\(\int (e x)^m (c+d x)^2 (a+b x^2)^2 \, dx\) [108]

Optimal result

Integrand size = 22, antiderivative size = 168 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right )^2 \, dx=\frac {a^2 c^2 (e x)^{1+m}}{e (1+m)}+\frac {2 a^2 c d (e x)^{2+m}}{e^2 (2+m)}+\frac {a \left (2 b c^2+a d^2\right ) (e x)^{3+m}}{e^3 (3+m)}+\frac {4 a b c d (e x)^{4+m}}{e^4 (4+m)}+\frac {b \left (b c^2+2 a d^2\right ) (e x)^{5+m}}{e^5 (5+m)}+\frac {2 b^2 c d (e x)^{6+m}}{e^6 (6+m)}+\frac {b^2 d^2 (e x)^{7+m}}{e^7 (7+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.72 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right )^2 \, dx=x (e x)^m \left (\frac {a^2 c^2}{1+m}+\frac {2 a^2 c d x}{2+m}+\frac {a \left (2 b c^2+a d^2\right ) x^2}{3+m}+\frac {4 a b c d x^3}{4+m}+\frac {b \left (b c^2+2 a d^2\right ) x^4}{5+m}+\frac {2 b^2 c d x^5}{6+m}+\frac {b^2 d^2 x^6}{7+m}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 168, 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.

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

\(\Big \downarrow \) 522

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(a^2*c^2*(e*x)^(1 + m))/(e*(1 + m)) + (2*a^2*c*d*(e*x)^(2 + m))/(e^2*(2 + 
m)) + (a*(2*b*c^2 + a*d^2)*(e*x)^(3 + m))/(e^3*(3 + m)) + (4*a*b*c*d*(e*x) 
^(4 + m))/(e^4*(4 + m)) + (b*(b*c^2 + 2*a*d^2)*(e*x)^(5 + m))/(e^5*(5 + m) 
) + (2*b^2*c*d*(e*x)^(6 + m))/(e^6*(6 + m)) + (b^2*d^2*(e*x)^(7 + m))/(e^7 
*(7 + 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 = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.99

method	result	size
norman	\(\frac {a \left (a \,d^{2}+2 b \,c^{2}\right ) x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}+\frac {a^{2} c^{2} x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}+\frac {b \left (2 a \,d^{2}+b \,c^{2}\right ) x^{5} {\mathrm e}^{m \ln \left (e x \right )}}{5+m}+\frac {b^{2} d^{2} x^{7} {\mathrm e}^{m \ln \left (e x \right )}}{7+m}+\frac {2 a^{2} c d \,x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}+\frac {2 c d \,b^{2} x^{6} {\mathrm e}^{m \ln \left (e x \right )}}{6+m}+\frac {4 a b c d \,x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}\)	\(167\)
gosper	\(\frac {x \left (b^{2} d^{2} m^{6} x^{6}+2 b^{2} c d \,m^{6} x^{5}+21 b^{2} d^{2} m^{5} x^{6}+2 a b \,d^{2} m^{6} x^{4}+b^{2} c^{2} m^{6} x^{4}+44 b^{2} c d \,m^{5} x^{5}+175 b^{2} d^{2} m^{4} x^{6}+4 a b c d \,m^{6} x^{3}+46 a b \,d^{2} m^{5} x^{4}+23 b^{2} c^{2} m^{5} x^{4}+380 b^{2} c d \,m^{4} x^{5}+735 b^{2} d^{2} m^{3} x^{6}+a^{2} d^{2} m^{6} x^{2}+2 a b \,c^{2} m^{6} x^{2}+96 a b c d \,m^{5} x^{3}+414 a b \,d^{2} m^{4} x^{4}+207 b^{2} c^{2} m^{4} x^{4}+1640 b^{2} c d \,m^{3} x^{5}+1624 b^{2} d^{2} m^{2} x^{6}+2 a^{2} c d \,m^{6} x +25 a^{2} d^{2} m^{5} x^{2}+50 a b \,c^{2} m^{5} x^{2}+904 a b c d \,m^{4} x^{3}+1850 a b \,d^{2} m^{3} x^{4}+925 b^{2} c^{2} m^{3} x^{4}+3698 b^{2} c d \,m^{2} x^{5}+1764 b^{2} d^{2} m \,x^{6}+a^{2} c^{2} m^{6}+52 a^{2} c d \,m^{5} x +247 a^{2} d^{2} m^{4} x^{2}+494 a b \,c^{2} m^{4} x^{2}+4224 a b c d \,m^{3} x^{3}+4288 a b \,d^{2} m^{2} x^{4}+2144 b^{2} c^{2} m^{2} x^{4}+4076 b^{2} c d m \,x^{5}+720 b^{2} d^{2} x^{6}+27 a^{2} c^{2} m^{5}+540 a^{2} c d \,m^{4} x +1219 a^{2} d^{2} m^{3} x^{2}+2438 a b \,c^{2} m^{3} x^{2}+10180 a b c d \,m^{2} x^{3}+4824 a b \,d^{2} m \,x^{4}+2412 b^{2} c^{2} m \,x^{4}+1680 b^{2} c d \,x^{5}+295 a^{2} c^{2} m^{4}+2840 a^{2} c d \,m^{3} x +3112 a^{2} d^{2} m^{2} x^{2}+6224 a b \,c^{2} m^{2} x^{2}+11808 a b c d m \,x^{3}+2016 a b \,d^{2} x^{4}+1008 b^{2} c^{2} x^{4}+1665 a^{2} c^{2} m^{3}+7858 a^{2} c d \,m^{2} x +3796 a^{2} d^{2} m \,x^{2}+7592 a b \,c^{2} m \,x^{2}+5040 a b c d \,x^{3}+5104 a^{2} c^{2} m^{2}+10548 a^{2} c d m x +1680 a^{2} d^{2} x^{2}+3360 a b \,c^{2} x^{2}+8028 a^{2} c^{2} m +5040 a^{2} c d x +5040 a^{2} c^{2}\right ) \left (e x \right )^{m}}{\left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(800\)
risch	\(\frac {x \left (b^{2} d^{2} m^{6} x^{6}+2 b^{2} c d \,m^{6} x^{5}+21 b^{2} d^{2} m^{5} x^{6}+2 a b \,d^{2} m^{6} x^{4}+b^{2} c^{2} m^{6} x^{4}+44 b^{2} c d \,m^{5} x^{5}+175 b^{2} d^{2} m^{4} x^{6}+4 a b c d \,m^{6} x^{3}+46 a b \,d^{2} m^{5} x^{4}+23 b^{2} c^{2} m^{5} x^{4}+380 b^{2} c d \,m^{4} x^{5}+735 b^{2} d^{2} m^{3} x^{6}+a^{2} d^{2} m^{6} x^{2}+2 a b \,c^{2} m^{6} x^{2}+96 a b c d \,m^{5} x^{3}+414 a b \,d^{2} m^{4} x^{4}+207 b^{2} c^{2} m^{4} x^{4}+1640 b^{2} c d \,m^{3} x^{5}+1624 b^{2} d^{2} m^{2} x^{6}+2 a^{2} c d \,m^{6} x +25 a^{2} d^{2} m^{5} x^{2}+50 a b \,c^{2} m^{5} x^{2}+904 a b c d \,m^{4} x^{3}+1850 a b \,d^{2} m^{3} x^{4}+925 b^{2} c^{2} m^{3} x^{4}+3698 b^{2} c d \,m^{2} x^{5}+1764 b^{2} d^{2} m \,x^{6}+a^{2} c^{2} m^{6}+52 a^{2} c d \,m^{5} x +247 a^{2} d^{2} m^{4} x^{2}+494 a b \,c^{2} m^{4} x^{2}+4224 a b c d \,m^{3} x^{3}+4288 a b \,d^{2} m^{2} x^{4}+2144 b^{2} c^{2} m^{2} x^{4}+4076 b^{2} c d m \,x^{5}+720 b^{2} d^{2} x^{6}+27 a^{2} c^{2} m^{5}+540 a^{2} c d \,m^{4} x +1219 a^{2} d^{2} m^{3} x^{2}+2438 a b \,c^{2} m^{3} x^{2}+10180 a b c d \,m^{2} x^{3}+4824 a b \,d^{2} m \,x^{4}+2412 b^{2} c^{2} m \,x^{4}+1680 b^{2} c d \,x^{5}+295 a^{2} c^{2} m^{4}+2840 a^{2} c d \,m^{3} x +3112 a^{2} d^{2} m^{2} x^{2}+6224 a b \,c^{2} m^{2} x^{2}+11808 a b c d m \,x^{3}+2016 a b \,d^{2} x^{4}+1008 b^{2} c^{2} x^{4}+1665 a^{2} c^{2} m^{3}+7858 a^{2} c d \,m^{2} x +3796 a^{2} d^{2} m \,x^{2}+7592 a b \,c^{2} m \,x^{2}+5040 a b c d \,x^{3}+5104 a^{2} c^{2} m^{2}+10548 a^{2} c d m x +1680 a^{2} d^{2} x^{2}+3360 a b \,c^{2} x^{2}+8028 a^{2} c^{2} m +5040 a^{2} c d x +5040 a^{2} c^{2}\right ) \left (e x \right )^{m}}{\left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(800\)
orering	\(\frac {x \left (b^{2} d^{2} m^{6} x^{6}+2 b^{2} c d \,m^{6} x^{5}+21 b^{2} d^{2} m^{5} x^{6}+2 a b \,d^{2} m^{6} x^{4}+b^{2} c^{2} m^{6} x^{4}+44 b^{2} c d \,m^{5} x^{5}+175 b^{2} d^{2} m^{4} x^{6}+4 a b c d \,m^{6} x^{3}+46 a b \,d^{2} m^{5} x^{4}+23 b^{2} c^{2} m^{5} x^{4}+380 b^{2} c d \,m^{4} x^{5}+735 b^{2} d^{2} m^{3} x^{6}+a^{2} d^{2} m^{6} x^{2}+2 a b \,c^{2} m^{6} x^{2}+96 a b c d \,m^{5} x^{3}+414 a b \,d^{2} m^{4} x^{4}+207 b^{2} c^{2} m^{4} x^{4}+1640 b^{2} c d \,m^{3} x^{5}+1624 b^{2} d^{2} m^{2} x^{6}+2 a^{2} c d \,m^{6} x +25 a^{2} d^{2} m^{5} x^{2}+50 a b \,c^{2} m^{5} x^{2}+904 a b c d \,m^{4} x^{3}+1850 a b \,d^{2} m^{3} x^{4}+925 b^{2} c^{2} m^{3} x^{4}+3698 b^{2} c d \,m^{2} x^{5}+1764 b^{2} d^{2} m \,x^{6}+a^{2} c^{2} m^{6}+52 a^{2} c d \,m^{5} x +247 a^{2} d^{2} m^{4} x^{2}+494 a b \,c^{2} m^{4} x^{2}+4224 a b c d \,m^{3} x^{3}+4288 a b \,d^{2} m^{2} x^{4}+2144 b^{2} c^{2} m^{2} x^{4}+4076 b^{2} c d m \,x^{5}+720 b^{2} d^{2} x^{6}+27 a^{2} c^{2} m^{5}+540 a^{2} c d \,m^{4} x +1219 a^{2} d^{2} m^{3} x^{2}+2438 a b \,c^{2} m^{3} x^{2}+10180 a b c d \,m^{2} x^{3}+4824 a b \,d^{2} m \,x^{4}+2412 b^{2} c^{2} m \,x^{4}+1680 b^{2} c d \,x^{5}+295 a^{2} c^{2} m^{4}+2840 a^{2} c d \,m^{3} x +3112 a^{2} d^{2} m^{2} x^{2}+6224 a b \,c^{2} m^{2} x^{2}+11808 a b c d m \,x^{3}+2016 a b \,d^{2} x^{4}+1008 b^{2} c^{2} x^{4}+1665 a^{2} c^{2} m^{3}+7858 a^{2} c d \,m^{2} x +3796 a^{2} d^{2} m \,x^{2}+7592 a b \,c^{2} m \,x^{2}+5040 a b c d \,x^{3}+5104 a^{2} c^{2} m^{2}+10548 a^{2} c d m x +1680 a^{2} d^{2} x^{2}+3360 a b \,c^{2} x^{2}+8028 a^{2} c^{2} m +5040 a^{2} c d x +5040 a^{2} c^{2}\right ) \left (e x \right )^{m}}{\left (7+m \right ) \left (6+m \right ) \left (5+m \right ) \left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\)	\(800\)
parallelrisch	\(\text {Expression too large to display}\)	\(1130\)

Input:

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

Output:

a*(a*d^2+2*b*c^2)/(3+m)*x^3*exp(m*ln(e*x))+a^2*c^2/(1+m)*x*exp(m*ln(e*x))+ 
b*(2*a*d^2+b*c^2)/(5+m)*x^5*exp(m*ln(e*x))+b^2*d^2/(7+m)*x^7*exp(m*ln(e*x) 
)+2*a^2*c*d/(2+m)*x^2*exp(m*ln(e*x))+2*c*d*b^2/(6+m)*x^6*exp(m*ln(e*x))+4* 
a*b*c*d/(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. 669 vs. \(2 (168) = 336\).

Time = 0.12 (sec) , antiderivative size = 669, normalized size of antiderivative = 3.98 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right )^2 \, dx=\frac {{\left ({\left (b^{2} d^{2} m^{6} + 21 \, b^{2} d^{2} m^{5} + 175 \, b^{2} d^{2} m^{4} + 735 \, b^{2} d^{2} m^{3} + 1624 \, b^{2} d^{2} m^{2} + 1764 \, b^{2} d^{2} m + 720 \, b^{2} d^{2}\right )} x^{7} + 2 \, {\left (b^{2} c d m^{6} + 22 \, b^{2} c d m^{5} + 190 \, b^{2} c d m^{4} + 820 \, b^{2} c d m^{3} + 1849 \, b^{2} c d m^{2} + 2038 \, b^{2} c d m + 840 \, b^{2} c d\right )} x^{6} + {\left ({\left (b^{2} c^{2} + 2 \, a b d^{2}\right )} m^{6} + 23 \, {\left (b^{2} c^{2} + 2 \, a b d^{2}\right )} m^{5} + 207 \, {\left (b^{2} c^{2} + 2 \, a b d^{2}\right )} m^{4} + 1008 \, b^{2} c^{2} + 2016 \, a b d^{2} + 925 \, {\left (b^{2} c^{2} + 2 \, a b d^{2}\right )} m^{3} + 2144 \, {\left (b^{2} c^{2} + 2 \, a b d^{2}\right )} m^{2} + 2412 \, {\left (b^{2} c^{2} + 2 \, a b d^{2}\right )} m\right )} x^{5} + 4 \, {\left (a b c d m^{6} + 24 \, a b c d m^{5} + 226 \, a b c d m^{4} + 1056 \, a b c d m^{3} + 2545 \, a b c d m^{2} + 2952 \, a b c d m + 1260 \, a b c d\right )} x^{4} + {\left ({\left (2 \, a b c^{2} + a^{2} d^{2}\right )} m^{6} + 25 \, {\left (2 \, a b c^{2} + a^{2} d^{2}\right )} m^{5} + 247 \, {\left (2 \, a b c^{2} + a^{2} d^{2}\right )} m^{4} + 3360 \, a b c^{2} + 1680 \, a^{2} d^{2} + 1219 \, {\left (2 \, a b c^{2} + a^{2} d^{2}\right )} m^{3} + 3112 \, {\left (2 \, a b c^{2} + a^{2} d^{2}\right )} m^{2} + 3796 \, {\left (2 \, a b c^{2} + a^{2} d^{2}\right )} m\right )} x^{3} + 2 \, {\left (a^{2} c d m^{6} + 26 \, a^{2} c d m^{5} + 270 \, a^{2} c d m^{4} + 1420 \, a^{2} c d m^{3} + 3929 \, a^{2} c d m^{2} + 5274 \, a^{2} c d m + 2520 \, a^{2} c d\right )} x^{2} + {\left (a^{2} c^{2} m^{6} + 27 \, a^{2} c^{2} m^{5} + 295 \, a^{2} c^{2} m^{4} + 1665 \, a^{2} c^{2} m^{3} + 5104 \, a^{2} c^{2} m^{2} + 8028 \, a^{2} c^{2} m + 5040 \, a^{2} c^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} \] Input:

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

Output:

((b^2*d^2*m^6 + 21*b^2*d^2*m^5 + 175*b^2*d^2*m^4 + 735*b^2*d^2*m^3 + 1624* 
b^2*d^2*m^2 + 1764*b^2*d^2*m + 720*b^2*d^2)*x^7 + 2*(b^2*c*d*m^6 + 22*b^2* 
c*d*m^5 + 190*b^2*c*d*m^4 + 820*b^2*c*d*m^3 + 1849*b^2*c*d*m^2 + 2038*b^2* 
c*d*m + 840*b^2*c*d)*x^6 + ((b^2*c^2 + 2*a*b*d^2)*m^6 + 23*(b^2*c^2 + 2*a* 
b*d^2)*m^5 + 207*(b^2*c^2 + 2*a*b*d^2)*m^4 + 1008*b^2*c^2 + 2016*a*b*d^2 + 
 925*(b^2*c^2 + 2*a*b*d^2)*m^3 + 2144*(b^2*c^2 + 2*a*b*d^2)*m^2 + 2412*(b^ 
2*c^2 + 2*a*b*d^2)*m)*x^5 + 4*(a*b*c*d*m^6 + 24*a*b*c*d*m^5 + 226*a*b*c*d* 
m^4 + 1056*a*b*c*d*m^3 + 2545*a*b*c*d*m^2 + 2952*a*b*c*d*m + 1260*a*b*c*d) 
*x^4 + ((2*a*b*c^2 + a^2*d^2)*m^6 + 25*(2*a*b*c^2 + a^2*d^2)*m^5 + 247*(2* 
a*b*c^2 + a^2*d^2)*m^4 + 3360*a*b*c^2 + 1680*a^2*d^2 + 1219*(2*a*b*c^2 + a 
^2*d^2)*m^3 + 3112*(2*a*b*c^2 + a^2*d^2)*m^2 + 3796*(2*a*b*c^2 + a^2*d^2)* 
m)*x^3 + 2*(a^2*c*d*m^6 + 26*a^2*c*d*m^5 + 270*a^2*c*d*m^4 + 1420*a^2*c*d* 
m^3 + 3929*a^2*c*d*m^2 + 5274*a^2*c*d*m + 2520*a^2*c*d)*x^2 + (a^2*c^2*m^6 
 + 27*a^2*c^2*m^5 + 295*a^2*c^2*m^4 + 1665*a^2*c^2*m^3 + 5104*a^2*c^2*m^2 
+ 8028*a^2*c^2*m + 5040*a^2*c^2)*x)*(e*x)^m/(m^7 + 28*m^6 + 322*m^5 + 1960 
*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4094 vs. \(2 (158) = 316\).

Time = 0.58 (sec) , antiderivative size = 4094, normalized size of antiderivative = 24.37 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise(((-a**2*c**2/(6*x**6) - 2*a**2*c*d/(5*x**5) - a**2*d**2/(4*x**4) 
 - a*b*c**2/(2*x**4) - 4*a*b*c*d/(3*x**3) - a*b*d**2/x**2 - b**2*c**2/(2*x 
**2) - 2*b**2*c*d/x + b**2*d**2*log(x))/e**7, Eq(m, -7)), ((-a**2*c**2/(5* 
x**5) - a**2*c*d/(2*x**4) - a**2*d**2/(3*x**3) - 2*a*b*c**2/(3*x**3) - 2*a 
*b*c*d/x**2 - 2*a*b*d**2/x - b**2*c**2/x + 2*b**2*c*d*log(x) + b**2*d**2*x 
)/e**6, Eq(m, -6)), ((-a**2*c**2/(4*x**4) - 2*a**2*c*d/(3*x**3) - a**2*d** 
2/(2*x**2) - a*b*c**2/x**2 - 4*a*b*c*d/x + 2*a*b*d**2*log(x) + b**2*c**2*l 
og(x) + 2*b**2*c*d*x + b**2*d**2*x**2/2)/e**5, Eq(m, -5)), ((-a**2*c**2/(3 
*x**3) - a**2*c*d/x**2 - a**2*d**2/x - 2*a*b*c**2/x + 4*a*b*c*d*log(x) + 2 
*a*b*d**2*x + b**2*c**2*x + b**2*c*d*x**2 + b**2*d**2*x**3/3)/e**4, Eq(m, 
-4)), ((-a**2*c**2/(2*x**2) - 2*a**2*c*d/x + a**2*d**2*log(x) + 2*a*b*c**2 
*log(x) + 4*a*b*c*d*x + a*b*d**2*x**2 + b**2*c**2*x**2/2 + 2*b**2*c*d*x**3 
/3 + b**2*d**2*x**4/4)/e**3, Eq(m, -3)), ((-a**2*c**2/x + 2*a**2*c*d*log(x 
) + a**2*d**2*x + 2*a*b*c**2*x + 2*a*b*c*d*x**2 + 2*a*b*d**2*x**3/3 + b**2 
*c**2*x**3/3 + b**2*c*d*x**4/2 + b**2*d**2*x**5/5)/e**2, Eq(m, -2)), ((a** 
2*c**2*log(x) + 2*a**2*c*d*x + a**2*d**2*x**2/2 + a*b*c**2*x**2 + 4*a*b*c* 
d*x**3/3 + a*b*d**2*x**4/2 + b**2*c**2*x**4/4 + 2*b**2*c*d*x**5/5 + b**2*d 
**2*x**6/6)/e, Eq(m, -1)), (a**2*c**2*m**6*x*(e*x)**m/(m**7 + 28*m**6 + 32 
2*m**5 + 1960*m**4 + 6769*m**3 + 13132*m**2 + 13068*m + 5040) + 27*a**2*c* 
*2*m**5*x*(e*x)**m/(m**7 + 28*m**6 + 322*m**5 + 1960*m**4 + 6769*m**3 +...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.13 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right )^2 \, dx=\frac {b^{2} d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, b^{2} c d e^{m} x^{6} x^{m}}{m + 6} + \frac {b^{2} c^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, a b d^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {4 \, a b c d e^{m} x^{4} x^{m}}{m + 4} + \frac {2 \, a b c^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {a^{2} d^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, a^{2} c d e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} a^{2} c^{2}}{e {\left (m + 1\right )}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1129 vs. \(2 (168) = 336\).

Time = 0.14 (sec) , antiderivative size = 1129, normalized size of antiderivative = 6.72 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

((e*x)^m*b^2*d^2*m^6*x^7 + 2*(e*x)^m*b^2*c*d*m^6*x^6 + 21*(e*x)^m*b^2*d^2* 
m^5*x^7 + (e*x)^m*b^2*c^2*m^6*x^5 + 2*(e*x)^m*a*b*d^2*m^6*x^5 + 44*(e*x)^m 
*b^2*c*d*m^5*x^6 + 175*(e*x)^m*b^2*d^2*m^4*x^7 + 4*(e*x)^m*a*b*c*d*m^6*x^4 
 + 23*(e*x)^m*b^2*c^2*m^5*x^5 + 46*(e*x)^m*a*b*d^2*m^5*x^5 + 380*(e*x)^m*b 
^2*c*d*m^4*x^6 + 735*(e*x)^m*b^2*d^2*m^3*x^7 + 2*(e*x)^m*a*b*c^2*m^6*x^3 + 
 (e*x)^m*a^2*d^2*m^6*x^3 + 96*(e*x)^m*a*b*c*d*m^5*x^4 + 207*(e*x)^m*b^2*c^ 
2*m^4*x^5 + 414*(e*x)^m*a*b*d^2*m^4*x^5 + 1640*(e*x)^m*b^2*c*d*m^3*x^6 + 1 
624*(e*x)^m*b^2*d^2*m^2*x^7 + 2*(e*x)^m*a^2*c*d*m^6*x^2 + 50*(e*x)^m*a*b*c 
^2*m^5*x^3 + 25*(e*x)^m*a^2*d^2*m^5*x^3 + 904*(e*x)^m*a*b*c*d*m^4*x^4 + 92 
5*(e*x)^m*b^2*c^2*m^3*x^5 + 1850*(e*x)^m*a*b*d^2*m^3*x^5 + 3698*(e*x)^m*b^ 
2*c*d*m^2*x^6 + 1764*(e*x)^m*b^2*d^2*m*x^7 + (e*x)^m*a^2*c^2*m^6*x + 52*(e 
*x)^m*a^2*c*d*m^5*x^2 + 494*(e*x)^m*a*b*c^2*m^4*x^3 + 247*(e*x)^m*a^2*d^2* 
m^4*x^3 + 4224*(e*x)^m*a*b*c*d*m^3*x^4 + 2144*(e*x)^m*b^2*c^2*m^2*x^5 + 42 
88*(e*x)^m*a*b*d^2*m^2*x^5 + 4076*(e*x)^m*b^2*c*d*m*x^6 + 720*(e*x)^m*b^2* 
d^2*x^7 + 27*(e*x)^m*a^2*c^2*m^5*x + 540*(e*x)^m*a^2*c*d*m^4*x^2 + 2438*(e 
*x)^m*a*b*c^2*m^3*x^3 + 1219*(e*x)^m*a^2*d^2*m^3*x^3 + 10180*(e*x)^m*a*b*c 
*d*m^2*x^4 + 2412*(e*x)^m*b^2*c^2*m*x^5 + 4824*(e*x)^m*a*b*d^2*m*x^5 + 168 
0*(e*x)^m*b^2*c*d*x^6 + 295*(e*x)^m*a^2*c^2*m^4*x + 2840*(e*x)^m*a^2*c*d*m 
^3*x^2 + 6224*(e*x)^m*a*b*c^2*m^2*x^3 + 3112*(e*x)^m*a^2*d^2*m^2*x^3 + 118 
08*(e*x)^m*a*b*c*d*m*x^4 + 1008*(e*x)^m*b^2*c^2*x^5 + 2016*(e*x)^m*a*b*...
 

Mupad [B] (verification not implemented)

Time = 9.08 (sec) , antiderivative size = 558, normalized size of antiderivative = 3.32 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right )^2 \, dx=\frac {a^2\,c^2\,x\,{\left (e\,x\right )}^m\,\left (m^6+27\,m^5+295\,m^4+1665\,m^3+5104\,m^2+8028\,m+5040\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {b^2\,d^2\,x^7\,{\left (e\,x\right )}^m\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {a\,x^3\,{\left (e\,x\right )}^m\,\left (2\,b\,c^2+a\,d^2\right )\,\left (m^6+25\,m^5+247\,m^4+1219\,m^3+3112\,m^2+3796\,m+1680\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {b\,x^5\,{\left (e\,x\right )}^m\,\left (b\,c^2+2\,a\,d^2\right )\,\left (m^6+23\,m^5+207\,m^4+925\,m^3+2144\,m^2+2412\,m+1008\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {2\,a^2\,c\,d\,x^2\,{\left (e\,x\right )}^m\,\left (m^6+26\,m^5+270\,m^4+1420\,m^3+3929\,m^2+5274\,m+2520\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {2\,b^2\,c\,d\,x^6\,{\left (e\,x\right )}^m\,\left (m^6+22\,m^5+190\,m^4+820\,m^3+1849\,m^2+2038\,m+840\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}+\frac {4\,a\,b\,c\,d\,x^4\,{\left (e\,x\right )}^m\,\left (m^6+24\,m^5+226\,m^4+1056\,m^3+2545\,m^2+2952\,m+1260\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040} \] Input:

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

Output:

(a^2*c^2*x*(e*x)^m*(8028*m + 5104*m^2 + 1665*m^3 + 295*m^4 + 27*m^5 + m^6 
+ 5040))/(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m 
^7 + 5040) + (b^2*d^2*x^7*(e*x)^m*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 
 21*m^5 + m^6 + 720))/(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 
 + 28*m^6 + m^7 + 5040) + (a*x^3*(e*x)^m*(a*d^2 + 2*b*c^2)*(3796*m + 3112* 
m^2 + 1219*m^3 + 247*m^4 + 25*m^5 + m^6 + 1680))/(13068*m + 13132*m^2 + 67 
69*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040) + (b*x^5*(e*x)^m*(2*a*d 
^2 + b*c^2)*(2412*m + 2144*m^2 + 925*m^3 + 207*m^4 + 23*m^5 + m^6 + 1008)) 
/(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 504 
0) + (2*a^2*c*d*x^2*(e*x)^m*(5274*m + 3929*m^2 + 1420*m^3 + 270*m^4 + 26*m 
^5 + m^6 + 2520))/(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 2 
8*m^6 + m^7 + 5040) + (2*b^2*c*d*x^6*(e*x)^m*(2038*m + 1849*m^2 + 820*m^3 
+ 190*m^4 + 22*m^5 + m^6 + 840))/(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^ 
4 + 322*m^5 + 28*m^6 + m^7 + 5040) + (4*a*b*c*d*x^4*(e*x)^m*(2952*m + 2545 
*m^2 + 1056*m^3 + 226*m^4 + 24*m^5 + m^6 + 1260))/(13068*m + 13132*m^2 + 6 
769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 800, normalized size of antiderivative = 4.76 \[ \int (e x)^m (c+d x)^2 \left (a+b x^2\right )^2 \, dx=\frac {x^{m} e^{m} x \left (b^{2} d^{2} m^{6} x^{6}+2 b^{2} c d \,m^{6} x^{5}+21 b^{2} d^{2} m^{5} x^{6}+2 a b \,d^{2} m^{6} x^{4}+b^{2} c^{2} m^{6} x^{4}+44 b^{2} c d \,m^{5} x^{5}+175 b^{2} d^{2} m^{4} x^{6}+4 a b c d \,m^{6} x^{3}+46 a b \,d^{2} m^{5} x^{4}+23 b^{2} c^{2} m^{5} x^{4}+380 b^{2} c d \,m^{4} x^{5}+735 b^{2} d^{2} m^{3} x^{6}+a^{2} d^{2} m^{6} x^{2}+2 a b \,c^{2} m^{6} x^{2}+96 a b c d \,m^{5} x^{3}+414 a b \,d^{2} m^{4} x^{4}+207 b^{2} c^{2} m^{4} x^{4}+1640 b^{2} c d \,m^{3} x^{5}+1624 b^{2} d^{2} m^{2} x^{6}+2 a^{2} c d \,m^{6} x +25 a^{2} d^{2} m^{5} x^{2}+50 a b \,c^{2} m^{5} x^{2}+904 a b c d \,m^{4} x^{3}+1850 a b \,d^{2} m^{3} x^{4}+925 b^{2} c^{2} m^{3} x^{4}+3698 b^{2} c d \,m^{2} x^{5}+1764 b^{2} d^{2} m \,x^{6}+a^{2} c^{2} m^{6}+52 a^{2} c d \,m^{5} x +247 a^{2} d^{2} m^{4} x^{2}+494 a b \,c^{2} m^{4} x^{2}+4224 a b c d \,m^{3} x^{3}+4288 a b \,d^{2} m^{2} x^{4}+2144 b^{2} c^{2} m^{2} x^{4}+4076 b^{2} c d m \,x^{5}+720 b^{2} d^{2} x^{6}+27 a^{2} c^{2} m^{5}+540 a^{2} c d \,m^{4} x +1219 a^{2} d^{2} m^{3} x^{2}+2438 a b \,c^{2} m^{3} x^{2}+10180 a b c d \,m^{2} x^{3}+4824 a b \,d^{2} m \,x^{4}+2412 b^{2} c^{2} m \,x^{4}+1680 b^{2} c d \,x^{5}+295 a^{2} c^{2} m^{4}+2840 a^{2} c d \,m^{3} x +3112 a^{2} d^{2} m^{2} x^{2}+6224 a b \,c^{2} m^{2} x^{2}+11808 a b c d m \,x^{3}+2016 a b \,d^{2} x^{4}+1008 b^{2} c^{2} x^{4}+1665 a^{2} c^{2} m^{3}+7858 a^{2} c d \,m^{2} x +3796 a^{2} d^{2} m \,x^{2}+7592 a b \,c^{2} m \,x^{2}+5040 a b c d \,x^{3}+5104 a^{2} c^{2} m^{2}+10548 a^{2} c d m x +1680 a^{2} d^{2} x^{2}+3360 a b \,c^{2} x^{2}+8028 a^{2} c^{2} m +5040 a^{2} c d x +5040 a^{2} c^{2}\right )}{m^{7}+28 m^{6}+322 m^{5}+1960 m^{4}+6769 m^{3}+13132 m^{2}+13068 m +5040} \] Input:

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

Output:

(x**m*e**m*x*(a**2*c**2*m**6 + 27*a**2*c**2*m**5 + 295*a**2*c**2*m**4 + 16 
65*a**2*c**2*m**3 + 5104*a**2*c**2*m**2 + 8028*a**2*c**2*m + 5040*a**2*c** 
2 + 2*a**2*c*d*m**6*x + 52*a**2*c*d*m**5*x + 540*a**2*c*d*m**4*x + 2840*a* 
*2*c*d*m**3*x + 7858*a**2*c*d*m**2*x + 10548*a**2*c*d*m*x + 5040*a**2*c*d* 
x + a**2*d**2*m**6*x**2 + 25*a**2*d**2*m**5*x**2 + 247*a**2*d**2*m**4*x**2 
 + 1219*a**2*d**2*m**3*x**2 + 3112*a**2*d**2*m**2*x**2 + 3796*a**2*d**2*m* 
x**2 + 1680*a**2*d**2*x**2 + 2*a*b*c**2*m**6*x**2 + 50*a*b*c**2*m**5*x**2 
+ 494*a*b*c**2*m**4*x**2 + 2438*a*b*c**2*m**3*x**2 + 6224*a*b*c**2*m**2*x* 
*2 + 7592*a*b*c**2*m*x**2 + 3360*a*b*c**2*x**2 + 4*a*b*c*d*m**6*x**3 + 96* 
a*b*c*d*m**5*x**3 + 904*a*b*c*d*m**4*x**3 + 4224*a*b*c*d*m**3*x**3 + 10180 
*a*b*c*d*m**2*x**3 + 11808*a*b*c*d*m*x**3 + 5040*a*b*c*d*x**3 + 2*a*b*d**2 
*m**6*x**4 + 46*a*b*d**2*m**5*x**4 + 414*a*b*d**2*m**4*x**4 + 1850*a*b*d** 
2*m**3*x**4 + 4288*a*b*d**2*m**2*x**4 + 4824*a*b*d**2*m*x**4 + 2016*a*b*d* 
*2*x**4 + b**2*c**2*m**6*x**4 + 23*b**2*c**2*m**5*x**4 + 207*b**2*c**2*m** 
4*x**4 + 925*b**2*c**2*m**3*x**4 + 2144*b**2*c**2*m**2*x**4 + 2412*b**2*c* 
*2*m*x**4 + 1008*b**2*c**2*x**4 + 2*b**2*c*d*m**6*x**5 + 44*b**2*c*d*m**5* 
x**5 + 380*b**2*c*d*m**4*x**5 + 1640*b**2*c*d*m**3*x**5 + 3698*b**2*c*d*m* 
*2*x**5 + 4076*b**2*c*d*m*x**5 + 1680*b**2*c*d*x**5 + b**2*d**2*m**6*x**6 
+ 21*b**2*d**2*m**5*x**6 + 175*b**2*d**2*m**4*x**6 + 735*b**2*d**2*m**3*x* 
*6 + 1624*b**2*d**2*m**2*x**6 + 1764*b**2*d**2*m*x**6 + 720*b**2*d**2*x...