^{2} m \,x^{4}+20160 a \,b^{3} x^{5}+4025 a^{4} m^{4}+15289 a^{3} b \,m^{3} x +86076 a^{3} b \,m^{2} x^{2}+74628 a^{2} b^{2} m \,x^{3}+24192 a^{2} b^{2} x^{4}+18424 a^{4} m^{3}+36706 a^{3} b \,m^{2} x +96144 a^{3} b m \,x^{2}+30240 a^{2} b^{2} x^{3}+48860 a^{4} m^{2}+44712 a^{3} b m x +40320 a^{3} b \,x^{2}+69264 a^{4} m +20160 a^{3} b x +40320 a^{4}\right )}{m^{8}+36 m^{7}+546 m^{6}+4536 m^{5}+22449 m^{4}+67284 m^{3}+118124 m^{2}+109584 m +40320} \] Input:

int((e*x)^m*(B*x+A)*(b*x^2+a)^3,x)
 

Output:

(x**m*e**m*x*(a**4*m**7 + 35*a**4*m**6 + 511*a**4*m**5 + 4025*a**4*m**4 + 
18424*a**4*m**3 + 48860*a**4*m**2 + 69264*a**4*m + 40320*a**4 + 3*a**3*b*m 
**7*x**2 + a**3*b*m**7*x + 99*a**3*b*m**6*x**2 + 34*a**3*b*m**6*x + 1341*a 
**3*b*m**5*x**2 + 478*a**3*b*m**5*x + 9585*a**3*b*m**4*x**2 + 3580*a**3*b* 
m**4*x + 38592*a**3*b*m**3*x**2 + 15289*a**3*b*m**3*x + 86076*a**3*b*m**2* 
x**2 + 36706*a**3*b*m**2*x + 96144*a**3*b*m*x**2 + 44712*a**3*b*m*x + 4032 
0*a**3*b*x**2 + 20160*a**3*b*x + 3*a**2*b**2*m**7*x**4 + 3*a**2*b**2*m**7* 
x**3 + 93*a**2*b**2*m**6*x**4 + 96*a**2*b**2*m**6*x**3 + 1173*a**2*b**2*m* 
*5*x**4 + 1254*a**2*b**2*m**5*x**3 + 7743*a**2*b**2*m**4*x**4 + 8592*a**2* 
b**2*m**4*x**3 + 28632*a**2*b**2*m**3*x**4 + 32979*a**2*b**2*m**3*x**3 + 5 
8692*a**2*b**2*m**2*x**4 + 69936*a**2*b**2*m**2*x**3 + 60912*a**2*b**2*m*x 
**4 + 74628*a**2*b**2*m*x**3 + 24192*a**2*b**2*x**4 + 30240*a**2*b**2*x**3 
 + a*b**3*m**7*x**6 + 3*a*b**3*m**7*x**5 + 29*a*b**3*m**6*x**6 + 90*a*b**3 
*m**6*x**5 + 343*a*b**3*m**5*x**6 + 1098*a*b**3*m**5*x**5 + 2135*a*b**3*m* 
*4*x**6 + 7020*a*b**3*m**4*x**5 + 7504*a*b**3*m**3*x**6 + 25227*a*b**3*m** 
3*x**5 + 14756*a*b**3*m**2*x**6 + 50490*a*b**3*m**2*x**5 + 14832*a*b**3*m* 
x**6 + 51432*a*b**3*m*x**5 + 5760*a*b**3*x**6 + 20160*a*b**3*x**5 + b**4*m 
**7*x**7 + 28*b**4*m**6*x**7 + 322*b**4*m**5*x**7 + 1960*b**4*m**4*x**7 + 
6769*b**4*m**3*x**7 + 13132*b**4*m**2*x**7 + 13068*b**4*m*x**7 + 5040*b**4 
*x**7))/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**...