\(\int (e x)^m (c+d x)^3 (a+b x^2)^2 \, dx\) [107]

Optimal result

Integrand size = 22, antiderivative size = 215 \[ \int (e x)^m (c+d x)^3 \left (a+b x^2\right )^2 \, dx=\frac {a^2 c^3 (e x)^{1+m}}{e (1+m)}+\frac {3 a^2 c^2 d (e x)^{2+m}}{e^2 (2+m)}+\frac {a c \left (2 b c^2+3 a d^2\right ) (e x)^{3+m}}{e^3 (3+m)}+\frac {a d \left (6 b c^2+a d^2\right ) (e x)^{4+m}}{e^4 (4+m)}+\frac {b c \left (b c^2+6 a d^2\right ) (e x)^{5+m}}{e^5 (5+m)}+\frac {b d \left (3 b c^2+2 a d^2\right ) (e x)^{6+m}}{e^6 (6+m)}+\frac {3 b^2 c d^2 (e x)^{7+m}}{e^7 (7+m)}+\frac {b^2 d^3 (e x)^{8+m}}{e^8 (8+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.75 \[ \int (e x)^m (c+d x)^3 \left (a+b x^2\right )^2 \, dx=x (e x)^m \left (\frac {a^2 c^3}{1+m}+\frac {3 a^2 c^2 d x}{2+m}+\frac {a c \left (2 b c^2+3 a d^2\right ) x^2}{3+m}+\frac {a d \left (6 b c^2+a d^2\right ) x^3}{4+m}+\frac {b c \left (b c^2+6 a d^2\right ) x^4}{5+m}+\frac {b d \left (3 b c^2+2 a d^2\right ) x^5}{6+m}+\frac {3 b^2 c d^2 x^6}{7+m}+\frac {b^2 d^3 x^7}{8+m}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

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

\(\Big \downarrow \) 522

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(a^2*c^3*(e*x)^(1 + m))/(e*(1 + m)) + (3*a^2*c^2*d*(e*x)^(2 + m))/(e^2*(2 
+ m)) + (a*c*(2*b*c^2 + 3*a*d^2)*(e*x)^(3 + m))/(e^3*(3 + m)) + (a*d*(6*b* 
c^2 + a*d^2)*(e*x)^(4 + m))/(e^4*(4 + m)) + (b*c*(b*c^2 + 6*a*d^2)*(e*x)^( 
5 + m))/(e^5*(5 + m)) + (b*d*(3*b*c^2 + 2*a*d^2)*(e*x)^(6 + m))/(e^6*(6 + 
m)) + (3*b^2*c*d^2*(e*x)^(7 + m))/(e^7*(7 + m)) + (b^2*d^3*(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 = 0.31 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00

method	result	size
norman	\(\frac {a^{2} c^{3} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {b^{2} d^{3} x^{8} {\mathrm e}^{m \ln \left (e x \right )}}{8+m}+\frac {a c \left (3 a \,d^{2}+2 b \,c^{2}\right ) x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}+\frac {a d \left (a \,d^{2}+6 b \,c^{2}\right ) x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}+\frac {b c \left (6 a \,d^{2}+b \,c^{2}\right ) x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {b d \left (2 a \,d^{2}+3 b \,c^{2}\right ) x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}+\frac {3 a^{2} c^{2} d \,x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}+\frac {3 b^{2} c \,d^{2} x^{7} {\mathrm e}^{m \ln \left (e x \right )}}{7+m}\)	\(214\)
gosper	\(\text {Expression too large to display}\)	\(1305\)
risch	\(\text {Expression too large to display}\)	\(1305\)
orering	\(\text {Expression too large to display}\)	\(1305\)
parallelrisch	\(\text {Expression too large to display}\)	\(1803\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1085 vs. \(2 (215) = 430\).

Time = 0.10 (sec) , antiderivative size = 1085, normalized size of antiderivative = 5.05 \[ \int (e x)^m (c+d x)^3 \left (a+b x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

((b^2*d^3*m^7 + 28*b^2*d^3*m^6 + 322*b^2*d^3*m^5 + 1960*b^2*d^3*m^4 + 6769 
*b^2*d^3*m^3 + 13132*b^2*d^3*m^2 + 13068*b^2*d^3*m + 5040*b^2*d^3)*x^8 + 3 
*(b^2*c*d^2*m^7 + 29*b^2*c*d^2*m^6 + 343*b^2*c*d^2*m^5 + 2135*b^2*c*d^2*m^ 
4 + 7504*b^2*c*d^2*m^3 + 14756*b^2*c*d^2*m^2 + 14832*b^2*c*d^2*m + 5760*b^ 
2*c*d^2)*x^7 + ((3*b^2*c^2*d + 2*a*b*d^3)*m^7 + 30*(3*b^2*c^2*d + 2*a*b*d^ 
3)*m^6 + 366*(3*b^2*c^2*d + 2*a*b*d^3)*m^5 + 20160*b^2*c^2*d + 13440*a*b*d 
^3 + 2340*(3*b^2*c^2*d + 2*a*b*d^3)*m^4 + 8409*(3*b^2*c^2*d + 2*a*b*d^3)*m 
^3 + 16830*(3*b^2*c^2*d + 2*a*b*d^3)*m^2 + 17144*(3*b^2*c^2*d + 2*a*b*d^3) 
*m)*x^6 + ((b^2*c^3 + 6*a*b*c*d^2)*m^7 + 31*(b^2*c^3 + 6*a*b*c*d^2)*m^6 + 
391*(b^2*c^3 + 6*a*b*c*d^2)*m^5 + 8064*b^2*c^3 + 48384*a*b*c*d^2 + 2581*(b 
^2*c^3 + 6*a*b*c*d^2)*m^4 + 9544*(b^2*c^3 + 6*a*b*c*d^2)*m^3 + 19564*(b^2* 
c^3 + 6*a*b*c*d^2)*m^2 + 20304*(b^2*c^3 + 6*a*b*c*d^2)*m)*x^5 + ((6*a*b*c^ 
2*d + a^2*d^3)*m^7 + 32*(6*a*b*c^2*d + a^2*d^3)*m^6 + 418*(6*a*b*c^2*d + a 
^2*d^3)*m^5 + 60480*a*b*c^2*d + 10080*a^2*d^3 + 2864*(6*a*b*c^2*d + a^2*d^ 
3)*m^4 + 10993*(6*a*b*c^2*d + a^2*d^3)*m^3 + 23312*(6*a*b*c^2*d + a^2*d^3) 
*m^2 + 24876*(6*a*b*c^2*d + a^2*d^3)*m)*x^4 + ((2*a*b*c^3 + 3*a^2*c*d^2)*m 
^7 + 33*(2*a*b*c^3 + 3*a^2*c*d^2)*m^6 + 447*(2*a*b*c^3 + 3*a^2*c*d^2)*m^5 
+ 26880*a*b*c^3 + 40320*a^2*c*d^2 + 3195*(2*a*b*c^3 + 3*a^2*c*d^2)*m^4 + 1 
2864*(2*a*b*c^3 + 3*a^2*c*d^2)*m^3 + 28692*(2*a*b*c^3 + 3*a^2*c*d^2)*m^2 + 
 32048*(2*a*b*c^3 + 3*a^2*c*d^2)*m)*x^3 + 3*(a^2*c^2*d*m^7 + 34*a^2*c^2...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6928 vs. \(2 (204) = 408\).

Time = 0.79 (sec) , antiderivative size = 6928, normalized size of antiderivative = 32.22 \[ \int (e x)^m (c+d x)^3 \left (a+b x^2\right )^2 \, dx=\text {Too large to display} \] Input:

integrate((e*x)**m*(d*x+c)**3*(b*x**2+a)**2,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.23 \[ \int (e x)^m (c+d x)^3 \left (a+b x^2\right )^2 \, dx=\frac {b^{2} d^{3} e^{m} x^{8} x^{m}}{m + 8} + \frac {3 \, b^{2} c d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, b^{2} c^{2} d e^{m} x^{6} x^{m}}{m + 6} + \frac {2 \, a b d^{3} e^{m} x^{6} x^{m}}{m + 6} + \frac {b^{2} c^{3} e^{m} x^{5} x^{m}}{m + 5} + \frac {6 \, a b c d^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {6 \, a b c^{2} d e^{m} x^{4} x^{m}}{m + 4} + \frac {a^{2} d^{3} e^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, a b c^{3} e^{m} x^{3} x^{m}}{m + 3} + \frac {3 \, a^{2} c d^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {3 \, a^{2} c^{2} d e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} a^{2} c^{3}}{e {\left (m + 1\right )}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1802 vs. \(2 (215) = 430\).

Time = 0.15 (sec) , antiderivative size = 1802, normalized size of antiderivative = 8.38 \[ \int (e x)^m (c+d x)^3 \left (a+b x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

((e*x)^m*b^2*d^3*m^7*x^8 + 3*(e*x)^m*b^2*c*d^2*m^7*x^7 + 28*(e*x)^m*b^2*d^ 
3*m^6*x^8 + 3*(e*x)^m*b^2*c^2*d*m^7*x^6 + 2*(e*x)^m*a*b*d^3*m^7*x^6 + 87*( 
e*x)^m*b^2*c*d^2*m^6*x^7 + 322*(e*x)^m*b^2*d^3*m^5*x^8 + (e*x)^m*b^2*c^3*m 
^7*x^5 + 6*(e*x)^m*a*b*c*d^2*m^7*x^5 + 90*(e*x)^m*b^2*c^2*d*m^6*x^6 + 60*( 
e*x)^m*a*b*d^3*m^6*x^6 + 1029*(e*x)^m*b^2*c*d^2*m^5*x^7 + 1960*(e*x)^m*b^2 
*d^3*m^4*x^8 + 6*(e*x)^m*a*b*c^2*d*m^7*x^4 + (e*x)^m*a^2*d^3*m^7*x^4 + 31* 
(e*x)^m*b^2*c^3*m^6*x^5 + 186*(e*x)^m*a*b*c*d^2*m^6*x^5 + 1098*(e*x)^m*b^2 
*c^2*d*m^5*x^6 + 732*(e*x)^m*a*b*d^3*m^5*x^6 + 6405*(e*x)^m*b^2*c*d^2*m^4* 
x^7 + 6769*(e*x)^m*b^2*d^3*m^3*x^8 + 2*(e*x)^m*a*b*c^3*m^7*x^3 + 3*(e*x)^m 
*a^2*c*d^2*m^7*x^3 + 192*(e*x)^m*a*b*c^2*d*m^6*x^4 + 32*(e*x)^m*a^2*d^3*m^ 
6*x^4 + 391*(e*x)^m*b^2*c^3*m^5*x^5 + 2346*(e*x)^m*a*b*c*d^2*m^5*x^5 + 702 
0*(e*x)^m*b^2*c^2*d*m^4*x^6 + 4680*(e*x)^m*a*b*d^3*m^4*x^6 + 22512*(e*x)^m 
*b^2*c*d^2*m^3*x^7 + 13132*(e*x)^m*b^2*d^3*m^2*x^8 + 3*(e*x)^m*a^2*c^2*d*m 
^7*x^2 + 66*(e*x)^m*a*b*c^3*m^6*x^3 + 99*(e*x)^m*a^2*c*d^2*m^6*x^3 + 2508* 
(e*x)^m*a*b*c^2*d*m^5*x^4 + 418*(e*x)^m*a^2*d^3*m^5*x^4 + 2581*(e*x)^m*b^2 
*c^3*m^4*x^5 + 15486*(e*x)^m*a*b*c*d^2*m^4*x^5 + 25227*(e*x)^m*b^2*c^2*d*m 
^3*x^6 + 16818*(e*x)^m*a*b*d^3*m^3*x^6 + 44268*(e*x)^m*b^2*c*d^2*m^2*x^7 + 
 13068*(e*x)^m*b^2*d^3*m*x^8 + (e*x)^m*a^2*c^3*m^7*x + 102*(e*x)^m*a^2*c^2 
*d*m^6*x^2 + 894*(e*x)^m*a*b*c^3*m^5*x^3 + 1341*(e*x)^m*a^2*c*d^2*m^5*x^3 
+ 17184*(e*x)^m*a*b*c^2*d*m^4*x^4 + 2864*(e*x)^m*a^2*d^3*m^4*x^4 + 9544...
 

Mupad [B] (verification not implemented)

Time = 9.25 (sec) , antiderivative size = 741, normalized size of antiderivative = 3.45 \[ \int (e x)^m (c+d x)^3 \left (a+b x^2\right )^2 \, dx=\frac {a^2\,c^3\,x\,{\left (e\,x\right )}^m\,\left (m^7+35\,m^6+511\,m^5+4025\,m^4+18424\,m^3+48860\,m^2+69264\,m+40320\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {b^2\,d^3\,x^8\,{\left (e\,x\right )}^m\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,a^2\,c^2\,d\,x^2\,{\left (e\,x\right )}^m\,\left (m^7+34\,m^6+478\,m^5+3580\,m^4+15289\,m^3+36706\,m^2+44712\,m+20160\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,b^2\,c\,d^2\,x^7\,{\left (e\,x\right )}^m\,\left (m^7+29\,m^6+343\,m^5+2135\,m^4+7504\,m^3+14756\,m^2+14832\,m+5760\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {a\,c\,x^3\,{\left (e\,x\right )}^m\,\left (2\,b\,c^2+3\,a\,d^2\right )\,\left (m^7+33\,m^6+447\,m^5+3195\,m^4+12864\,m^3+28692\,m^2+32048\,m+13440\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {b\,c\,x^5\,{\left (e\,x\right )}^m\,\left (b\,c^2+6\,a\,d^2\right )\,\left (m^7+31\,m^6+391\,m^5+2581\,m^4+9544\,m^3+19564\,m^2+20304\,m+8064\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {a\,d\,x^4\,{\left (e\,x\right )}^m\,\left (6\,b\,c^2+a\,d^2\right )\,\left (m^7+32\,m^6+418\,m^5+2864\,m^4+10993\,m^3+23312\,m^2+24876\,m+10080\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {b\,d\,x^6\,{\left (e\,x\right )}^m\,\left (3\,b\,c^2+2\,a\,d^2\right )\,\left (m^7+30\,m^6+366\,m^5+2340\,m^4+8409\,m^3+16830\,m^2+17144\,m+6720\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320} \] Input:

int((e*x)^m*(a + b*x^2)^2*(c + d*x)^3,x)
 

Output:

(a^2*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^2*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))/(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*a^2*c^2*d*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*b^2*c*d 
^2*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*c*x^3*(e*x)^m*(3*a*d^2 + 2*b*c^2) 
*(32048*m + 28692*m^2 + 12864*m^3 + 3195*m^4 + 447*m^5 + 33*m^6 + m^7 + 13 
440))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 
+ 36*m^7 + m^8 + 40320) + (b*c*x^5*(e*x)^m*(6*a*d^2 + b*c^2)*(20304*m + 19 
564*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) + (a*d*x^4*(e*x)^m*(a*d^2 + 6*b*c^2)*(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) + (b*d 
*x^6*(e*x)^m*(2*a*d^2 + 3*b*c^2)*(17144*m + 16830*m^2 + 8409*m^3 + 2340...
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 1305, normalized size of antiderivative = 6.07 \[ \int (e x)^m (c+d x)^3 \left (a+b x^2\right )^2 \, dx =\text {Too large to display} \] Input:

int((e*x)^m*(d*x+c)^3*(b*x^2+a)^2,x)
 

Output:

(x**m*e**m*x*(a**2*c**3*m**7 + 35*a**2*c**3*m**6 + 511*a**2*c**3*m**5 + 40 
25*a**2*c**3*m**4 + 18424*a**2*c**3*m**3 + 48860*a**2*c**3*m**2 + 69264*a* 
*2*c**3*m + 40320*a**2*c**3 + 3*a**2*c**2*d*m**7*x + 102*a**2*c**2*d*m**6* 
x + 1434*a**2*c**2*d*m**5*x + 10740*a**2*c**2*d*m**4*x + 45867*a**2*c**2*d 
*m**3*x + 110118*a**2*c**2*d*m**2*x + 134136*a**2*c**2*d*m*x + 60480*a**2* 
c**2*d*x + 3*a**2*c*d**2*m**7*x**2 + 99*a**2*c*d**2*m**6*x**2 + 1341*a**2* 
c*d**2*m**5*x**2 + 9585*a**2*c*d**2*m**4*x**2 + 38592*a**2*c*d**2*m**3*x** 
2 + 86076*a**2*c*d**2*m**2*x**2 + 96144*a**2*c*d**2*m*x**2 + 40320*a**2*c* 
d**2*x**2 + a**2*d**3*m**7*x**3 + 32*a**2*d**3*m**6*x**3 + 418*a**2*d**3*m 
**5*x**3 + 2864*a**2*d**3*m**4*x**3 + 10993*a**2*d**3*m**3*x**3 + 23312*a* 
*2*d**3*m**2*x**3 + 24876*a**2*d**3*m*x**3 + 10080*a**2*d**3*x**3 + 2*a*b* 
c**3*m**7*x**2 + 66*a*b*c**3*m**6*x**2 + 894*a*b*c**3*m**5*x**2 + 6390*a*b 
*c**3*m**4*x**2 + 25728*a*b*c**3*m**3*x**2 + 57384*a*b*c**3*m**2*x**2 + 64 
096*a*b*c**3*m*x**2 + 26880*a*b*c**3*x**2 + 6*a*b*c**2*d*m**7*x**3 + 192*a 
*b*c**2*d*m**6*x**3 + 2508*a*b*c**2*d*m**5*x**3 + 17184*a*b*c**2*d*m**4*x* 
*3 + 65958*a*b*c**2*d*m**3*x**3 + 139872*a*b*c**2*d*m**2*x**3 + 149256*a*b 
*c**2*d*m*x**3 + 60480*a*b*c**2*d*x**3 + 6*a*b*c*d**2*m**7*x**4 + 186*a*b* 
c*d**2*m**6*x**4 + 2346*a*b*c*d**2*m**5*x**4 + 15486*a*b*c*d**2*m**4*x**4 
+ 57264*a*b*c*d**2*m**3*x**4 + 117384*a*b*c*d**2*m**2*x**4 + 121824*a*b*c* 
d**2*m*x**4 + 48384*a*b*c*d**2*x**4 + 2*a*b*d**3*m**7*x**5 + 60*a*b*d**...