\(\int (d+e x)^3 (f+g x)^{3/2} (a+c x^2)^2 \, dx\) [70]

Optimal result

Integrand size = 26, antiderivative size = 474 \[ \int (d+e x)^3 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx=-\frac {2 (e f-d g)^3 \left (c f^2+a g^2\right )^2 (f+g x)^{5/2}}{5 g^8}+\frac {2 (e f-d g)^2 \left (c f^2+a g^2\right ) \left (3 a e g^2+c f (7 e f-4 d g)\right ) (f+g x)^{7/2}}{7 g^8}-\frac {2 (e f-d g) \left (3 a^2 e^2 g^4+2 a c g^2 \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )+3 c^2 f^2 \left (7 e^2 f^2-8 d e f g+2 d^2 g^2\right )\right ) (f+g x)^{9/2}}{9 g^8}+\frac {2 \left (a^2 e^3 g^4+2 a c e g^2 \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )+c^2 f \left (35 e^3 f^3-60 d e^2 f^2 g+30 d^2 e f g^2-4 d^3 g^3\right )\right ) (f+g x)^{11/2}}{11 g^8}-\frac {2 c \left (2 a e^2 g^2 (5 e f-3 d g)+c \left (35 e^3 f^3-45 d e^2 f^2 g+15 d^2 e f g^2-d^3 g^3\right )\right ) (f+g x)^{13/2}}{13 g^8}+\frac {2 c e \left (2 a e^2 g^2+3 c \left (7 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) (f+g x)^{15/2}}{15 g^8}-\frac {2 c^2 e^2 (7 e f-3 d g) (f+g x)^{17/2}}{17 g^8}+\frac {2 c^2 e^3 (f+g x)^{19/2}}{19 g^8} \] Output:

-2/5*(-d*g+e*f)^3*(a*g^2+c*f^2)^2*(g*x+f)^(5/2)/g^8+2/7*(-d*g+e*f)^2*(a*g^ 
2+c*f^2)*(3*a*e*g^2+c*f*(-4*d*g+7*e*f))*(g*x+f)^(7/2)/g^8-2/9*(-d*g+e*f)*( 
3*a^2*e^2*g^4+2*a*c*g^2*(d^2*g^2-8*d*e*f*g+10*e^2*f^2)+3*c^2*f^2*(2*d^2*g^ 
2-8*d*e*f*g+7*e^2*f^2))*(g*x+f)^(9/2)/g^8+2/11*(a^2*e^3*g^4+2*a*c*e*g^2*(3 
*d^2*g^2-12*d*e*f*g+10*e^2*f^2)+c^2*f*(-4*d^3*g^3+30*d^2*e*f*g^2-60*d*e^2* 
f^2*g+35*e^3*f^3))*(g*x+f)^(11/2)/g^8-2/13*c*(2*a*e^2*g^2*(-3*d*g+5*e*f)+c 
*(-d^3*g^3+15*d^2*e*f*g^2-45*d*e^2*f^2*g+35*e^3*f^3))*(g*x+f)^(13/2)/g^8+2 
/15*c*e*(2*a*e^2*g^2+3*c*(d^2*g^2-6*d*e*f*g+7*e^2*f^2))*(g*x+f)^(15/2)/g^8 
-2/17*c^2*e^2*(-3*d*g+7*e*f)*(g*x+f)^(17/2)/g^8+2/19*c^2*e^3*(g*x+f)^(19/2 
)/g^8
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.17 \[ \int (d+e x)^3 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx=\frac {2 (f+g x)^{5/2} \left (12597 a^2 g^4 \left (231 d^3 g^3+99 d^2 e g^2 (-2 f+5 g x)+11 d e^2 g \left (8 f^2-20 f g x+35 g^2 x^2\right )+e^3 \left (-16 f^3+40 f^2 g x-70 f g^2 x^2+105 g^3 x^3\right )\right )+646 a c g^2 \left (143 d^3 g^3 \left (8 f^2-20 f g x+35 g^2 x^2\right )+117 d^2 e g^2 \left (-16 f^3+40 f^2 g x-70 f g^2 x^2+105 g^3 x^3\right )+9 d e^2 g \left (128 f^4-320 f^3 g x+560 f^2 g^2 x^2-840 f g^3 x^3+1155 g^4 x^4\right )+e^3 \left (-256 f^5+640 f^4 g x-1120 f^3 g^2 x^2+1680 f^2 g^3 x^3-2310 f g^4 x^4+3003 g^5 x^5\right )\right )-3 c^2 \left (-323 d^3 g^3 \left (128 f^4-320 f^3 g x+560 f^2 g^2 x^2-840 f g^3 x^3+1155 g^4 x^4\right )+323 d^2 e g^2 \left (256 f^5-640 f^4 g x+1120 f^3 g^2 x^2-1680 f^2 g^3 x^3+2310 f g^4 x^4-3003 g^5 x^5\right )-57 d e^2 g \left (1024 f^6-2560 f^5 g x+4480 f^4 g^2 x^2-6720 f^3 g^3 x^3+9240 f^2 g^4 x^4-12012 f g^5 x^5+15015 g^6 x^6\right )+7 e^3 \left (2048 f^7-5120 f^6 g x+8960 f^5 g^2 x^2-13440 f^4 g^3 x^3+18480 f^3 g^4 x^4-24024 f^2 g^5 x^5+30030 f g^6 x^6-36465 g^7 x^7\right )\right )\right )}{14549535 g^8} \] Input:

Integrate[(d + e*x)^3*(f + g*x)^(3/2)*(a + c*x^2)^2,x]
 

Output:

(2*(f + g*x)^(5/2)*(12597*a^2*g^4*(231*d^3*g^3 + 99*d^2*e*g^2*(-2*f + 5*g* 
x) + 11*d*e^2*g*(8*f^2 - 20*f*g*x + 35*g^2*x^2) + e^3*(-16*f^3 + 40*f^2*g* 
x - 70*f*g^2*x^2 + 105*g^3*x^3)) + 646*a*c*g^2*(143*d^3*g^3*(8*f^2 - 20*f* 
g*x + 35*g^2*x^2) + 117*d^2*e*g^2*(-16*f^3 + 40*f^2*g*x - 70*f*g^2*x^2 + 1 
05*g^3*x^3) + 9*d*e^2*g*(128*f^4 - 320*f^3*g*x + 560*f^2*g^2*x^2 - 840*f*g 
^3*x^3 + 1155*g^4*x^4) + e^3*(-256*f^5 + 640*f^4*g*x - 1120*f^3*g^2*x^2 + 
1680*f^2*g^3*x^3 - 2310*f*g^4*x^4 + 3003*g^5*x^5)) - 3*c^2*(-323*d^3*g^3*( 
128*f^4 - 320*f^3*g*x + 560*f^2*g^2*x^2 - 840*f*g^3*x^3 + 1155*g^4*x^4) + 
323*d^2*e*g^2*(256*f^5 - 640*f^4*g*x + 1120*f^3*g^2*x^2 - 1680*f^2*g^3*x^3 
 + 2310*f*g^4*x^4 - 3003*g^5*x^5) - 57*d*e^2*g*(1024*f^6 - 2560*f^5*g*x + 
4480*f^4*g^2*x^2 - 6720*f^3*g^3*x^3 + 9240*f^2*g^4*x^4 - 12012*f*g^5*x^5 + 
 15015*g^6*x^6) + 7*e^3*(2048*f^7 - 5120*f^6*g*x + 8960*f^5*g^2*x^2 - 1344 
0*f^4*g^3*x^3 + 18480*f^3*g^4*x^4 - 24024*f^2*g^5*x^5 + 30030*f*g^6*x^6 - 
36465*g^7*x^7))))/(14549535*g^8)
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {652, 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.

\(\displaystyle \int \left (a+c x^2\right )^2 (d+e x)^3 (f+g x)^{3/2} \, dx\)

\(\Big \downarrow \) 652

\(\displaystyle \int \left (\frac {(f+g x)^{7/2} (e f-d g) \left (-3 a^2 e^2 g^4-2 a c g^2 \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )-3 c^2 f^2 \left (2 d^2 g^2-8 d e f g+7 e^2 f^2\right )\right )}{g^7}+\frac {(f+g x)^{9/2} \left (a^2 e^3 g^4+2 a c e g^2 \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )+c^2 f \left (-4 d^3 g^3+30 d^2 e f g^2-60 d e^2 f^2 g+35 e^3 f^3\right )\right )}{g^7}+\frac {c e (f+g x)^{13/2} \left (2 a e^2 g^2+3 c \left (d^2 g^2-6 d e f g+7 e^2 f^2\right )\right )}{g^7}+\frac {c (f+g x)^{11/2} \left (-2 a e^2 g^2 (5 e f-3 d g)-c \left (-d^3 g^3+15 d^2 e f g^2-45 d e^2 f^2 g+35 e^3 f^3\right )\right )}{g^7}+\frac {(f+g x)^{5/2} \left (a g^2+c f^2\right ) (e f-d g)^2 \left (3 a e g^2+c f (7 e f-4 d g)\right )}{g^7}+\frac {(f+g x)^{3/2} \left (a g^2+c f^2\right )^2 (d g-e f)^3}{g^7}-\frac {c^2 e^2 (f+g x)^{15/2} (7 e f-3 d g)}{g^7}+\frac {c^2 e^3 (f+g x)^{17/2}}{g^7}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 (f+g x)^{9/2} (e f-d g) \left (3 a^2 e^2 g^4+2 a c g^2 \left (d^2 g^2-8 d e f g+10 e^2 f^2\right )+3 c^2 f^2 \left (2 d^2 g^2-8 d e f g+7 e^2 f^2\right )\right )}{9 g^8}+\frac {2 (f+g x)^{11/2} \left (a^2 e^3 g^4+2 a c e g^2 \left (3 d^2 g^2-12 d e f g+10 e^2 f^2\right )+c^2 f \left (-4 d^3 g^3+30 d^2 e f g^2-60 d e^2 f^2 g+35 e^3 f^3\right )\right )}{11 g^8}+\frac {2 c e (f+g x)^{15/2} \left (2 a e^2 g^2+3 c \left (d^2 g^2-6 d e f g+7 e^2 f^2\right )\right )}{15 g^8}-\frac {2 c (f+g x)^{13/2} \left (2 a e^2 g^2 (5 e f-3 d g)+c \left (-d^3 g^3+15 d^2 e f g^2-45 d e^2 f^2 g+35 e^3 f^3\right )\right )}{13 g^8}+\frac {2 (f+g x)^{7/2} \left (a g^2+c f^2\right ) (e f-d g)^2 \left (3 a e g^2+c f (7 e f-4 d g)\right )}{7 g^8}-\frac {2 (f+g x)^{5/2} \left (a g^2+c f^2\right )^2 (e f-d g)^3}{5 g^8}-\frac {2 c^2 e^2 (f+g x)^{17/2} (7 e f-3 d g)}{17 g^8}+\frac {2 c^2 e^3 (f+g x)^{19/2}}{19 g^8}\)

Input:

Int[(d + e*x)^3*(f + g*x)^(3/2)*(a + c*x^2)^2,x]
 

Output:

(-2*(e*f - d*g)^3*(c*f^2 + a*g^2)^2*(f + g*x)^(5/2))/(5*g^8) + (2*(e*f - d 
*g)^2*(c*f^2 + a*g^2)*(3*a*e*g^2 + c*f*(7*e*f - 4*d*g))*(f + g*x)^(7/2))/( 
7*g^8) - (2*(e*f - d*g)*(3*a^2*e^2*g^4 + 2*a*c*g^2*(10*e^2*f^2 - 8*d*e*f*g 
 + d^2*g^2) + 3*c^2*f^2*(7*e^2*f^2 - 8*d*e*f*g + 2*d^2*g^2))*(f + g*x)^(9/ 
2))/(9*g^8) + (2*(a^2*e^3*g^4 + 2*a*c*e*g^2*(10*e^2*f^2 - 12*d*e*f*g + 3*d 
^2*g^2) + c^2*f*(35*e^3*f^3 - 60*d*e^2*f^2*g + 30*d^2*e*f*g^2 - 4*d^3*g^3) 
)*(f + g*x)^(11/2))/(11*g^8) - (2*c*(2*a*e^2*g^2*(5*e*f - 3*d*g) + c*(35*e 
^3*f^3 - 45*d*e^2*f^2*g + 15*d^2*e*f*g^2 - d^3*g^3))*(f + g*x)^(13/2))/(13 
*g^8) + (2*c*e*(2*a*e^2*g^2 + 3*c*(7*e^2*f^2 - 6*d*e*f*g + d^2*g^2))*(f + 
g*x)^(15/2))/(15*g^8) - (2*c^2*e^2*(7*e*f - 3*d*g)*(f + g*x)^(17/2))/(17*g 
^8) + (2*c^2*e^3*(f + g*x)^(19/2))/(19*g^8)
 

Defintions of rubi rules used

rule 652

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

rule 2009

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

Maple [A] (verified)

Time = 1.50 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.05

method	result	size
pseudoelliptic	\(\frac {2 \left (g x +f \right )^{\frac {5}{2}} \left (\left (\left (\frac {5}{19} c^{2} x^{7}+\frac {5}{11} a^{2} x^{3}+\frac {2}{3} a c \,x^{5}\right ) e^{3}+\frac {5 x^{2} \left (\frac {9}{17} c^{2} x^{4}+\frac {18}{13} a c \,x^{2}+a^{2}\right ) d \,e^{2}}{3}+\frac {15 \left (\frac {7}{15} c^{2} x^{4}+\frac {14}{11} a c \,x^{2}+a^{2}\right ) x \,d^{2} e}{7}+d^{3} \left (\frac {5}{13} c^{2} x^{4}+\frac {10}{9} a c \,x^{2}+a^{2}\right )\right ) g^{7}-\frac {6 \left (\frac {35 x^{2} \left (\frac {231}{323} c^{2} x^{4}+\frac {22}{13} a c \,x^{2}+a^{2}\right ) e^{3}}{99}+\frac {10 x \left (\frac {63}{85} c^{2} x^{4}+\frac {252}{143} a c \,x^{2}+a^{2}\right ) d \,e^{2}}{9}+d^{2} \left (\frac {35}{39} c^{2} x^{4}+\frac {70}{33} a c \,x^{2}+a^{2}\right ) e +\frac {20 c x \left (\frac {63 c \,x^{2}}{143}+a \right ) d^{3}}{27}\right ) f \,g^{6}}{7}+\frac {8 f^{2} \left (\left (\frac {147}{323} c^{2} x^{5}+\frac {140}{143} a c \,x^{3}+\frac {5}{11} a^{2} x \right ) e^{3}+d \left (\frac {315}{221} c^{2} x^{4}+\frac {420}{143} a c \,x^{2}+a^{2}\right ) e^{2}+\frac {30 \left (\frac {7 c \,x^{2}}{13}+a \right ) c x \,d^{2} e}{11}+\frac {2 \left (\frac {105 c \,x^{2}}{143}+a \right ) c \,d^{3}}{3}\right ) g^{5}}{21}-\frac {16 \left (\left (\frac {8085}{4199} c^{2} x^{4}+\frac {140}{39} a c \,x^{2}+a^{2}\right ) e^{3}+\frac {120 c x \left (\frac {21 c \,x^{2}}{34}+a \right ) d \,e^{2}}{13}+6 \left (\frac {35 c \,x^{2}}{39}+a \right ) c \,d^{2} e +\frac {20 c^{2} d^{3} x}{13}\right ) f^{3} g^{4}}{231}+\frac {256 f^{4} c \left (\frac {5 x \left (\frac {441 c \,x^{2}}{646}+a \right ) e^{3}}{9}+d \left (\frac {35 c \,x^{2}}{34}+a \right ) e^{2}+\frac {5 c \,d^{2} e x}{6}+\frac {c \,d^{3}}{6}\right ) g^{3}}{1001}-\frac {512 \left (\left (\frac {735 c \,x^{2}}{646}+a \right ) e^{2}+\frac {45 c d x e}{17}+\frac {3 c \,d^{2}}{2}\right ) e \,f^{5} c \,g^{2}}{9009}+\frac {1024 \left (\frac {35 e x}{57}+d \right ) e^{2} f^{6} c^{2} g}{17017}-\frac {2048 c^{2} e^{3} f^{7}}{138567}\right )}{5 g^{8}}\)	\(499\)
derivativedivides	\(\frac {\frac {2 c^{2} e^{3} \left (g x +f \right )^{\frac {19}{2}}}{19}+\frac {2 \left (3 \left (d g -e f \right ) e^{2} c^{2}-4 f \,c^{2} e^{3}\right ) \left (g x +f \right )^{\frac {17}{2}}}{17}+\frac {2 \left (3 \left (d g -e f \right )^{2} e \,c^{2}-12 \left (d g -e f \right ) e^{2} c^{2} f +e^{3} \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )\right ) \left (g x +f \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (d g -e f \right )^{3} c^{2}-12 \left (d g -e f \right )^{2} e \,c^{2} f +3 \left (d g -e f \right ) e^{2} \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-4 e^{3} \left (a \,g^{2}+c \,f^{2}\right ) c f \right ) \left (g x +f \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-4 \left (d g -e f \right )^{3} c^{2} f +3 \left (d g -e f \right )^{2} e \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-12 \left (d g -e f \right ) e^{2} \left (a \,g^{2}+c \,f^{2}\right ) c f +e^{3} \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (d g -e f \right )^{3} \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-12 \left (d g -e f \right )^{2} e \left (a \,g^{2}+c \,f^{2}\right ) c f +3 \left (d g -e f \right ) e^{2} \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-4 \left (d g -e f \right )^{3} \left (a \,g^{2}+c \,f^{2}\right ) c f +3 \left (d g -e f \right )^{2} e \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (d g -e f \right )^{3} \left (a \,g^{2}+c \,f^{2}\right )^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}}{g^{8}}\)	\(516\)
default	\(\frac {\frac {2 c^{2} e^{3} \left (g x +f \right )^{\frac {19}{2}}}{19}+\frac {2 \left (3 \left (d g -e f \right ) e^{2} c^{2}-4 f \,c^{2} e^{3}\right ) \left (g x +f \right )^{\frac {17}{2}}}{17}+\frac {2 \left (3 \left (d g -e f \right )^{2} e \,c^{2}-12 \left (d g -e f \right ) e^{2} c^{2} f +e^{3} \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )\right ) \left (g x +f \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (d g -e f \right )^{3} c^{2}-12 \left (d g -e f \right )^{2} e \,c^{2} f +3 \left (d g -e f \right ) e^{2} \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-4 e^{3} \left (a \,g^{2}+c \,f^{2}\right ) c f \right ) \left (g x +f \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-4 \left (d g -e f \right )^{3} c^{2} f +3 \left (d g -e f \right )^{2} e \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-12 \left (d g -e f \right ) e^{2} \left (a \,g^{2}+c \,f^{2}\right ) c f +e^{3} \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (d g -e f \right )^{3} \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-12 \left (d g -e f \right )^{2} e \left (a \,g^{2}+c \,f^{2}\right ) c f +3 \left (d g -e f \right ) e^{2} \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-4 \left (d g -e f \right )^{3} \left (a \,g^{2}+c \,f^{2}\right ) c f +3 \left (d g -e f \right )^{2} e \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (d g -e f \right )^{3} \left (a \,g^{2}+c \,f^{2}\right )^{2} \left (g x +f \right )^{\frac {5}{2}}}{5}}{g^{8}}\)	\(516\)
gosper	\(\frac {2 \left (g x +f \right )^{\frac {5}{2}} \left (765765 c^{2} e^{3} x^{7} g^{7}+2567565 c^{2} d \,e^{2} g^{7} x^{6}-630630 c^{2} e^{3} f \,g^{6} x^{6}+1939938 a c \,e^{3} g^{7} x^{5}+2909907 c^{2} d^{2} e \,g^{7} x^{5}-2054052 c^{2} d \,e^{2} f \,g^{6} x^{5}+504504 c^{2} e^{3} f^{2} g^{5} x^{5}+6715170 a c d \,e^{2} g^{7} x^{4}-1492260 a c \,e^{3} f \,g^{6} x^{4}+1119195 c^{2} d^{3} g^{7} x^{4}-2238390 c^{2} d^{2} e f \,g^{6} x^{4}+1580040 c^{2} d \,e^{2} f^{2} g^{5} x^{4}-388080 c^{2} e^{3} f^{3} g^{4} x^{4}+1322685 a^{2} e^{3} g^{7} x^{3}+7936110 a c \,d^{2} e \,g^{7} x^{3}-4883760 a c d \,e^{2} f \,g^{6} x^{3}+1085280 a c \,e^{3} f^{2} g^{5} x^{3}-813960 c^{2} d^{3} f \,g^{6} x^{3}+1627920 c^{2} d^{2} e \,f^{2} g^{5} x^{3}-1149120 c^{2} d \,e^{2} f^{3} g^{4} x^{3}+282240 c^{2} e^{3} f^{4} g^{3} x^{3}+4849845 a^{2} d \,e^{2} g^{7} x^{2}-881790 a^{2} e^{3} f \,g^{6} x^{2}+3233230 a c \,d^{3} g^{7} x^{2}-5290740 a c \,d^{2} e f \,g^{6} x^{2}+3255840 a c d \,e^{2} f^{2} g^{5} x^{2}-723520 a c \,e^{3} f^{3} g^{4} x^{2}+542640 c^{2} d^{3} f^{2} g^{5} x^{2}-1085280 c^{2} d^{2} e \,f^{3} g^{4} x^{2}+766080 c^{2} d \,e^{2} f^{4} g^{3} x^{2}-188160 c^{2} e^{3} f^{5} g^{2} x^{2}+6235515 a^{2} d^{2} e \,g^{7} x -2771340 a^{2} d \,e^{2} f \,g^{6} x +503880 a^{2} e^{3} f^{2} g^{5} x -1847560 a c \,d^{3} f \,g^{6} x +3023280 a c \,d^{2} e \,f^{2} g^{5} x -1860480 a c d \,e^{2} f^{3} g^{4} x +413440 a c \,e^{3} f^{4} g^{3} x -310080 c^{2} d^{3} f^{3} g^{4} x +620160 c^{2} d^{2} e \,f^{4} g^{3} x -437760 c^{2} d \,e^{2} f^{5} g^{2} x +107520 c^{2} e^{3} f^{6} g x +2909907 a^{2} d^{3} g^{7}-2494206 a^{2} d^{2} e f \,g^{6}+1108536 a^{2} d \,e^{2} f^{2} g^{5}-201552 a^{2} e^{3} f^{3} g^{4}+739024 a c \,d^{3} f^{2} g^{5}-1209312 a c \,d^{2} e \,f^{3} g^{4}+744192 a c d \,e^{2} f^{4} g^{3}-165376 a c \,e^{3} f^{5} g^{2}+124032 c^{2} d^{3} f^{4} g^{3}-248064 c^{2} d^{2} e \,f^{5} g^{2}+175104 c^{2} d \,e^{2} f^{6} g -43008 c^{2} e^{3} f^{7}\right )}{14549535 g^{8}}\)	\(818\)
orering	\(\frac {2 \left (g x +f \right )^{\frac {5}{2}} \left (765765 c^{2} e^{3} x^{7} g^{7}+2567565 c^{2} d \,e^{2} g^{7} x^{6}-630630 c^{2} e^{3} f \,g^{6} x^{6}+1939938 a c \,e^{3} g^{7} x^{5}+2909907 c^{2} d^{2} e \,g^{7} x^{5}-2054052 c^{2} d \,e^{2} f \,g^{6} x^{5}+504504 c^{2} e^{3} f^{2} g^{5} x^{5}+6715170 a c d \,e^{2} g^{7} x^{4}-1492260 a c \,e^{3} f \,g^{6} x^{4}+1119195 c^{2} d^{3} g^{7} x^{4}-2238390 c^{2} d^{2} e f \,g^{6} x^{4}+1580040 c^{2} d \,e^{2} f^{2} g^{5} x^{4}-388080 c^{2} e^{3} f^{3} g^{4} x^{4}+1322685 a^{2} e^{3} g^{7} x^{3}+7936110 a c \,d^{2} e \,g^{7} x^{3}-4883760 a c d \,e^{2} f \,g^{6} x^{3}+1085280 a c \,e^{3} f^{2} g^{5} x^{3}-813960 c^{2} d^{3} f \,g^{6} x^{3}+1627920 c^{2} d^{2} e \,f^{2} g^{5} x^{3}-1149120 c^{2} d \,e^{2} f^{3} g^{4} x^{3}+282240 c^{2} e^{3} f^{4} g^{3} x^{3}+4849845 a^{2} d \,e^{2} g^{7} x^{2}-881790 a^{2} e^{3} f \,g^{6} x^{2}+3233230 a c \,d^{3} g^{7} x^{2}-5290740 a c \,d^{2} e f \,g^{6} x^{2}+3255840 a c d \,e^{2} f^{2} g^{5} x^{2}-723520 a c \,e^{3} f^{3} g^{4} x^{2}+542640 c^{2} d^{3} f^{2} g^{5} x^{2}-1085280 c^{2} d^{2} e \,f^{3} g^{4} x^{2}+766080 c^{2} d \,e^{2} f^{4} g^{3} x^{2}-188160 c^{2} e^{3} f^{5} g^{2} x^{2}+6235515 a^{2} d^{2} e \,g^{7} x -2771340 a^{2} d \,e^{2} f \,g^{6} x +503880 a^{2} e^{3} f^{2} g^{5} x -1847560 a c \,d^{3} f \,g^{6} x +3023280 a c \,d^{2} e \,f^{2} g^{5} x -1860480 a c d \,e^{2} f^{3} g^{4} x +413440 a c \,e^{3} f^{4} g^{3} x -310080 c^{2} d^{3} f^{3} g^{4} x +620160 c^{2} d^{2} e \,f^{4} g^{3} x -437760 c^{2} d \,e^{2} f^{5} g^{2} x +107520 c^{2} e^{3} f^{6} g x +2909907 a^{2} d^{3} g^{7}-2494206 a^{2} d^{2} e f \,g^{6}+1108536 a^{2} d \,e^{2} f^{2} g^{5}-201552 a^{2} e^{3} f^{3} g^{4}+739024 a c \,d^{3} f^{2} g^{5}-1209312 a c \,d^{2} e \,f^{3} g^{4}+744192 a c d \,e^{2} f^{4} g^{3}-165376 a c \,e^{3} f^{5} g^{2}+124032 c^{2} d^{3} f^{4} g^{3}-248064 c^{2} d^{2} e \,f^{5} g^{2}+175104 c^{2} d \,e^{2} f^{6} g -43008 c^{2} e^{3} f^{7}\right )}{14549535 g^{8}}\)	\(818\)
trager	\(\text {Expression too large to display}\)	\(1226\)
risch	\(\text {Expression too large to display}\)	\(1226\)

Input:

int((e*x+d)^3*(g*x+f)^(3/2)*(c*x^2+a)^2,x,method=_RETURNVERBOSE)
 

Output:

2/5*(g*x+f)^(5/2)*(((5/19*c^2*x^7+5/11*a^2*x^3+2/3*a*c*x^5)*e^3+5/3*x^2*(9 
/17*c^2*x^4+18/13*a*c*x^2+a^2)*d*e^2+15/7*(7/15*c^2*x^4+14/11*a*c*x^2+a^2) 
*x*d^2*e+d^3*(5/13*c^2*x^4+10/9*a*c*x^2+a^2))*g^7-6/7*(35/99*x^2*(231/323* 
c^2*x^4+22/13*a*c*x^2+a^2)*e^3+10/9*x*(63/85*c^2*x^4+252/143*a*c*x^2+a^2)* 
d*e^2+d^2*(35/39*c^2*x^4+70/33*a*c*x^2+a^2)*e+20/27*c*x*(63/143*c*x^2+a)*d 
^3)*f*g^6+8/21*f^2*((147/323*c^2*x^5+140/143*a*c*x^3+5/11*a^2*x)*e^3+d*(31 
5/221*c^2*x^4+420/143*a*c*x^2+a^2)*e^2+30/11*(7/13*c*x^2+a)*c*x*d^2*e+2/3* 
(105/143*c*x^2+a)*c*d^3)*g^5-16/231*((8085/4199*c^2*x^4+140/39*a*c*x^2+a^2 
)*e^3+120/13*c*x*(21/34*c*x^2+a)*d*e^2+6*(35/39*c*x^2+a)*c*d^2*e+20/13*c^2 
*d^3*x)*f^3*g^4+256/1001*f^4*c*(5/9*x*(441/646*c*x^2+a)*e^3+d*(35/34*c*x^2 
+a)*e^2+5/6*c*d^2*e*x+1/6*c*d^3)*g^3-512/9009*((735/646*c*x^2+a)*e^2+45/17 
*c*d*x*e+3/2*c*d^2)*e*f^5*c*g^2+1024/17017*(35/57*e*x+d)*e^2*f^6*c^2*g-204 
8/138567*c^2*e^3*f^7)/g^8
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1028 vs. \(2 (442) = 884\).

Time = 0.10 (sec) , antiderivative size = 1028, normalized size of antiderivative = 2.17 \[ \int (d+e x)^3 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^3*(g*x+f)^(3/2)*(c*x^2+a)^2,x, algorithm="fricas")
 

Output:

2/14549535*(765765*c^2*e^3*g^9*x^9 - 43008*c^2*e^3*f^9 + 175104*c^2*d*e^2* 
f^8*g - 2494206*a^2*d^2*e*f^3*g^6 + 2909907*a^2*d^3*f^2*g^7 - 82688*(3*c^2 
*d^2*e + 2*a*c*e^3)*f^7*g^2 + 124032*(c^2*d^3 + 6*a*c*d*e^2)*f^6*g^3 - 201 
552*(6*a*c*d^2*e + a^2*e^3)*f^5*g^4 + 369512*(2*a*c*d^3 + 3*a^2*d*e^2)*f^4 
*g^5 + 45045*(20*c^2*e^3*f*g^8 + 57*c^2*d*e^2*g^9)*x^8 + 3003*(3*c^2*e^3*f 
^2*g^7 + 1026*c^2*d*e^2*f*g^8 + 323*(3*c^2*d^2*e + 2*a*c*e^3)*g^9)*x^7 - 2 
31*(42*c^2*e^3*f^3*g^6 - 171*c^2*d*e^2*f^2*g^7 - 5168*(3*c^2*d^2*e + 2*a*c 
*e^3)*f*g^8 - 4845*(c^2*d^3 + 6*a*c*d*e^2)*g^9)*x^6 + 63*(168*c^2*e^3*f^4* 
g^5 - 684*c^2*d*e^2*f^3*g^6 + 323*(3*c^2*d^2*e + 2*a*c*e^3)*f^2*g^7 + 2261 
0*(c^2*d^3 + 6*a*c*d*e^2)*f*g^8 + 20995*(6*a*c*d^2*e + a^2*e^3)*g^9)*x^5 - 
 35*(336*c^2*e^3*f^5*g^4 - 1368*c^2*d*e^2*f^4*g^5 + 646*(3*c^2*d^2*e + 2*a 
*c*e^3)*f^3*g^6 - 969*(c^2*d^3 + 6*a*c*d*e^2)*f^2*g^7 - 50388*(6*a*c*d^2*e 
 + a^2*e^3)*f*g^8 - 46189*(2*a*c*d^3 + 3*a^2*d*e^2)*g^9)*x^4 + 5*(2688*c^2 
*e^3*f^6*g^3 - 10944*c^2*d*e^2*f^5*g^4 + 1247103*a^2*d^2*e*g^9 + 5168*(3*c 
^2*d^2*e + 2*a*c*e^3)*f^4*g^5 - 7752*(c^2*d^3 + 6*a*c*d*e^2)*f^3*g^6 + 125 
97*(6*a*c*d^2*e + a^2*e^3)*f^2*g^7 + 461890*(2*a*c*d^3 + 3*a^2*d*e^2)*f*g^ 
8)*x^3 - 3*(5376*c^2*e^3*f^7*g^2 - 21888*c^2*d*e^2*f^6*g^3 - 3325608*a^2*d 
^2*e*f*g^8 - 969969*a^2*d^3*g^9 + 10336*(3*c^2*d^2*e + 2*a*c*e^3)*f^5*g^4 
- 15504*(c^2*d^3 + 6*a*c*d*e^2)*f^4*g^5 + 25194*(6*a*c*d^2*e + a^2*e^3)*f^ 
3*g^6 - 46189*(2*a*c*d^3 + 3*a^2*d*e^2)*f^2*g^7)*x^2 + (21504*c^2*e^3*f...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1032 vs. \(2 (495) = 990\).

Time = 1.58 (sec) , antiderivative size = 1032, normalized size of antiderivative = 2.18 \[ \int (d+e x)^3 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)**3*(g*x+f)**(3/2)*(c*x**2+a)**2,x)
 

Output:

Piecewise((2*(c**2*e**3*(f + g*x)**(19/2)/(19*g**7) + (f + g*x)**(17/2)*(3 
*c**2*d*e**2*g - 7*c**2*e**3*f)/(17*g**7) + (f + g*x)**(15/2)*(2*a*c*e**3* 
g**2 + 3*c**2*d**2*e*g**2 - 18*c**2*d*e**2*f*g + 21*c**2*e**3*f**2)/(15*g* 
*7) + (f + g*x)**(13/2)*(6*a*c*d*e**2*g**3 - 10*a*c*e**3*f*g**2 + c**2*d** 
3*g**3 - 15*c**2*d**2*e*f*g**2 + 45*c**2*d*e**2*f**2*g - 35*c**2*e**3*f**3 
)/(13*g**7) + (f + g*x)**(11/2)*(a**2*e**3*g**4 + 6*a*c*d**2*e*g**4 - 24*a 
*c*d*e**2*f*g**3 + 20*a*c*e**3*f**2*g**2 - 4*c**2*d**3*f*g**3 + 30*c**2*d* 
*2*e*f**2*g**2 - 60*c**2*d*e**2*f**3*g + 35*c**2*e**3*f**4)/(11*g**7) + (f 
 + g*x)**(9/2)*(3*a**2*d*e**2*g**5 - 3*a**2*e**3*f*g**4 + 2*a*c*d**3*g**5 
- 18*a*c*d**2*e*f*g**4 + 36*a*c*d*e**2*f**2*g**3 - 20*a*c*e**3*f**3*g**2 + 
 6*c**2*d**3*f**2*g**3 - 30*c**2*d**2*e*f**3*g**2 + 45*c**2*d*e**2*f**4*g 
- 21*c**2*e**3*f**5)/(9*g**7) + (f + g*x)**(7/2)*(3*a**2*d**2*e*g**6 - 6*a 
**2*d*e**2*f*g**5 + 3*a**2*e**3*f**2*g**4 - 4*a*c*d**3*f*g**5 + 18*a*c*d** 
2*e*f**2*g**4 - 24*a*c*d*e**2*f**3*g**3 + 10*a*c*e**3*f**4*g**2 - 4*c**2*d 
**3*f**3*g**3 + 15*c**2*d**2*e*f**4*g**2 - 18*c**2*d*e**2*f**5*g + 7*c**2* 
e**3*f**6)/(7*g**7) + (f + g*x)**(5/2)*(a**2*d**3*g**7 - 3*a**2*d**2*e*f*g 
**6 + 3*a**2*d*e**2*f**2*g**5 - a**2*e**3*f**3*g**4 + 2*a*c*d**3*f**2*g**5 
 - 6*a*c*d**2*e*f**3*g**4 + 6*a*c*d*e**2*f**4*g**3 - 2*a*c*e**3*f**5*g**2 
+ c**2*d**3*f**4*g**3 - 3*c**2*d**2*e*f**5*g**2 + 3*c**2*d*e**2*f**6*g - c 
**2*e**3*f**7)/(5*g**7))/g, Ne(g, 0)), (f**(3/2)*(a**2*d**3*x + 3*a**2*...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 704, normalized size of antiderivative = 1.49 \[ \int (d+e x)^3 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^3*(g*x+f)^(3/2)*(c*x^2+a)^2,x, algorithm="maxima")
 

Output:

2/14549535*(765765*(g*x + f)^(19/2)*c^2*e^3 - 855855*(7*c^2*e^3*f - 3*c^2* 
d*e^2*g)*(g*x + f)^(17/2) + 969969*(21*c^2*e^3*f^2 - 18*c^2*d*e^2*f*g + (3 
*c^2*d^2*e + 2*a*c*e^3)*g^2)*(g*x + f)^(15/2) - 1119195*(35*c^2*e^3*f^3 - 
45*c^2*d*e^2*f^2*g + 5*(3*c^2*d^2*e + 2*a*c*e^3)*f*g^2 - (c^2*d^3 + 6*a*c* 
d*e^2)*g^3)*(g*x + f)^(13/2) + 1322685*(35*c^2*e^3*f^4 - 60*c^2*d*e^2*f^3* 
g + 10*(3*c^2*d^2*e + 2*a*c*e^3)*f^2*g^2 - 4*(c^2*d^3 + 6*a*c*d*e^2)*f*g^3 
 + (6*a*c*d^2*e + a^2*e^3)*g^4)*(g*x + f)^(11/2) - 1616615*(21*c^2*e^3*f^5 
 - 45*c^2*d*e^2*f^4*g + 10*(3*c^2*d^2*e + 2*a*c*e^3)*f^3*g^2 - 6*(c^2*d^3 
+ 6*a*c*d*e^2)*f^2*g^3 + 3*(6*a*c*d^2*e + a^2*e^3)*f*g^4 - (2*a*c*d^3 + 3* 
a^2*d*e^2)*g^5)*(g*x + f)^(9/2) + 2078505*(7*c^2*e^3*f^6 - 18*c^2*d*e^2*f^ 
5*g + 3*a^2*d^2*e*g^6 + 5*(3*c^2*d^2*e + 2*a*c*e^3)*f^4*g^2 - 4*(c^2*d^3 + 
 6*a*c*d*e^2)*f^3*g^3 + 3*(6*a*c*d^2*e + a^2*e^3)*f^2*g^4 - 2*(2*a*c*d^3 + 
 3*a^2*d*e^2)*f*g^5)*(g*x + f)^(7/2) - 2909907*(c^2*e^3*f^7 - 3*c^2*d*e^2* 
f^6*g + 3*a^2*d^2*e*f*g^6 - a^2*d^3*g^7 + (3*c^2*d^2*e + 2*a*c*e^3)*f^5*g^ 
2 - (c^2*d^3 + 6*a*c*d*e^2)*f^4*g^3 + (6*a*c*d^2*e + a^2*e^3)*f^3*g^4 - (2 
*a*c*d^3 + 3*a^2*d*e^2)*f^2*g^5)*(g*x + f)^(5/2))/g^8
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2674 vs. \(2 (442) = 884\).

Time = 0.14 (sec) , antiderivative size = 2674, normalized size of antiderivative = 5.64 \[ \int (d+e x)^3 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^3*(g*x+f)^(3/2)*(c*x^2+a)^2,x, algorithm="giac")
 

Output:

2/14549535*(14549535*sqrt(g*x + f)*a^2*d^3*f^2 + 9699690*((g*x + f)^(3/2) 
- 3*sqrt(g*x + f)*f)*a^2*d^3*f + 14549535*((g*x + f)^(3/2) - 3*sqrt(g*x + 
f)*f)*a^2*d^2*e*f^2/g + 969969*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 
 15*sqrt(g*x + f)*f^2)*a^2*d^3 + 1939938*(3*(g*x + f)^(5/2) - 10*(g*x + f) 
^(3/2)*f + 15*sqrt(g*x + f)*f^2)*a*c*d^3*f^2/g^2 + 2909907*(3*(g*x + f)^(5 
/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*a^2*d*e^2*f^2/g^2 + 581 
9814*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*a^2 
*d^2*e*f/g + 2494206*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f + 35*(g*x + 
 f)^(3/2)*f^2 - 35*sqrt(g*x + f)*f^3)*a*c*d^2*e*f^2/g^3 + 415701*(5*(g*x + 
 f)^(7/2) - 21*(g*x + f)^(5/2)*f + 35*(g*x + f)^(3/2)*f^2 - 35*sqrt(g*x + 
f)*f^3)*a^2*e^3*f^2/g^3 + 1662804*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)* 
f + 35*(g*x + f)^(3/2)*f^2 - 35*sqrt(g*x + f)*f^3)*a*c*d^3*f/g^2 + 2494206 
*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f + 35*(g*x + f)^(3/2)*f^2 - 35*s 
qrt(g*x + f)*f^3)*a^2*d*e^2*f/g^2 + 1247103*(5*(g*x + f)^(7/2) - 21*(g*x + 
 f)^(5/2)*f + 35*(g*x + f)^(3/2)*f^2 - 35*sqrt(g*x + f)*f^3)*a^2*d^2*e/g + 
 46189*(35*(g*x + f)^(9/2) - 180*(g*x + f)^(7/2)*f + 378*(g*x + f)^(5/2)*f 
^2 - 420*(g*x + f)^(3/2)*f^3 + 315*sqrt(g*x + f)*f^4)*c^2*d^3*f^2/g^4 + 27 
7134*(35*(g*x + f)^(9/2) - 180*(g*x + f)^(7/2)*f + 378*(g*x + f)^(5/2)*f^2 
 - 420*(g*x + f)^(3/2)*f^3 + 315*sqrt(g*x + f)*f^4)*a*c*d*e^2*f^2/g^4 + 55 
4268*(35*(g*x + f)^(9/2) - 180*(g*x + f)^(7/2)*f + 378*(g*x + f)^(5/2)*...
 

Mupad [B] (verification not implemented)

Time = 6.08 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.01 \[ \int (d+e x)^3 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx=\frac {{\left (f+g\,x\right )}^{13/2}\,\left (2\,c^2\,d^3\,g^3-30\,c^2\,d^2\,e\,f\,g^2+90\,c^2\,d\,e^2\,f^2\,g-70\,c^2\,e^3\,f^3+12\,a\,c\,d\,e^2\,g^3-20\,a\,c\,e^3\,f\,g^2\right )}{13\,g^8}+\frac {{\left (f+g\,x\right )}^{11/2}\,\left (2\,a^2\,e^3\,g^4+12\,a\,c\,d^2\,e\,g^4-48\,a\,c\,d\,e^2\,f\,g^3+40\,a\,c\,e^3\,f^2\,g^2-8\,c^2\,d^3\,f\,g^3+60\,c^2\,d^2\,e\,f^2\,g^2-120\,c^2\,d\,e^2\,f^3\,g+70\,c^2\,e^3\,f^4\right )}{11\,g^8}+\frac {2\,c^2\,e^3\,{\left (f+g\,x\right )}^{19/2}}{19\,g^8}+\frac {2\,{\left (f+g\,x\right )}^{5/2}\,{\left (c\,f^2+a\,g^2\right )}^2\,{\left (d\,g-e\,f\right )}^3}{5\,g^8}+\frac {2\,{\left (f+g\,x\right )}^{9/2}\,\left (d\,g-e\,f\right )\,\left (3\,a^2\,e^2\,g^4+2\,a\,c\,d^2\,g^4-16\,a\,c\,d\,e\,f\,g^3+20\,a\,c\,e^2\,f^2\,g^2+6\,c^2\,d^2\,f^2\,g^2-24\,c^2\,d\,e\,f^3\,g+21\,c^2\,e^2\,f^4\right )}{9\,g^8}+\frac {2\,{\left (f+g\,x\right )}^{7/2}\,\left (c\,f^2+a\,g^2\right )\,{\left (d\,g-e\,f\right )}^2\,\left (7\,c\,e\,f^2-4\,c\,d\,f\,g+3\,a\,e\,g^2\right )}{7\,g^8}+\frac {2\,c^2\,e^2\,{\left (f+g\,x\right )}^{17/2}\,\left (3\,d\,g-7\,e\,f\right )}{17\,g^8}+\frac {2\,c\,e\,{\left (f+g\,x\right )}^{15/2}\,\left (3\,c\,d^2\,g^2-18\,c\,d\,e\,f\,g+21\,c\,e^2\,f^2+2\,a\,e^2\,g^2\right )}{15\,g^8} \] Input:

int((f + g*x)^(3/2)*(a + c*x^2)^2*(d + e*x)^3,x)
 

Output:

((f + g*x)^(13/2)*(2*c^2*d^3*g^3 - 70*c^2*e^3*f^3 + 12*a*c*d*e^2*g^3 - 20* 
a*c*e^3*f*g^2 + 90*c^2*d*e^2*f^2*g - 30*c^2*d^2*e*f*g^2))/(13*g^8) + ((f + 
 g*x)^(11/2)*(2*a^2*e^3*g^4 + 70*c^2*e^3*f^4 - 8*c^2*d^3*f*g^3 + 12*a*c*d^ 
2*e*g^4 + 40*a*c*e^3*f^2*g^2 - 120*c^2*d*e^2*f^3*g + 60*c^2*d^2*e*f^2*g^2 
- 48*a*c*d*e^2*f*g^3))/(11*g^8) + (2*c^2*e^3*(f + g*x)^(19/2))/(19*g^8) + 
(2*(f + g*x)^(5/2)*(a*g^2 + c*f^2)^2*(d*g - e*f)^3)/(5*g^8) + (2*(f + g*x) 
^(9/2)*(d*g - e*f)*(3*a^2*e^2*g^4 + 21*c^2*e^2*f^4 + 6*c^2*d^2*f^2*g^2 + 2 
*a*c*d^2*g^4 - 24*c^2*d*e*f^3*g + 20*a*c*e^2*f^2*g^2 - 16*a*c*d*e*f*g^3))/ 
(9*g^8) + (2*(f + g*x)^(7/2)*(a*g^2 + c*f^2)*(d*g - e*f)^2*(3*a*e*g^2 + 7* 
c*e*f^2 - 4*c*d*f*g))/(7*g^8) + (2*c^2*e^2*(f + g*x)^(17/2)*(3*d*g - 7*e*f 
))/(17*g^8) + (2*c*e*(f + g*x)^(15/2)*(2*a*e^2*g^2 + 3*c*d^2*g^2 + 21*c*e^ 
2*f^2 - 18*c*d*e*f*g))/(15*g^8)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1224, normalized size of antiderivative = 2.58 \[ \int (d+e x)^3 (f+g x)^{3/2} \left (a+c x^2\right )^2 \, dx =\text {Too large to display} \] Input:

int((e*x+d)^3*(g*x+f)^(3/2)*(c*x^2+a)^2,x)
 

Output:

(2*sqrt(f + g*x)*(2909907*a**2*d**3*f**2*g**7 + 5819814*a**2*d**3*f*g**8*x 
 + 2909907*a**2*d**3*g**9*x**2 - 2494206*a**2*d**2*e*f**3*g**6 + 1247103*a 
**2*d**2*e*f**2*g**7*x + 9976824*a**2*d**2*e*f*g**8*x**2 + 6235515*a**2*d* 
*2*e*g**9*x**3 + 1108536*a**2*d*e**2*f**4*g**5 - 554268*a**2*d*e**2*f**3*g 
**6*x + 415701*a**2*d*e**2*f**2*g**7*x**2 + 6928350*a**2*d*e**2*f*g**8*x** 
3 + 4849845*a**2*d*e**2*g**9*x**4 - 201552*a**2*e**3*f**5*g**4 + 100776*a* 
*2*e**3*f**4*g**5*x - 75582*a**2*e**3*f**3*g**6*x**2 + 62985*a**2*e**3*f** 
2*g**7*x**3 + 1763580*a**2*e**3*f*g**8*x**4 + 1322685*a**2*e**3*g**9*x**5 
+ 739024*a*c*d**3*f**4*g**5 - 369512*a*c*d**3*f**3*g**6*x + 277134*a*c*d** 
3*f**2*g**7*x**2 + 4618900*a*c*d**3*f*g**8*x**3 + 3233230*a*c*d**3*g**9*x* 
*4 - 1209312*a*c*d**2*e*f**5*g**4 + 604656*a*c*d**2*e*f**4*g**5*x - 453492 
*a*c*d**2*e*f**3*g**6*x**2 + 377910*a*c*d**2*e*f**2*g**7*x**3 + 10581480*a 
*c*d**2*e*f*g**8*x**4 + 7936110*a*c*d**2*e*g**9*x**5 + 744192*a*c*d*e**2*f 
**6*g**3 - 372096*a*c*d*e**2*f**5*g**4*x + 279072*a*c*d*e**2*f**4*g**5*x** 
2 - 232560*a*c*d*e**2*f**3*g**6*x**3 + 203490*a*c*d*e**2*f**2*g**7*x**4 + 
8546580*a*c*d*e**2*f*g**8*x**5 + 6715170*a*c*d*e**2*g**9*x**6 - 165376*a*c 
*e**3*f**7*g**2 + 82688*a*c*e**3*f**6*g**3*x - 62016*a*c*e**3*f**5*g**4*x* 
*2 + 51680*a*c*e**3*f**4*g**5*x**3 - 45220*a*c*e**3*f**3*g**6*x**4 + 40698 
*a*c*e**3*f**2*g**7*x**5 + 2387616*a*c*e**3*f*g**8*x**6 + 1939938*a*c*e**3 
*g**9*x**7 + 124032*c**2*d**3*f**6*g**3 - 62016*c**2*d**3*f**5*g**4*x +...