\(\int (e x)^m (c+d x)^2 (a+b x^2) \, dx\) [101]

Optimal result

Integrand size = 20, antiderivative size = 108 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right ) \, dx=\frac {a c^2 (e x)^{1+m}}{e (1+m)}+\frac {2 a c d (e x)^{2+m}}{e^2 (2+m)}+\frac {\left (b c^2+a d^2\right ) (e x)^{3+m}}{e^3 (3+m)}+\frac {2 b c d (e x)^{4+m}}{e^4 (4+m)}+\frac {b d^2 (e x)^{5+m}}{e^5 (5+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right ) \, dx=x (e x)^m \left (\frac {a c^2}{1+m}+\frac {2 a c d x}{2+m}+\frac {\left (b c^2+a d^2\right ) x^2}{3+m}+\frac {2 b c d x^3}{4+m}+\frac {b d^2 x^4}{5+m}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 108, 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.

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

\(\Big \downarrow \) 522

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(a*c^2*(e*x)^(1 + m))/(e*(1 + m)) + (2*a*c*d*(e*x)^(2 + m))/(e^2*(2 + m)) 
+ ((b*c^2 + a*d^2)*(e*x)^(3 + m))/(e^3*(3 + m)) + (2*b*c*d*(e*x)^(4 + m))/ 
(e^4*(4 + m)) + (b*d^2*(e*x)^(5 + m))/(e^5*(5 + 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 = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {\left (a \,d^{2}+b \,c^{2}\right ) x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}+\frac {a \,c^{2} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {b \,d^{2} x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {2 a c d \,x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}+\frac {2 d b c \,x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}\)	\(107\)
gosper	\(\frac {x \left (b \,d^{2} m^{4} x^{4}+2 b c d \,m^{4} x^{3}+10 b \,d^{2} m^{3} x^{4}+a \,d^{2} m^{4} x^{2}+b \,c^{2} m^{4} x^{2}+22 b c d \,m^{3} x^{3}+35 b \,d^{2} m^{2} x^{4}+2 a c d \,m^{4} x +12 a \,d^{2} m^{3} x^{2}+12 b \,c^{2} m^{3} x^{2}+82 b c d \,m^{2} x^{3}+50 b \,d^{2} m \,x^{4}+a \,c^{2} m^{4}+26 a c d \,m^{3} x +49 a \,d^{2} m^{2} x^{2}+49 b \,c^{2} m^{2} x^{2}+122 b c d m \,x^{3}+24 b \,d^{2} x^{4}+14 a \,c^{2} m^{3}+118 a c d \,m^{2} x +78 a \,d^{2} m \,x^{2}+78 b \,c^{2} m \,x^{2}+60 b c d \,x^{3}+71 a \,c^{2} m^{2}+214 a c d m x +40 a \,d^{2} x^{2}+40 b \,c^{2} x^{2}+154 a \,c^{2} m +120 a d x c +120 a \,c^{2}\right ) \left (e x \right )^{m}}{\left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(325\)
risch	\(\frac {x \left (b \,d^{2} m^{4} x^{4}+2 b c d \,m^{4} x^{3}+10 b \,d^{2} m^{3} x^{4}+a \,d^{2} m^{4} x^{2}+b \,c^{2} m^{4} x^{2}+22 b c d \,m^{3} x^{3}+35 b \,d^{2} m^{2} x^{4}+2 a c d \,m^{4} x +12 a \,d^{2} m^{3} x^{2}+12 b \,c^{2} m^{3} x^{2}+82 b c d \,m^{2} x^{3}+50 b \,d^{2} m \,x^{4}+a \,c^{2} m^{4}+26 a c d \,m^{3} x +49 a \,d^{2} m^{2} x^{2}+49 b \,c^{2} m^{2} x^{2}+122 b c d m \,x^{3}+24 b \,d^{2} x^{4}+14 a \,c^{2} m^{3}+118 a c d \,m^{2} x +78 a \,d^{2} m \,x^{2}+78 b \,c^{2} m \,x^{2}+60 b c d \,x^{3}+71 a \,c^{2} m^{2}+214 a c d m x +40 a \,d^{2} x^{2}+40 b \,c^{2} x^{2}+154 a \,c^{2} m +120 a d x c +120 a \,c^{2}\right ) \left (e x \right )^{m}}{\left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(325\)
orering	\(\frac {x \left (b \,d^{2} m^{4} x^{4}+2 b c d \,m^{4} x^{3}+10 b \,d^{2} m^{3} x^{4}+a \,d^{2} m^{4} x^{2}+b \,c^{2} m^{4} x^{2}+22 b c d \,m^{3} x^{3}+35 b \,d^{2} m^{2} x^{4}+2 a c d \,m^{4} x +12 a \,d^{2} m^{3} x^{2}+12 b \,c^{2} m^{3} x^{2}+82 b c d \,m^{2} x^{3}+50 b \,d^{2} m \,x^{4}+a \,c^{2} m^{4}+26 a c d \,m^{3} x +49 a \,d^{2} m^{2} x^{2}+49 b \,c^{2} m^{2} x^{2}+122 b c d m \,x^{3}+24 b \,d^{2} x^{4}+14 a \,c^{2} m^{3}+118 a c d \,m^{2} x +78 a \,d^{2} m \,x^{2}+78 b \,c^{2} m \,x^{2}+60 b c d \,x^{3}+71 a \,c^{2} m^{2}+214 a c d m x +40 a \,d^{2} x^{2}+40 b \,c^{2} x^{2}+154 a \,c^{2} m +120 a d x c +120 a \,c^{2}\right ) \left (e x \right )^{m}}{\left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(325\)
parallelrisch	\(\frac {2 x^{2} \left (e x \right )^{m} a c d \,m^{4}+24 x^{5} \left (e x \right )^{m} b \,d^{2}+40 x^{3} \left (e x \right )^{m} a \,d^{2}+40 x^{3} \left (e x \right )^{m} b \,c^{2}+120 x \left (e x \right )^{m} a \,c^{2}+22 x^{4} \left (e x \right )^{m} b c d \,m^{3}+50 x^{5} \left (e x \right )^{m} b \,d^{2} m +122 x^{4} \left (e x \right )^{m} b c d m +2 x^{4} \left (e x \right )^{m} b c d \,m^{4}+120 x^{2} \left (e x \right )^{m} a c d +154 x \left (e x \right )^{m} a \,c^{2} m +10 x^{5} \left (e x \right )^{m} b \,d^{2} m^{3}+35 x^{5} \left (e x \right )^{m} b \,d^{2} m^{2}+x^{3} \left (e x \right )^{m} a \,d^{2} m^{4}+x^{3} \left (e x \right )^{m} b \,c^{2} m^{4}+26 x^{2} \left (e x \right )^{m} a c d \,m^{3}+49 x^{3} \left (e x \right )^{m} a \,d^{2} m^{2}+49 x^{3} \left (e x \right )^{m} b \,c^{2} m^{2}+82 x^{4} \left (e x \right )^{m} b c d \,m^{2}+118 x^{2} \left (e x \right )^{m} a c d \,m^{2}+214 x^{2} \left (e x \right )^{m} a c d m +12 x^{3} \left (e x \right )^{m} a \,d^{2} m^{3}+12 x^{3} \left (e x \right )^{m} b \,c^{2} m^{3}+x^{5} \left (e x \right )^{m} b \,d^{2} m^{4}+60 x^{4} \left (e x \right )^{m} b c d +78 x^{3} \left (e x \right )^{m} a \,d^{2} m +78 x^{3} \left (e x \right )^{m} b \,c^{2} m +14 x \left (e x \right )^{m} a \,c^{2} m^{3}+71 x \left (e x \right )^{m} a \,c^{2} m^{2}+x \left (e x \right )^{m} a \,c^{2} m^{4}}{\left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(484\)

Input:

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

Output:

(a*d^2+b*c^2)/(3+m)*x^3*exp(m*ln(e*x))+a*c^2/(1+m)*x*exp(m*ln(e*x))+b*d^2/ 
(5+m)*x^5*exp(m*ln(e*x))+2*a*c*d/(2+m)*x^2*exp(m*ln(e*x))+2*d*b*c/(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. 276 vs. \(2 (108) = 216\).

Time = 0.09 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.56 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right ) \, dx=\frac {{\left ({\left (b d^{2} m^{4} + 10 \, b d^{2} m^{3} + 35 \, b d^{2} m^{2} + 50 \, b d^{2} m + 24 \, b d^{2}\right )} x^{5} + 2 \, {\left (b c d m^{4} + 11 \, b c d m^{3} + 41 \, b c d m^{2} + 61 \, b c d m + 30 \, b c d\right )} x^{4} + {\left ({\left (b c^{2} + a d^{2}\right )} m^{4} + 12 \, {\left (b c^{2} + a d^{2}\right )} m^{3} + 40 \, b c^{2} + 40 \, a d^{2} + 49 \, {\left (b c^{2} + a d^{2}\right )} m^{2} + 78 \, {\left (b c^{2} + a d^{2}\right )} m\right )} x^{3} + 2 \, {\left (a c d m^{4} + 13 \, a c d m^{3} + 59 \, a c d m^{2} + 107 \, a c d m + 60 \, a c d\right )} x^{2} + {\left (a c^{2} m^{4} + 14 \, a c^{2} m^{3} + 71 \, a c^{2} m^{2} + 154 \, a c^{2} m + 120 \, a c^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \] Input:

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

Output:

((b*d^2*m^4 + 10*b*d^2*m^3 + 35*b*d^2*m^2 + 50*b*d^2*m + 24*b*d^2)*x^5 + 2 
*(b*c*d*m^4 + 11*b*c*d*m^3 + 41*b*c*d*m^2 + 61*b*c*d*m + 30*b*c*d)*x^4 + ( 
(b*c^2 + a*d^2)*m^4 + 12*(b*c^2 + a*d^2)*m^3 + 40*b*c^2 + 40*a*d^2 + 49*(b 
*c^2 + a*d^2)*m^2 + 78*(b*c^2 + a*d^2)*m)*x^3 + 2*(a*c*d*m^4 + 13*a*c*d*m^ 
3 + 59*a*c*d*m^2 + 107*a*c*d*m + 60*a*c*d)*x^2 + (a*c^2*m^4 + 14*a*c^2*m^3 
 + 71*a*c^2*m^2 + 154*a*c^2*m + 120*a*c^2)*x)*(e*x)^m/(m^5 + 15*m^4 + 85*m 
^3 + 225*m^2 + 274*m + 120)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1516 vs. \(2 (99) = 198\).

Time = 0.39 (sec) , antiderivative size = 1516, normalized size of antiderivative = 14.04 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise(((-a*c**2/(4*x**4) - 2*a*c*d/(3*x**3) - a*d**2/(2*x**2) - b*c**2 
/(2*x**2) - 2*b*c*d/x + b*d**2*log(x))/e**5, Eq(m, -5)), ((-a*c**2/(3*x**3 
) - a*c*d/x**2 - a*d**2/x - b*c**2/x + 2*b*c*d*log(x) + b*d**2*x)/e**4, Eq 
(m, -4)), ((-a*c**2/(2*x**2) - 2*a*c*d/x + a*d**2*log(x) + b*c**2*log(x) + 
 2*b*c*d*x + b*d**2*x**2/2)/e**3, Eq(m, -3)), ((-a*c**2/x + 2*a*c*d*log(x) 
 + a*d**2*x + b*c**2*x + b*c*d*x**2 + b*d**2*x**3/3)/e**2, Eq(m, -2)), ((a 
*c**2*log(x) + 2*a*c*d*x + a*d**2*x**2/2 + b*c**2*x**2/2 + 2*b*c*d*x**3/3 
+ b*d**2*x**4/4)/e, Eq(m, -1)), (a*c**2*m**4*x*(e*x)**m/(m**5 + 15*m**4 + 
85*m**3 + 225*m**2 + 274*m + 120) + 14*a*c**2*m**3*x*(e*x)**m/(m**5 + 15*m 
**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 71*a*c**2*m**2*x*(e*x)**m/(m**5 
+ 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 154*a*c**2*m*x*(e*x)**m/(m 
**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 120*a*c**2*x*(e*x)**m/ 
(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 2*a*c*d*m**4*x**2*(e 
*x)**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 26*a*c*d*m**3 
*x**2*(e*x)**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 118*a 
*c*d*m**2*x**2*(e*x)**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120 
) + 214*a*c*d*m*x**2*(e*x)**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m 
 + 120) + 120*a*c*d*x**2*(e*x)**m/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 2 
74*m + 120) + a*d**2*m**4*x**3*(e*x)**m/(m**5 + 15*m**4 + 85*m**3 + 225*m* 
*2 + 274*m + 120) + 12*a*d**2*m**3*x**3*(e*x)**m/(m**5 + 15*m**4 + 85*m...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.07 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right ) \, dx=\frac {b d^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, b c d e^{m} x^{4} x^{m}}{m + 4} + \frac {b c^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {a d^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, a c d e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} a c^{2}}{e {\left (m + 1\right )}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (108) = 216\).

Time = 0.13 (sec) , antiderivative size = 483, normalized size of antiderivative = 4.47 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right ) \, dx=\frac {\left (e x\right )^{m} b d^{2} m^{4} x^{5} + 2 \, \left (e x\right )^{m} b c d m^{4} x^{4} + 10 \, \left (e x\right )^{m} b d^{2} m^{3} x^{5} + \left (e x\right )^{m} b c^{2} m^{4} x^{3} + \left (e x\right )^{m} a d^{2} m^{4} x^{3} + 22 \, \left (e x\right )^{m} b c d m^{3} x^{4} + 35 \, \left (e x\right )^{m} b d^{2} m^{2} x^{5} + 2 \, \left (e x\right )^{m} a c d m^{4} x^{2} + 12 \, \left (e x\right )^{m} b c^{2} m^{3} x^{3} + 12 \, \left (e x\right )^{m} a d^{2} m^{3} x^{3} + 82 \, \left (e x\right )^{m} b c d m^{2} x^{4} + 50 \, \left (e x\right )^{m} b d^{2} m x^{5} + \left (e x\right )^{m} a c^{2} m^{4} x + 26 \, \left (e x\right )^{m} a c d m^{3} x^{2} + 49 \, \left (e x\right )^{m} b c^{2} m^{2} x^{3} + 49 \, \left (e x\right )^{m} a d^{2} m^{2} x^{3} + 122 \, \left (e x\right )^{m} b c d m x^{4} + 24 \, \left (e x\right )^{m} b d^{2} x^{5} + 14 \, \left (e x\right )^{m} a c^{2} m^{3} x + 118 \, \left (e x\right )^{m} a c d m^{2} x^{2} + 78 \, \left (e x\right )^{m} b c^{2} m x^{3} + 78 \, \left (e x\right )^{m} a d^{2} m x^{3} + 60 \, \left (e x\right )^{m} b c d x^{4} + 71 \, \left (e x\right )^{m} a c^{2} m^{2} x + 214 \, \left (e x\right )^{m} a c d m x^{2} + 40 \, \left (e x\right )^{m} b c^{2} x^{3} + 40 \, \left (e x\right )^{m} a d^{2} x^{3} + 154 \, \left (e x\right )^{m} a c^{2} m x + 120 \, \left (e x\right )^{m} a c d x^{2} + 120 \, \left (e x\right )^{m} a c^{2} x}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \] Input:

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

Output:

((e*x)^m*b*d^2*m^4*x^5 + 2*(e*x)^m*b*c*d*m^4*x^4 + 10*(e*x)^m*b*d^2*m^3*x^ 
5 + (e*x)^m*b*c^2*m^4*x^3 + (e*x)^m*a*d^2*m^4*x^3 + 22*(e*x)^m*b*c*d*m^3*x 
^4 + 35*(e*x)^m*b*d^2*m^2*x^5 + 2*(e*x)^m*a*c*d*m^4*x^2 + 12*(e*x)^m*b*c^2 
*m^3*x^3 + 12*(e*x)^m*a*d^2*m^3*x^3 + 82*(e*x)^m*b*c*d*m^2*x^4 + 50*(e*x)^ 
m*b*d^2*m*x^5 + (e*x)^m*a*c^2*m^4*x + 26*(e*x)^m*a*c*d*m^3*x^2 + 49*(e*x)^ 
m*b*c^2*m^2*x^3 + 49*(e*x)^m*a*d^2*m^2*x^3 + 122*(e*x)^m*b*c*d*m*x^4 + 24* 
(e*x)^m*b*d^2*x^5 + 14*(e*x)^m*a*c^2*m^3*x + 118*(e*x)^m*a*c*d*m^2*x^2 + 7 
8*(e*x)^m*b*c^2*m*x^3 + 78*(e*x)^m*a*d^2*m*x^3 + 60*(e*x)^m*b*c*d*x^4 + 71 
*(e*x)^m*a*c^2*m^2*x + 214*(e*x)^m*a*c*d*m*x^2 + 40*(e*x)^m*b*c^2*x^3 + 40 
*(e*x)^m*a*d^2*x^3 + 154*(e*x)^m*a*c^2*m*x + 120*(e*x)^m*a*c*d*x^2 + 120*( 
e*x)^m*a*c^2*x)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)
 

Mupad [B] (verification not implemented)

Time = 8.87 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.47 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right ) \, dx={\left (e\,x\right )}^m\,\left (\frac {x^3\,\left (b\,c^2+a\,d^2\right )\,\left (m^4+12\,m^3+49\,m^2+78\,m+40\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {a\,c^2\,x\,\left (m^4+14\,m^3+71\,m^2+154\,m+120\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {b\,d^2\,x^5\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {2\,a\,c\,d\,x^2\,\left (m^4+13\,m^3+59\,m^2+107\,m+60\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {2\,b\,c\,d\,x^4\,\left (m^4+11\,m^3+41\,m^2+61\,m+30\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}\right ) \] Input:

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

Output:

(e*x)^m*((x^3*(a*d^2 + b*c^2)*(78*m + 49*m^2 + 12*m^3 + m^4 + 40))/(274*m 
+ 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120) + (a*c^2*x*(154*m + 71*m^2 + 14*m 
^3 + m^4 + 120))/(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120) + (b*d^2* 
x^5*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/(274*m + 225*m^2 + 85*m^3 + 15*m^ 
4 + m^5 + 120) + (2*a*c*d*x^2*(107*m + 59*m^2 + 13*m^3 + m^4 + 60))/(274*m 
 + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120) + (2*b*c*d*x^4*(61*m + 41*m^2 + 
11*m^3 + m^4 + 30))/(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 325, normalized size of antiderivative = 3.01 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right ) \, dx=\frac {x^{m} e^{m} x \left (b \,d^{2} m^{4} x^{4}+2 b c d \,m^{4} x^{3}+10 b \,d^{2} m^{3} x^{4}+a \,d^{2} m^{4} x^{2}+b \,c^{2} m^{4} x^{2}+22 b c d \,m^{3} x^{3}+35 b \,d^{2} m^{2} x^{4}+2 a c d \,m^{4} x +12 a \,d^{2} m^{3} x^{2}+12 b \,c^{2} m^{3} x^{2}+82 b c d \,m^{2} x^{3}+50 b \,d^{2} m \,x^{4}+a \,c^{2} m^{4}+26 a c d \,m^{3} x +49 a \,d^{2} m^{2} x^{2}+49 b \,c^{2} m^{2} x^{2}+122 b c d m \,x^{3}+24 b \,d^{2} x^{4}+14 a \,c^{2} m^{3}+118 a c d \,m^{2} x +78 a \,d^{2} m \,x^{2}+78 b \,c^{2} m \,x^{2}+60 b c d \,x^{3}+71 a \,c^{2} m^{2}+214 a c d m x +40 a \,d^{2} x^{2}+40 b \,c^{2} x^{2}+154 a \,c^{2} m +120 a c d x +120 a \,c^{2}\right )}{m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120} \] Input:

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

Output:

(x**m*e**m*x*(a*c**2*m**4 + 14*a*c**2*m**3 + 71*a*c**2*m**2 + 154*a*c**2*m 
 + 120*a*c**2 + 2*a*c*d*m**4*x + 26*a*c*d*m**3*x + 118*a*c*d*m**2*x + 214* 
a*c*d*m*x + 120*a*c*d*x + a*d**2*m**4*x**2 + 12*a*d**2*m**3*x**2 + 49*a*d* 
*2*m**2*x**2 + 78*a*d**2*m*x**2 + 40*a*d**2*x**2 + b*c**2*m**4*x**2 + 12*b 
*c**2*m**3*x**2 + 49*b*c**2*m**2*x**2 + 78*b*c**2*m*x**2 + 40*b*c**2*x**2 
+ 2*b*c*d*m**4*x**3 + 22*b*c*d*m**3*x**3 + 82*b*c*d*m**2*x**3 + 122*b*c*d* 
m*x**3 + 60*b*c*d*x**3 + b*d**2*m**4*x**4 + 10*b*d**2*m**3*x**4 + 35*b*d** 
2*m**2*x**4 + 50*b*d**2*m*x**4 + 24*b*d**2*x**4))/(m**5 + 15*m**4 + 85*m** 
3 + 225*m**2 + 274*m + 120)