\(\int (d+e x)^4 (a+c x^2)^4 \, dx\) [491]

Optimal result

Integrand size = 17, antiderivative size = 270 \[ \int (d+e x)^4 \left (a+c x^2\right )^4 \, dx=a^4 d^4 x+\frac {2}{3} a^3 d^2 \left (2 c d^2+3 a e^2\right ) x^3+a^4 d e^3 x^4+\frac {1}{5} a^2 \left (6 c^2 d^4+24 a c d^2 e^2+a^2 e^4\right ) x^5+\frac {8}{3} a^3 c d e^3 x^6+\frac {4}{7} a c \left (c^2 d^4+9 a c d^2 e^2+a^2 e^4\right ) x^7+3 a^2 c^2 d e^3 x^8+\frac {1}{9} c^2 \left (c^2 d^4+24 a c d^2 e^2+6 a^2 e^4\right ) x^9+\frac {8}{5} a c^3 d e^3 x^{10}+\frac {2}{11} c^3 e^2 \left (3 c d^2+2 a e^2\right ) x^{11}+\frac {1}{3} c^4 d e^3 x^{12}+\frac {1}{13} c^4 e^4 x^{13}+\frac {2 d^3 e \left (a+c x^2\right )^5}{5 c} \]

[Out]

a^4*d^4*x+2/3*a^3*d^2*(3*a*e^2+2*c*d^2)*x^3+a^4*d*e^3*x^4+1/5*a^2*(a^2*e^4+24*a*c*d^2*e^2+6*c^2*d^4)*x^5+8/3*a
^3*c*d*e^3*x^6+4/7*a*c*(a^2*e^4+9*a*c*d^2*e^2+c^2*d^4)*x^7+3*a^2*c^2*d*e^3*x^8+1/9*c^2*(6*a^2*e^4+24*a*c*d^2*e
^2+c^2*d^4)*x^9+8/5*a*c^3*d*e^3*x^10+2/11*c^3*e^2*(2*a*e^2+3*c*d^2)*x^11+1/3*c^4*d*e^3*x^12+1/13*c^4*e^4*x^13+
2/5*d^3*e*(c*x^2+a)^5/c

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {710, 1824} \[ \int (d+e x)^4 \left (a+c x^2\right )^4 \, dx=a^4 d^4 x+a^4 d e^3 x^4+\frac {2}{3} a^3 d^2 x^3 \left (3 a e^2+2 c d^2\right )+\frac {8}{3} a^3 c d e^3 x^6+\frac {1}{9} c^2 x^9 \left (6 a^2 e^4+24 a c d^2 e^2+c^2 d^4\right )+\frac {4}{7} a c x^7 \left (a^2 e^4+9 a c d^2 e^2+c^2 d^4\right )+\frac {1}{5} a^2 x^5 \left (a^2 e^4+24 a c d^2 e^2+6 c^2 d^4\right )+3 a^2 c^2 d e^3 x^8+\frac {2}{11} c^3 e^2 x^{11} \left (2 a e^2+3 c d^2\right )+\frac {8}{5} a c^3 d e^3 x^{10}+\frac {2 d^3 e \left (a+c x^2\right )^5}{5 c}+\frac {1}{3} c^4 d e^3 x^{12}+\frac {1}{13} c^4 e^4 x^{13} \]

[In]

Int[(d + e*x)^4*(a + c*x^2)^4,x]

[Out]

a^4*d^4*x + (2*a^3*d^2*(2*c*d^2 + 3*a*e^2)*x^3)/3 + a^4*d*e^3*x^4 + (a^2*(6*c^2*d^4 + 24*a*c*d^2*e^2 + a^2*e^4
)*x^5)/5 + (8*a^3*c*d*e^3*x^6)/3 + (4*a*c*(c^2*d^4 + 9*a*c*d^2*e^2 + a^2*e^4)*x^7)/7 + 3*a^2*c^2*d*e^3*x^8 + (
c^2*(c^2*d^4 + 24*a*c*d^2*e^2 + 6*a^2*e^4)*x^9)/9 + (8*a*c^3*d*e^3*x^10)/5 + (2*c^3*e^2*(3*c*d^2 + 2*a*e^2)*x^
11)/11 + (c^4*d*e^3*x^12)/3 + (c^4*e^4*x^13)/13 + (2*d^3*e*(a + c*x^2)^5)/(5*c)

Rule 710

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*m*d^(m - 1)*((a + c*x^2)^(p + 1)/
(2*c*(p + 1))), x] + Int[((d + e*x)^m - e*m*d^(m - 1)*x)*(a + c*x^2)^p, x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*
d^2 + a*e^2, 0] && IGtQ[p, 1] && IGtQ[m, 0] && LeQ[m, p]

Rule 1824

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps \begin{align*} \text {integral}& = \frac {2 d^3 e \left (a+c x^2\right )^5}{5 c}+\int \left (a+c x^2\right )^4 \left (-4 d^3 e x+(d+e x)^4\right ) \, dx \\ & = \frac {2 d^3 e \left (a+c x^2\right )^5}{5 c}+\int \left (a^4 d^4+2 a^3 d^2 \left (2 c d^2+3 a e^2\right ) x^2+4 a^4 d e^3 x^3+a^2 \left (6 c^2 d^4+24 a c d^2 e^2+a^2 e^4\right ) x^4+16 a^3 c d e^3 x^5+4 a c \left (c^2 d^4+9 a c d^2 e^2+a^2 e^4\right ) x^6+24 a^2 c^2 d e^3 x^7+c^2 \left (c^2 d^4+24 a c d^2 e^2+6 a^2 e^4\right ) x^8+16 a c^3 d e^3 x^9+2 c^3 e^2 \left (3 c d^2+2 a e^2\right ) x^{10}+4 c^4 d e^3 x^{11}+c^4 e^4 x^{12}\right ) \, dx \\ & = a^4 d^4 x+\frac {2}{3} a^3 d^2 \left (2 c d^2+3 a e^2\right ) x^3+a^4 d e^3 x^4+\frac {1}{5} a^2 \left (6 c^2 d^4+24 a c d^2 e^2+a^2 e^4\right ) x^5+\frac {8}{3} a^3 c d e^3 x^6+\frac {4}{7} a c \left (c^2 d^4+9 a c d^2 e^2+a^2 e^4\right ) x^7+3 a^2 c^2 d e^3 x^8+\frac {1}{9} c^2 \left (c^2 d^4+24 a c d^2 e^2+6 a^2 e^4\right ) x^9+\frac {8}{5} a c^3 d e^3 x^{10}+\frac {2}{11} c^3 e^2 \left (3 c d^2+2 a e^2\right ) x^{11}+\frac {1}{3} c^4 d e^3 x^{12}+\frac {1}{13} c^4 e^4 x^{13}+\frac {2 d^3 e \left (a+c x^2\right )^5}{5 c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.11 \[ \int (d+e x)^4 \left (a+c x^2\right )^4 \, dx=a^4 d^4 x+2 a^4 d^3 e x^2+\frac {2}{3} a^3 d^2 \left (2 c d^2+3 a e^2\right ) x^3+a^3 d e \left (4 c d^2+a e^2\right ) x^4+\frac {1}{5} a^2 \left (6 c^2 d^4+24 a c d^2 e^2+a^2 e^4\right ) x^5+\frac {4}{3} a^2 c d e \left (3 c d^2+2 a e^2\right ) x^6+\frac {4}{7} a c \left (c^2 d^4+9 a c d^2 e^2+a^2 e^4\right ) x^7+a c^2 d e \left (2 c d^2+3 a e^2\right ) x^8+\frac {1}{9} c^2 \left (c^2 d^4+24 a c d^2 e^2+6 a^2 e^4\right ) x^9+\frac {2}{5} c^3 d e \left (c d^2+4 a e^2\right ) x^{10}+\frac {2}{11} c^3 e^2 \left (3 c d^2+2 a e^2\right ) x^{11}+\frac {1}{3} c^4 d e^3 x^{12}+\frac {1}{13} c^4 e^4 x^{13} \]

[In]

Integrate[(d + e*x)^4*(a + c*x^2)^4,x]

[Out]

a^4*d^4*x + 2*a^4*d^3*e*x^2 + (2*a^3*d^2*(2*c*d^2 + 3*a*e^2)*x^3)/3 + a^3*d*e*(4*c*d^2 + a*e^2)*x^4 + (a^2*(6*
c^2*d^4 + 24*a*c*d^2*e^2 + a^2*e^4)*x^5)/5 + (4*a^2*c*d*e*(3*c*d^2 + 2*a*e^2)*x^6)/3 + (4*a*c*(c^2*d^4 + 9*a*c
*d^2*e^2 + a^2*e^4)*x^7)/7 + a*c^2*d*e*(2*c*d^2 + 3*a*e^2)*x^8 + (c^2*(c^2*d^4 + 24*a*c*d^2*e^2 + 6*a^2*e^4)*x
^9)/9 + (2*c^3*d*e*(c*d^2 + 4*a*e^2)*x^10)/5 + (2*c^3*e^2*(3*c*d^2 + 2*a*e^2)*x^11)/11 + (c^4*d*e^3*x^12)/3 +
(c^4*e^4*x^13)/13

Maple [A] (verified)

Time = 2.50 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.13

method	result	size
norman	\(\frac {c^{4} e^{4} x^{13}}{13}+\frac {c^{4} d \,e^{3} x^{12}}{3}+\left (\frac {4}{11} e^{4} c^{3} a +\frac {6}{11} d^{2} e^{2} c^{4}\right ) x^{11}+\left (\frac {8}{5} d \,e^{3} c^{3} a +\frac {2}{5} c^{4} d^{3} e \right ) x^{10}+\left (\frac {2}{3} e^{4} a^{2} c^{2}+\frac {8}{3} d^{2} e^{2} c^{3} a +\frac {1}{9} c^{4} d^{4}\right ) x^{9}+\left (3 d \,e^{3} a^{2} c^{2}+2 d^{3} e \,c^{3} a \right ) x^{8}+\left (\frac {4}{7} e^{4} c \,a^{3}+\frac {36}{7} d^{2} e^{2} a^{2} c^{2}+\frac {4}{7} d^{4} c^{3} a \right ) x^{7}+\left (\frac {8}{3} d \,e^{3} c \,a^{3}+4 d^{3} e \,a^{2} c^{2}\right ) x^{6}+\left (\frac {1}{5} e^{4} a^{4}+\frac {24}{5} d^{2} e^{2} c \,a^{3}+\frac {6}{5} d^{4} a^{2} c^{2}\right ) x^{5}+\left (d \,e^{3} a^{4}+4 d^{3} e c \,a^{3}\right ) x^{4}+\left (2 d^{2} e^{2} a^{4}+\frac {4}{3} a^{3} c \,d^{4}\right ) x^{3}+2 d^{3} e \,a^{4} x^{2}+a^{4} d^{4} x\)	\(305\)
default	\(\frac {c^{4} e^{4} x^{13}}{13}+\frac {c^{4} d \,e^{3} x^{12}}{3}+\frac {\left (4 e^{4} c^{3} a +6 d^{2} e^{2} c^{4}\right ) x^{11}}{11}+\frac {\left (16 d \,e^{3} c^{3} a +4 c^{4} d^{3} e \right ) x^{10}}{10}+\frac {\left (6 e^{4} a^{2} c^{2}+24 d^{2} e^{2} c^{3} a +c^{4} d^{4}\right ) x^{9}}{9}+\frac {\left (24 d \,e^{3} a^{2} c^{2}+16 d^{3} e \,c^{3} a \right ) x^{8}}{8}+\frac {\left (4 e^{4} c \,a^{3}+36 d^{2} e^{2} a^{2} c^{2}+4 d^{4} c^{3} a \right ) x^{7}}{7}+\frac {\left (16 d \,e^{3} c \,a^{3}+24 d^{3} e \,a^{2} c^{2}\right ) x^{6}}{6}+\frac {\left (e^{4} a^{4}+24 d^{2} e^{2} c \,a^{3}+6 d^{4} a^{2} c^{2}\right ) x^{5}}{5}+\frac {\left (4 d \,e^{3} a^{4}+16 d^{3} e c \,a^{3}\right ) x^{4}}{4}+\frac {\left (6 d^{2} e^{2} a^{4}+4 a^{3} c \,d^{4}\right ) x^{3}}{3}+2 d^{3} e \,a^{4} x^{2}+a^{4} d^{4} x\)	\(313\)
gosper	\(\frac {6}{11} x^{11} d^{2} e^{2} c^{4}+\frac {2}{5} x^{10} c^{4} d^{3} e +\frac {2}{3} x^{9} e^{4} a^{2} c^{2}+a^{4} d \,e^{3} x^{4}+\frac {24}{5} x^{5} d^{2} e^{2} c \,a^{3}+2 a \,c^{3} d^{3} e \,x^{8}+4 x^{6} d^{3} e \,a^{2} c^{2}+\frac {8}{3} x^{9} d^{2} e^{2} c^{3} a +a^{4} d^{4} x +4 a^{3} c \,d^{3} e \,x^{4}+\frac {36}{7} x^{7} d^{2} e^{2} a^{2} c^{2}+\frac {4}{7} x^{7} e^{4} c \,a^{3}+\frac {4}{7} x^{7} d^{4} c^{3} a +\frac {6}{5} x^{5} d^{4} a^{2} c^{2}+2 x^{3} d^{2} e^{2} a^{4}+\frac {4}{3} x^{3} a^{3} c \,d^{4}+\frac {1}{9} x^{9} c^{4} d^{4}+2 d^{3} e \,a^{4} x^{2}+\frac {4}{11} x^{11} e^{4} c^{3} a +\frac {1}{13} c^{4} e^{4} x^{13}+\frac {1}{5} x^{5} e^{4} a^{4}+\frac {8}{3} a^{3} c d \,e^{3} x^{6}+3 a^{2} c^{2} d \,e^{3} x^{8}+\frac {8}{5} a \,c^{3} d \,e^{3} x^{10}+\frac {1}{3} c^{4} d \,e^{3} x^{12}\)	\(323\)
risch	\(\frac {6}{11} x^{11} d^{2} e^{2} c^{4}+\frac {2}{5} x^{10} c^{4} d^{3} e +\frac {2}{3} x^{9} e^{4} a^{2} c^{2}+a^{4} d \,e^{3} x^{4}+\frac {24}{5} x^{5} d^{2} e^{2} c \,a^{3}+2 a \,c^{3} d^{3} e \,x^{8}+4 x^{6} d^{3} e \,a^{2} c^{2}+\frac {8}{3} x^{9} d^{2} e^{2} c^{3} a +a^{4} d^{4} x +4 a^{3} c \,d^{3} e \,x^{4}+\frac {36}{7} x^{7} d^{2} e^{2} a^{2} c^{2}+\frac {4}{7} x^{7} e^{4} c \,a^{3}+\frac {4}{7} x^{7} d^{4} c^{3} a +\frac {6}{5} x^{5} d^{4} a^{2} c^{2}+2 x^{3} d^{2} e^{2} a^{4}+\frac {4}{3} x^{3} a^{3} c \,d^{4}+\frac {1}{9} x^{9} c^{4} d^{4}+2 d^{3} e \,a^{4} x^{2}+\frac {4}{11} x^{11} e^{4} c^{3} a +\frac {1}{13} c^{4} e^{4} x^{13}+\frac {1}{5} x^{5} e^{4} a^{4}+\frac {8}{3} a^{3} c d \,e^{3} x^{6}+3 a^{2} c^{2} d \,e^{3} x^{8}+\frac {8}{5} a \,c^{3} d \,e^{3} x^{10}+\frac {1}{3} c^{4} d \,e^{3} x^{12}\)	\(323\)
parallelrisch	\(\frac {6}{11} x^{11} d^{2} e^{2} c^{4}+\frac {2}{5} x^{10} c^{4} d^{3} e +\frac {2}{3} x^{9} e^{4} a^{2} c^{2}+a^{4} d \,e^{3} x^{4}+\frac {24}{5} x^{5} d^{2} e^{2} c \,a^{3}+2 a \,c^{3} d^{3} e \,x^{8}+4 x^{6} d^{3} e \,a^{2} c^{2}+\frac {8}{3} x^{9} d^{2} e^{2} c^{3} a +a^{4} d^{4} x +4 a^{3} c \,d^{3} e \,x^{4}+\frac {36}{7} x^{7} d^{2} e^{2} a^{2} c^{2}+\frac {4}{7} x^{7} e^{4} c \,a^{3}+\frac {4}{7} x^{7} d^{4} c^{3} a +\frac {6}{5} x^{5} d^{4} a^{2} c^{2}+2 x^{3} d^{2} e^{2} a^{4}+\frac {4}{3} x^{3} a^{3} c \,d^{4}+\frac {1}{9} x^{9} c^{4} d^{4}+2 d^{3} e \,a^{4} x^{2}+\frac {4}{11} x^{11} e^{4} c^{3} a +\frac {1}{13} c^{4} e^{4} x^{13}+\frac {1}{5} x^{5} e^{4} a^{4}+\frac {8}{3} a^{3} c d \,e^{3} x^{6}+3 a^{2} c^{2} d \,e^{3} x^{8}+\frac {8}{5} a \,c^{3} d \,e^{3} x^{10}+\frac {1}{3} c^{4} d \,e^{3} x^{12}\)	\(323\)

[In]

int((e*x+d)^4*(c*x^2+a)^4,x,method=_RETURNVERBOSE)

[Out]

1/13*c^4*e^4*x^13+1/3*c^4*d*e^3*x^12+(4/11*e^4*c^3*a+6/11*d^2*e^2*c^4)*x^11+(8/5*d*e^3*c^3*a+2/5*c^4*d^3*e)*x^
10+(2/3*e^4*a^2*c^2+8/3*d^2*e^2*c^3*a+1/9*c^4*d^4)*x^9+(3*a^2*c^2*d*e^3+2*a*c^3*d^3*e)*x^8+(4/7*e^4*c*a^3+36/7
*d^2*e^2*a^2*c^2+4/7*d^4*c^3*a)*x^7+(8/3*d*e^3*c*a^3+4*d^3*e*a^2*c^2)*x^6+(1/5*e^4*a^4+24/5*d^2*e^2*c*a^3+6/5*
d^4*a^2*c^2)*x^5+(a^4*d*e^3+4*a^3*c*d^3*e)*x^4+(2*d^2*e^2*a^4+4/3*a^3*c*d^4)*x^3+2*d^3*e*a^4*x^2+a^4*d^4*x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.13 \[ \int (d+e x)^4 \left (a+c x^2\right )^4 \, dx=\frac {1}{13} \, c^{4} e^{4} x^{13} + \frac {1}{3} \, c^{4} d e^{3} x^{12} + \frac {2}{11} \, {\left (3 \, c^{4} d^{2} e^{2} + 2 \, a c^{3} e^{4}\right )} x^{11} + \frac {2}{5} \, {\left (c^{4} d^{3} e + 4 \, a c^{3} d e^{3}\right )} x^{10} + 2 \, a^{4} d^{3} e x^{2} + \frac {1}{9} \, {\left (c^{4} d^{4} + 24 \, a c^{3} d^{2} e^{2} + 6 \, a^{2} c^{2} e^{4}\right )} x^{9} + a^{4} d^{4} x + {\left (2 \, a c^{3} d^{3} e + 3 \, a^{2} c^{2} d e^{3}\right )} x^{8} + \frac {4}{7} \, {\left (a c^{3} d^{4} + 9 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{7} + \frac {4}{3} \, {\left (3 \, a^{2} c^{2} d^{3} e + 2 \, a^{3} c d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, a^{2} c^{2} d^{4} + 24 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} x^{5} + {\left (4 \, a^{3} c d^{3} e + a^{4} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (2 \, a^{3} c d^{4} + 3 \, a^{4} d^{2} e^{2}\right )} x^{3} \]

[In]

integrate((e*x+d)^4*(c*x^2+a)^4,x, algorithm="fricas")

[Out]

1/13*c^4*e^4*x^13 + 1/3*c^4*d*e^3*x^12 + 2/11*(3*c^4*d^2*e^2 + 2*a*c^3*e^4)*x^11 + 2/5*(c^4*d^3*e + 4*a*c^3*d*
e^3)*x^10 + 2*a^4*d^3*e*x^2 + 1/9*(c^4*d^4 + 24*a*c^3*d^2*e^2 + 6*a^2*c^2*e^4)*x^9 + a^4*d^4*x + (2*a*c^3*d^3*
e + 3*a^2*c^2*d*e^3)*x^8 + 4/7*(a*c^3*d^4 + 9*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^7 + 4/3*(3*a^2*c^2*d^3*e + 2*a^3*
c*d*e^3)*x^6 + 1/5*(6*a^2*c^2*d^4 + 24*a^3*c*d^2*e^2 + a^4*e^4)*x^5 + (4*a^3*c*d^3*e + a^4*d*e^3)*x^4 + 2/3*(2
*a^3*c*d^4 + 3*a^4*d^2*e^2)*x^3

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.26 \[ \int (d+e x)^4 \left (a+c x^2\right )^4 \, dx=a^{4} d^{4} x + 2 a^{4} d^{3} e x^{2} + \frac {c^{4} d e^{3} x^{12}}{3} + \frac {c^{4} e^{4} x^{13}}{13} + x^{11} \cdot \left (\frac {4 a c^{3} e^{4}}{11} + \frac {6 c^{4} d^{2} e^{2}}{11}\right ) + x^{10} \cdot \left (\frac {8 a c^{3} d e^{3}}{5} + \frac {2 c^{4} d^{3} e}{5}\right ) + x^{9} \cdot \left (\frac {2 a^{2} c^{2} e^{4}}{3} + \frac {8 a c^{3} d^{2} e^{2}}{3} + \frac {c^{4} d^{4}}{9}\right ) + x^{8} \cdot \left (3 a^{2} c^{2} d e^{3} + 2 a c^{3} d^{3} e\right ) + x^{7} \cdot \left (\frac {4 a^{3} c e^{4}}{7} + \frac {36 a^{2} c^{2} d^{2} e^{2}}{7} + \frac {4 a c^{3} d^{4}}{7}\right ) + x^{6} \cdot \left (\frac {8 a^{3} c d e^{3}}{3} + 4 a^{2} c^{2} d^{3} e\right ) + x^{5} \left (\frac {a^{4} e^{4}}{5} + \frac {24 a^{3} c d^{2} e^{2}}{5} + \frac {6 a^{2} c^{2} d^{4}}{5}\right ) + x^{4} \left (a^{4} d e^{3} + 4 a^{3} c d^{3} e\right ) + x^{3} \cdot \left (2 a^{4} d^{2} e^{2} + \frac {4 a^{3} c d^{4}}{3}\right ) \]

[In]

integrate((e*x+d)**4*(c*x**2+a)**4,x)

[Out]

a**4*d**4*x + 2*a**4*d**3*e*x**2 + c**4*d*e**3*x**12/3 + c**4*e**4*x**13/13 + x**11*(4*a*c**3*e**4/11 + 6*c**4
*d**2*e**2/11) + x**10*(8*a*c**3*d*e**3/5 + 2*c**4*d**3*e/5) + x**9*(2*a**2*c**2*e**4/3 + 8*a*c**3*d**2*e**2/3
+ c**4*d**4/9) + x**8*(3*a**2*c**2*d*e**3 + 2*a*c**3*d**3*e) + x**7*(4*a**3*c*e**4/7 + 36*a**2*c**2*d**2*e**2
/7 + 4*a*c**3*d**4/7) + x**6*(8*a**3*c*d*e**3/3 + 4*a**2*c**2*d**3*e) + x**5*(a**4*e**4/5 + 24*a**3*c*d**2*e**
2/5 + 6*a**2*c**2*d**4/5) + x**4*(a**4*d*e**3 + 4*a**3*c*d**3*e) + x**3*(2*a**4*d**2*e**2 + 4*a**3*c*d**4/3)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.13 \[ \int (d+e x)^4 \left (a+c x^2\right )^4 \, dx=\frac {1}{13} \, c^{4} e^{4} x^{13} + \frac {1}{3} \, c^{4} d e^{3} x^{12} + \frac {2}{11} \, {\left (3 \, c^{4} d^{2} e^{2} + 2 \, a c^{3} e^{4}\right )} x^{11} + \frac {2}{5} \, {\left (c^{4} d^{3} e + 4 \, a c^{3} d e^{3}\right )} x^{10} + 2 \, a^{4} d^{3} e x^{2} + \frac {1}{9} \, {\left (c^{4} d^{4} + 24 \, a c^{3} d^{2} e^{2} + 6 \, a^{2} c^{2} e^{4}\right )} x^{9} + a^{4} d^{4} x + {\left (2 \, a c^{3} d^{3} e + 3 \, a^{2} c^{2} d e^{3}\right )} x^{8} + \frac {4}{7} \, {\left (a c^{3} d^{4} + 9 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{7} + \frac {4}{3} \, {\left (3 \, a^{2} c^{2} d^{3} e + 2 \, a^{3} c d e^{3}\right )} x^{6} + \frac {1}{5} \, {\left (6 \, a^{2} c^{2} d^{4} + 24 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} x^{5} + {\left (4 \, a^{3} c d^{3} e + a^{4} d e^{3}\right )} x^{4} + \frac {2}{3} \, {\left (2 \, a^{3} c d^{4} + 3 \, a^{4} d^{2} e^{2}\right )} x^{3} \]

[In]

integrate((e*x+d)^4*(c*x^2+a)^4,x, algorithm="maxima")

[Out]

1/13*c^4*e^4*x^13 + 1/3*c^4*d*e^3*x^12 + 2/11*(3*c^4*d^2*e^2 + 2*a*c^3*e^4)*x^11 + 2/5*(c^4*d^3*e + 4*a*c^3*d*
e^3)*x^10 + 2*a^4*d^3*e*x^2 + 1/9*(c^4*d^4 + 24*a*c^3*d^2*e^2 + 6*a^2*c^2*e^4)*x^9 + a^4*d^4*x + (2*a*c^3*d^3*
e + 3*a^2*c^2*d*e^3)*x^8 + 4/7*(a*c^3*d^4 + 9*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^7 + 4/3*(3*a^2*c^2*d^3*e + 2*a^3*
c*d*e^3)*x^6 + 1/5*(6*a^2*c^2*d^4 + 24*a^3*c*d^2*e^2 + a^4*e^4)*x^5 + (4*a^3*c*d^3*e + a^4*d*e^3)*x^4 + 2/3*(2
*a^3*c*d^4 + 3*a^4*d^2*e^2)*x^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.19 \[ \int (d+e x)^4 \left (a+c x^2\right )^4 \, dx=\frac {1}{13} \, c^{4} e^{4} x^{13} + \frac {1}{3} \, c^{4} d e^{3} x^{12} + \frac {6}{11} \, c^{4} d^{2} e^{2} x^{11} + \frac {4}{11} \, a c^{3} e^{4} x^{11} + \frac {2}{5} \, c^{4} d^{3} e x^{10} + \frac {8}{5} \, a c^{3} d e^{3} x^{10} + \frac {1}{9} \, c^{4} d^{4} x^{9} + \frac {8}{3} \, a c^{3} d^{2} e^{2} x^{9} + \frac {2}{3} \, a^{2} c^{2} e^{4} x^{9} + 2 \, a c^{3} d^{3} e x^{8} + 3 \, a^{2} c^{2} d e^{3} x^{8} + \frac {4}{7} \, a c^{3} d^{4} x^{7} + \frac {36}{7} \, a^{2} c^{2} d^{2} e^{2} x^{7} + \frac {4}{7} \, a^{3} c e^{4} x^{7} + 4 \, a^{2} c^{2} d^{3} e x^{6} + \frac {8}{3} \, a^{3} c d e^{3} x^{6} + \frac {6}{5} \, a^{2} c^{2} d^{4} x^{5} + \frac {24}{5} \, a^{3} c d^{2} e^{2} x^{5} + \frac {1}{5} \, a^{4} e^{4} x^{5} + 4 \, a^{3} c d^{3} e x^{4} + a^{4} d e^{3} x^{4} + \frac {4}{3} \, a^{3} c d^{4} x^{3} + 2 \, a^{4} d^{2} e^{2} x^{3} + 2 \, a^{4} d^{3} e x^{2} + a^{4} d^{4} x \]

[In]

integrate((e*x+d)^4*(c*x^2+a)^4,x, algorithm="giac")

[Out]

1/13*c^4*e^4*x^13 + 1/3*c^4*d*e^3*x^12 + 6/11*c^4*d^2*e^2*x^11 + 4/11*a*c^3*e^4*x^11 + 2/5*c^4*d^3*e*x^10 + 8/
5*a*c^3*d*e^3*x^10 + 1/9*c^4*d^4*x^9 + 8/3*a*c^3*d^2*e^2*x^9 + 2/3*a^2*c^2*e^4*x^9 + 2*a*c^3*d^3*e*x^8 + 3*a^2
*c^2*d*e^3*x^8 + 4/7*a*c^3*d^4*x^7 + 36/7*a^2*c^2*d^2*e^2*x^7 + 4/7*a^3*c*e^4*x^7 + 4*a^2*c^2*d^3*e*x^6 + 8/3*
a^3*c*d*e^3*x^6 + 6/5*a^2*c^2*d^4*x^5 + 24/5*a^3*c*d^2*e^2*x^5 + 1/5*a^4*e^4*x^5 + 4*a^3*c*d^3*e*x^4 + a^4*d*e
^3*x^4 + 4/3*a^3*c*d^4*x^3 + 2*a^4*d^2*e^2*x^3 + 2*a^4*d^3*e*x^2 + a^4*d^4*x

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.07 \[ \int (d+e x)^4 \left (a+c x^2\right )^4 \, dx=x^5\,\left (\frac {a^4\,e^4}{5}+\frac {24\,a^3\,c\,d^2\,e^2}{5}+\frac {6\,a^2\,c^2\,d^4}{5}\right )+x^9\,\left (\frac {2\,a^2\,c^2\,e^4}{3}+\frac {8\,a\,c^3\,d^2\,e^2}{3}+\frac {c^4\,d^4}{9}\right )+x^3\,\left (2\,a^4\,d^2\,e^2+\frac {4\,c\,a^3\,d^4}{3}\right )+x^{11}\,\left (\frac {6\,c^4\,d^2\,e^2}{11}+\frac {4\,a\,c^3\,e^4}{11}\right )+a^4\,d^4\,x+\frac {c^4\,e^4\,x^{13}}{13}+2\,a^4\,d^3\,e\,x^2+\frac {c^4\,d\,e^3\,x^{12}}{3}+\frac {4\,a\,c\,x^7\,\left (a^2\,e^4+9\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{7}+a^3\,d\,e\,x^4\,\left (4\,c\,d^2+a\,e^2\right )+\frac {2\,c^3\,d\,e\,x^{10}\,\left (c\,d^2+4\,a\,e^2\right )}{5}+\frac {4\,a^2\,c\,d\,e\,x^6\,\left (3\,c\,d^2+2\,a\,e^2\right )}{3}+a\,c^2\,d\,e\,x^8\,\left (2\,c\,d^2+3\,a\,e^2\right ) \]

[In]

int((a + c*x^2)^4*(d + e*x)^4,x)

[Out]

x^5*((a^4*e^4)/5 + (6*a^2*c^2*d^4)/5 + (24*a^3*c*d^2*e^2)/5) + x^9*((c^4*d^4)/9 + (2*a^2*c^2*e^4)/3 + (8*a*c^3
*d^2*e^2)/3) + x^3*((4*a^3*c*d^4)/3 + 2*a^4*d^2*e^2) + x^11*((4*a*c^3*e^4)/11 + (6*c^4*d^2*e^2)/11) + a^4*d^4*
x + (c^4*e^4*x^13)/13 + 2*a^4*d^3*e*x^2 + (c^4*d*e^3*x^12)/3 + (4*a*c*x^7*(a^2*e^4 + c^2*d^4 + 9*a*c*d^2*e^2))
/7 + a^3*d*e*x^4*(a*e^2 + 4*c*d^2) + (2*c^3*d*e*x^10*(4*a*e^2 + c*d^2))/5 + (4*a^2*c*d*e*x^6*(2*a*e^2 + 3*c*d^
2))/3 + a*c^2*d*e*x^8*(3*a*e^2 + 2*c*d^2)