\(\int \frac {(d+e x)^3 (a+c x^2)^2}{(f+g x)^{3/2}} \, dx\) [84]

Optimal result

Integrand size = 26, antiderivative size = 470 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )^2}{(f+g x)^{3/2}} \, dx=\frac {2 (e f-d g)^3 \left (c f^2+a g^2\right )^2}{g^8 \sqrt {f+g x}}+\frac {2 (e f-d g)^2 \left (c f^2+a g^2\right ) \left (3 a e g^2+c f (7 e f-4 d g)\right ) \sqrt {f+g x}}{g^8}-\frac {2 (e f-d g) \left (3 a^2 e^2 g^4+2 a c g^2 \left (10 e^2 f^2-8 d e f g+d^2 g^2\right )+3 c^2 f^2 \left (7 e^2 f^2-8 d e f g+2 d^2 g^2\right )\right ) (f+g x)^{3/2}}{3 g^8}+\frac {2 \left (a^2 e^3 g^4+2 a c e g^2 \left (10 e^2 f^2-12 d e f g+3 d^2 g^2\right )+c^2 f \left (35 e^3 f^3-60 d e^2 f^2 g+30 d^2 e f g^2-4 d^3 g^3\right )\right ) (f+g x)^{5/2}}{5 g^8}-\frac {2 c \left (2 a e^2 g^2 (5 e f-3 d g)+c \left (35 e^3 f^3-45 d e^2 f^2 g+15 d^2 e f g^2-d^3 g^3\right )\right ) (f+g x)^{7/2}}{7 g^8}+\frac {2 c e \left (2 a e^2 g^2+3 c \left (7 e^2 f^2-6 d e f g+d^2 g^2\right )\right ) (f+g x)^{9/2}}{9 g^8}-\frac {2 c^2 e^2 (7 e f-3 d g) (f+g x)^{11/2}}{11 g^8}+\frac {2 c^2 e^3 (f+g x)^{13/2}}{13 g^8} \] Output:

2*(-d*g+e*f)^3*(a*g^2+c*f^2)^2/g^8/(g*x+f)^(1/2)+2*(-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)^(1/2)/g^8-2/3*(-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)^(3/2)/g^8+2/5*(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)^(5/2)/g^8-2/7*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)^(7/2)/g^8+2/9*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)^(9/2)/g^8-2/11*c^2*e 
^2*(-3*d*g+7*e*f)*(g*x+f)^(11/2)/g^8+2/13*c^2*e^3*(g*x+f)^(13/2)/g^8
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )^2}{(f+g x)^{3/2}} \, dx=\frac {18018 a^2 g^4 \left (-5 d^3 g^3+15 d^2 e g^2 (2 f+g x)+5 d e^2 g \left (-8 f^2-4 f g x+g^2 x^2\right )+e^3 \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )\right )+572 a c g^2 \left (105 d^3 g^3 \left (-8 f^2-4 f g x+g^2 x^2\right )+189 d^2 e g^2 \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )+27 d e^2 g \left (-128 f^4-64 f^3 g x+16 f^2 g^2 x^2-8 f g^3 x^3+5 g^4 x^4\right )+5 e^3 \left (256 f^5+128 f^4 g x-32 f^3 g^2 x^2+16 f^2 g^3 x^3-10 f g^4 x^4+7 g^5 x^5\right )\right )+6 c^2 \left (429 d^3 g^3 \left (-128 f^4-64 f^3 g x+16 f^2 g^2 x^2-8 f g^3 x^3+5 g^4 x^4\right )+715 d^2 e g^2 \left (256 f^5+128 f^4 g x-32 f^3 g^2 x^2+16 f^2 g^3 x^3-10 f g^4 x^4+7 g^5 x^5\right )+195 d e^2 g \left (-1024 f^6-512 f^5 g x+128 f^4 g^2 x^2-64 f^3 g^3 x^3+40 f^2 g^4 x^4-28 f g^5 x^5+21 g^6 x^6\right )+35 e^3 \left (2048 f^7+1024 f^6 g x-256 f^5 g^2 x^2+128 f^4 g^3 x^3-80 f^3 g^4 x^4+56 f^2 g^5 x^5-42 f g^6 x^6+33 g^7 x^7\right )\right )}{45045 g^8 \sqrt {f+g x}} \] Input:

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

Output:

(18018*a^2*g^4*(-5*d^3*g^3 + 15*d^2*e*g^2*(2*f + g*x) + 5*d*e^2*g*(-8*f^2 
- 4*f*g*x + g^2*x^2) + e^3*(16*f^3 + 8*f^2*g*x - 2*f*g^2*x^2 + g^3*x^3)) + 
 572*a*c*g^2*(105*d^3*g^3*(-8*f^2 - 4*f*g*x + g^2*x^2) + 189*d^2*e*g^2*(16 
*f^3 + 8*f^2*g*x - 2*f*g^2*x^2 + g^3*x^3) + 27*d*e^2*g*(-128*f^4 - 64*f^3* 
g*x + 16*f^2*g^2*x^2 - 8*f*g^3*x^3 + 5*g^4*x^4) + 5*e^3*(256*f^5 + 128*f^4 
*g*x - 32*f^3*g^2*x^2 + 16*f^2*g^3*x^3 - 10*f*g^4*x^4 + 7*g^5*x^5)) + 6*c^ 
2*(429*d^3*g^3*(-128*f^4 - 64*f^3*g*x + 16*f^2*g^2*x^2 - 8*f*g^3*x^3 + 5*g 
^4*x^4) + 715*d^2*e*g^2*(256*f^5 + 128*f^4*g*x - 32*f^3*g^2*x^2 + 16*f^2*g 
^3*x^3 - 10*f*g^4*x^4 + 7*g^5*x^5) + 195*d*e^2*g*(-1024*f^6 - 512*f^5*g*x 
+ 128*f^4*g^2*x^2 - 64*f^3*g^3*x^3 + 40*f^2*g^4*x^4 - 28*f*g^5*x^5 + 21*g^ 
6*x^6) + 35*e^3*(2048*f^7 + 1024*f^6*g*x - 256*f^5*g^2*x^2 + 128*f^4*g^3*x 
^3 - 80*f^3*g^4*x^4 + 56*f^2*g^5*x^5 - 42*f*g^6*x^6 + 33*g^7*x^7)))/(45045 
*g^8*Sqrt[f + g*x])
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 470, 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.

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

\(\Big \downarrow \) 652

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*(e*f - d*g)^3*(c*f^2 + a*g^2)^2)/(g^8*Sqrt[f + g*x]) + (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))*Sqrt[f + g*x])/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)^(3/2))/(3*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) 
^(5/2))/(5*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)^(7/2))/(7*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)^(9/2))/(9* 
g^8) - (2*c^2*e^2*(7*e*f - 3*d*g)*(f + g*x)^(11/2))/(11*g^8) + (2*c^2*e^3* 
(f + g*x)^(13/2))/(13*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 = 0.99 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(-\frac {2 \left (\left (\left (-\frac {1}{13} c^{2} x^{7}-\frac {1}{5} a^{2} x^{3}-\frac {2}{9} a c \,x^{5}\right ) e^{3}-x^{2} d \left (\frac {6}{7} a c \,x^{2}+\frac {3}{11} c^{2} x^{4}+a^{2}\right ) e^{2}-3 \left (\frac {2}{5} a c \,x^{2}+a^{2}+\frac {1}{9} c^{2} x^{4}\right ) x \,d^{2} e +d^{3} \left (-\frac {1}{7} c^{2} x^{4}-\frac {2}{3} a c \,x^{2}+a^{2}\right )\right ) g^{7}-6 f \left (\left (-\frac {7}{429} c^{2} x^{6}-\frac {10}{189} a c \,x^{4}-\frac {1}{15} a^{2} x^{2}\right ) e^{3}-\frac {2 x \left (\frac {1}{11} c^{2} x^{4}+\frac {12}{35} a c \,x^{2}+a^{2}\right ) d \,e^{2}}{3}+d^{2} \left (-\frac {5}{63} c^{2} x^{4}-\frac {2}{5} a c \,x^{2}+a^{2}\right ) e -\frac {4 c x \,d^{3} \left (\frac {3 c \,x^{2}}{35}+a \right )}{9}\right ) g^{6}+8 f^{2} \left (\left (-\frac {7}{429} c^{2} x^{5}-\frac {4}{63} a c \,x^{3}-\frac {1}{5} a^{2} x \right ) e^{3}+d \left (-\frac {5}{77} c^{2} x^{4}-\frac {12}{35} a c \,x^{2}+a^{2}\right ) e^{2}-\frac {6 c x \,d^{2} \left (\frac {5 c \,x^{2}}{63}+a \right ) e}{5}+\frac {2 \left (-\frac {3 c \,x^{2}}{35}+a \right ) c \,d^{3}}{3}\right ) g^{5}-\frac {16 f^{3} \left (\left (-\frac {25}{429} c^{2} x^{4}-\frac {20}{63} a c \,x^{2}+a^{2}\right ) e^{3}-\frac {24 c \left (\frac {5 c \,x^{2}}{66}+a \right ) x d \,e^{2}}{7}+6 c \left (-\frac {5 c \,x^{2}}{63}+a \right ) d^{2} e -\frac {4 c^{2} d^{3} x}{7}\right ) g^{4}}{5}+\frac {768 \left (-\frac {5 \left (\frac {21 c \,x^{2}}{286}+a \right ) x \,e^{3}}{27}+d \left (-\frac {5 c \,x^{2}}{66}+a \right ) e^{2}-\frac {5 c \,d^{2} e x}{18}+\frac {c \,d^{3}}{6}\right ) f^{4} c \,g^{3}}{35}-\frac {512 e \left (\left (-\frac {21 c \,x^{2}}{286}+a \right ) e^{2}-\frac {9 c d x e}{11}+\frac {3 c \,d^{2}}{2}\right ) f^{5} c \,g^{2}}{63}+\frac {1024 e^{2} f^{6} c^{2} \left (-\frac {7 e x}{39}+d \right ) g}{77}-\frac {2048 c^{2} e^{3} f^{7}}{429}\right )}{\sqrt {g x +f}\, g^{8}}\)	\(500\)
risch	\(\frac {2 \left (3465 g^{6} c^{2} e^{3} x^{6}+12285 g^{6} c^{2} d \,e^{2} x^{5}-7875 f \,g^{5} c^{2} x^{5} e^{3}+10010 g^{6} a c \,e^{3} x^{4}+15015 g^{6} d^{2} e \,c^{2} x^{4}-28665 f \,g^{5} c^{2} d \,x^{4} e^{2}+13755 f^{2} g^{4} c^{2} x^{4} e^{3}+38610 g^{6} a d \,e^{2} c \,x^{3}-24310 f \,g^{5} e^{3} a c \,x^{3}+6435 g^{6} c^{2} d^{3} x^{3}-36465 f \,g^{5} d^{2} c^{2} x^{3} e +52065 f^{2} g^{4} c^{2} d \,x^{3} e^{2}-22155 f^{3} g^{3} e^{3} x^{3} c^{2}+9009 g^{6} a^{2} e^{3} x^{2}+54054 g^{6} a c \,d^{2} e \,x^{2}-100386 f \,g^{5} d \,e^{2} a c \,x^{2}+47190 f^{2} g^{4} a c \,e^{3} x^{2}-16731 f \,g^{5} c^{2} d^{3} x^{2}+70785 f^{2} g^{4} d^{2} e \,c^{2} x^{2}-89505 f^{3} g^{3} d \,e^{2} c^{2} x^{2}+35595 f^{4} c^{2} g^{2} x^{2} e^{3}+45045 g^{6} a^{2} d \,e^{2} x -27027 f \,g^{5} e^{3} a^{2} x +30030 g^{6} a c \,d^{3} x -162162 f \,g^{5} d^{2} e a c x +223938 f^{2} g^{4} d \,e^{2} a c x -92950 f^{3} g^{3} a c \,e^{3} x +37323 f^{2} g^{4} c^{2} d^{3} x -139425 f^{3} g^{3} d^{2} e \,c^{2} x +164385 f^{4} c^{2} g^{2} d x \,e^{2}-62475 e^{3} f^{5} c^{2} g x +135135 d^{2} e \,a^{2} g^{6}-225225 a^{2} d \,e^{2} f \,g^{5}+99099 a^{2} e^{3} f^{2} g^{4}-150150 a c \,d^{3} f \,g^{5}+594594 a c \,d^{2} e \,f^{2} g^{4}-718146 a c d \,e^{2} f^{3} g^{3}+275990 a c \,e^{3} f^{4} g^{2}-119691 c^{2} d^{3} f^{3} g^{3}+413985 c^{2} d^{2} e \,f^{4} g^{2}-463905 c^{2} d \,e^{2} f^{5} g +169995 c^{2} e^{3} f^{6}\right ) \sqrt {g x +f}}{45045 g^{8}}-\frac {2 \left (a^{2} d^{3} g^{7}-3 a^{2} d^{2} e f \,g^{6}+3 a^{2} d \,e^{2} f^{2} g^{5}-a^{2} e^{3} f^{3} g^{4}+2 a c \,d^{3} f^{2} g^{5}-6 a c \,d^{2} e \,f^{3} g^{4}+6 a c d \,e^{2} f^{4} g^{3}-2 a c \,e^{3} f^{5} g^{2}+c^{2} d^{3} f^{4} g^{3}-3 c^{2} d^{2} e \,f^{5} g^{2}+3 c^{2} d \,e^{2} f^{6} g -c^{2} e^{3} f^{7}\right )}{g^{8} \sqrt {g x +f}}\)	\(794\)
gosper	\(-\frac {2 \left (-3465 c^{2} e^{3} x^{7} g^{7}-12285 c^{2} d \,e^{2} g^{7} x^{6}+4410 c^{2} e^{3} f \,g^{6} x^{6}-10010 a c \,e^{3} g^{7} x^{5}-15015 c^{2} d^{2} e \,g^{7} x^{5}+16380 c^{2} d \,e^{2} f \,g^{6} x^{5}-5880 c^{2} e^{3} f^{2} g^{5} x^{5}-38610 a c d \,e^{2} g^{7} x^{4}+14300 a c \,e^{3} f \,g^{6} x^{4}-6435 c^{2} d^{3} g^{7} x^{4}+21450 c^{2} d^{2} e f \,g^{6} x^{4}-23400 c^{2} d \,e^{2} f^{2} g^{5} x^{4}+8400 c^{2} e^{3} f^{3} g^{4} x^{4}-9009 a^{2} e^{3} g^{7} x^{3}-54054 a c \,d^{2} e \,g^{7} x^{3}+61776 a c d \,e^{2} f \,g^{6} x^{3}-22880 a c \,e^{3} f^{2} g^{5} x^{3}+10296 c^{2} d^{3} f \,g^{6} x^{3}-34320 c^{2} d^{2} e \,f^{2} g^{5} x^{3}+37440 c^{2} d \,e^{2} f^{3} g^{4} x^{3}-13440 c^{2} e^{3} f^{4} g^{3} x^{3}-45045 a^{2} d \,e^{2} g^{7} x^{2}+18018 a^{2} e^{3} f \,g^{6} x^{2}-30030 a c \,d^{3} g^{7} x^{2}+108108 a c \,d^{2} e f \,g^{6} x^{2}-123552 a c d \,e^{2} f^{2} g^{5} x^{2}+45760 a c \,e^{3} f^{3} g^{4} x^{2}-20592 c^{2} d^{3} f^{2} g^{5} x^{2}+68640 c^{2} d^{2} e \,f^{3} g^{4} x^{2}-74880 c^{2} d \,e^{2} f^{4} g^{3} x^{2}+26880 c^{2} e^{3} f^{5} g^{2} x^{2}-135135 a^{2} d^{2} e \,g^{7} x +180180 a^{2} d \,e^{2} f \,g^{6} x -72072 a^{2} e^{3} f^{2} g^{5} x +120120 a c \,d^{3} f \,g^{6} x -432432 a c \,d^{2} e \,f^{2} g^{5} x +494208 a c d \,e^{2} f^{3} g^{4} x -183040 a c \,e^{3} f^{4} g^{3} x +82368 c^{2} d^{3} f^{3} g^{4} x -274560 c^{2} d^{2} e \,f^{4} g^{3} x +299520 c^{2} d \,e^{2} f^{5} g^{2} x -107520 c^{2} e^{3} f^{6} g x +45045 a^{2} d^{3} g^{7}-270270 a^{2} d^{2} e f \,g^{6}+360360 a^{2} d \,e^{2} f^{2} g^{5}-144144 a^{2} e^{3} f^{3} g^{4}+240240 a c \,d^{3} f^{2} g^{5}-864864 a c \,d^{2} e \,f^{3} g^{4}+988416 a c d \,e^{2} f^{4} g^{3}-366080 a c \,e^{3} f^{5} g^{2}+164736 c^{2} d^{3} f^{4} g^{3}-549120 c^{2} d^{2} e \,f^{5} g^{2}+599040 c^{2} d \,e^{2} f^{6} g -215040 c^{2} e^{3} f^{7}\right )}{45045 \sqrt {g x +f}\, g^{8}}\)	\(818\)
trager	\(-\frac {2 \left (-3465 c^{2} e^{3} x^{7} g^{7}-12285 c^{2} d \,e^{2} g^{7} x^{6}+4410 c^{2} e^{3} f \,g^{6} x^{6}-10010 a c \,e^{3} g^{7} x^{5}-15015 c^{2} d^{2} e \,g^{7} x^{5}+16380 c^{2} d \,e^{2} f \,g^{6} x^{5}-5880 c^{2} e^{3} f^{2} g^{5} x^{5}-38610 a c d \,e^{2} g^{7} x^{4}+14300 a c \,e^{3} f \,g^{6} x^{4}-6435 c^{2} d^{3} g^{7} x^{4}+21450 c^{2} d^{2} e f \,g^{6} x^{4}-23400 c^{2} d \,e^{2} f^{2} g^{5} x^{4}+8400 c^{2} e^{3} f^{3} g^{4} x^{4}-9009 a^{2} e^{3} g^{7} x^{3}-54054 a c \,d^{2} e \,g^{7} x^{3}+61776 a c d \,e^{2} f \,g^{6} x^{3}-22880 a c \,e^{3} f^{2} g^{5} x^{3}+10296 c^{2} d^{3} f \,g^{6} x^{3}-34320 c^{2} d^{2} e \,f^{2} g^{5} x^{3}+37440 c^{2} d \,e^{2} f^{3} g^{4} x^{3}-13440 c^{2} e^{3} f^{4} g^{3} x^{3}-45045 a^{2} d \,e^{2} g^{7} x^{2}+18018 a^{2} e^{3} f \,g^{6} x^{2}-30030 a c \,d^{3} g^{7} x^{2}+108108 a c \,d^{2} e f \,g^{6} x^{2}-123552 a c d \,e^{2} f^{2} g^{5} x^{2}+45760 a c \,e^{3} f^{3} g^{4} x^{2}-20592 c^{2} d^{3} f^{2} g^{5} x^{2}+68640 c^{2} d^{2} e \,f^{3} g^{4} x^{2}-74880 c^{2} d \,e^{2} f^{4} g^{3} x^{2}+26880 c^{2} e^{3} f^{5} g^{2} x^{2}-135135 a^{2} d^{2} e \,g^{7} x +180180 a^{2} d \,e^{2} f \,g^{6} x -72072 a^{2} e^{3} f^{2} g^{5} x +120120 a c \,d^{3} f \,g^{6} x -432432 a c \,d^{2} e \,f^{2} g^{5} x +494208 a c d \,e^{2} f^{3} g^{4} x -183040 a c \,e^{3} f^{4} g^{3} x +82368 c^{2} d^{3} f^{3} g^{4} x -274560 c^{2} d^{2} e \,f^{4} g^{3} x +299520 c^{2} d \,e^{2} f^{5} g^{2} x -107520 c^{2} e^{3} f^{6} g x +45045 a^{2} d^{3} g^{7}-270270 a^{2} d^{2} e f \,g^{6}+360360 a^{2} d \,e^{2} f^{2} g^{5}-144144 a^{2} e^{3} f^{3} g^{4}+240240 a c \,d^{3} f^{2} g^{5}-864864 a c \,d^{2} e \,f^{3} g^{4}+988416 a c d \,e^{2} f^{4} g^{3}-366080 a c \,e^{3} f^{5} g^{2}+164736 c^{2} d^{3} f^{4} g^{3}-549120 c^{2} d^{2} e \,f^{5} g^{2}+599040 c^{2} d \,e^{2} f^{6} g -215040 c^{2} e^{3} f^{7}\right )}{45045 \sqrt {g x +f}\, g^{8}}\)	\(818\)
orering	\(-\frac {2 \left (-3465 c^{2} e^{3} x^{7} g^{7}-12285 c^{2} d \,e^{2} g^{7} x^{6}+4410 c^{2} e^{3} f \,g^{6} x^{6}-10010 a c \,e^{3} g^{7} x^{5}-15015 c^{2} d^{2} e \,g^{7} x^{5}+16380 c^{2} d \,e^{2} f \,g^{6} x^{5}-5880 c^{2} e^{3} f^{2} g^{5} x^{5}-38610 a c d \,e^{2} g^{7} x^{4}+14300 a c \,e^{3} f \,g^{6} x^{4}-6435 c^{2} d^{3} g^{7} x^{4}+21450 c^{2} d^{2} e f \,g^{6} x^{4}-23400 c^{2} d \,e^{2} f^{2} g^{5} x^{4}+8400 c^{2} e^{3} f^{3} g^{4} x^{4}-9009 a^{2} e^{3} g^{7} x^{3}-54054 a c \,d^{2} e \,g^{7} x^{3}+61776 a c d \,e^{2} f \,g^{6} x^{3}-22880 a c \,e^{3} f^{2} g^{5} x^{3}+10296 c^{2} d^{3} f \,g^{6} x^{3}-34320 c^{2} d^{2} e \,f^{2} g^{5} x^{3}+37440 c^{2} d \,e^{2} f^{3} g^{4} x^{3}-13440 c^{2} e^{3} f^{4} g^{3} x^{3}-45045 a^{2} d \,e^{2} g^{7} x^{2}+18018 a^{2} e^{3} f \,g^{6} x^{2}-30030 a c \,d^{3} g^{7} x^{2}+108108 a c \,d^{2} e f \,g^{6} x^{2}-123552 a c d \,e^{2} f^{2} g^{5} x^{2}+45760 a c \,e^{3} f^{3} g^{4} x^{2}-20592 c^{2} d^{3} f^{2} g^{5} x^{2}+68640 c^{2} d^{2} e \,f^{3} g^{4} x^{2}-74880 c^{2} d \,e^{2} f^{4} g^{3} x^{2}+26880 c^{2} e^{3} f^{5} g^{2} x^{2}-135135 a^{2} d^{2} e \,g^{7} x +180180 a^{2} d \,e^{2} f \,g^{6} x -72072 a^{2} e^{3} f^{2} g^{5} x +120120 a c \,d^{3} f \,g^{6} x -432432 a c \,d^{2} e \,f^{2} g^{5} x +494208 a c d \,e^{2} f^{3} g^{4} x -183040 a c \,e^{3} f^{4} g^{3} x +82368 c^{2} d^{3} f^{3} g^{4} x -274560 c^{2} d^{2} e \,f^{4} g^{3} x +299520 c^{2} d \,e^{2} f^{5} g^{2} x -107520 c^{2} e^{3} f^{6} g x +45045 a^{2} d^{3} g^{7}-270270 a^{2} d^{2} e f \,g^{6}+360360 a^{2} d \,e^{2} f^{2} g^{5}-144144 a^{2} e^{3} f^{3} g^{4}+240240 a c \,d^{3} f^{2} g^{5}-864864 a c \,d^{2} e \,f^{3} g^{4}+988416 a c d \,e^{2} f^{4} g^{3}-366080 a c \,e^{3} f^{5} g^{2}+164736 c^{2} d^{3} f^{4} g^{3}-549120 c^{2} d^{2} e \,f^{5} g^{2}+599040 c^{2} d \,e^{2} f^{6} g -215040 c^{2} e^{3} f^{7}\right )}{45045 \sqrt {g x +f}\, g^{8}}\)	\(818\)
derivativedivides	\(\frac {-\frac {40 a c \,e^{3} f^{3} g^{2} \left (g x +f \right )^{\frac {3}{2}}}{3}+30 c^{2} d^{2} e \,f^{4} g^{2} \sqrt {g x +f}-24 c^{2} d \,e^{2} f^{3} g \left (g x +f \right )^{\frac {5}{2}}-12 a c \,d^{2} e f \,g^{4} \left (g x +f \right )^{\frac {3}{2}}-2 a^{2} e^{3} f \,g^{4} \left (g x +f \right )^{\frac {3}{2}}-12 a^{2} d \,e^{2} f \,g^{5} \sqrt {g x +f}-8 c^{2} d^{3} f^{3} g^{3} \sqrt {g x +f}-8 a c \,d^{3} f \,g^{5} \sqrt {g x +f}+\frac {4 a c \,d^{3} g^{5} \left (g x +f \right )^{\frac {3}{2}}}{3}+2 a^{2} d \,e^{2} g^{5} \left (g x +f \right )^{\frac {3}{2}}+24 a c d \,e^{2} f^{2} g^{3} \left (g x +f \right )^{\frac {3}{2}}+36 a c \,d^{2} e \,f^{2} g^{4} \sqrt {g x +f}-\frac {48 a c d \,e^{2} f \,g^{3} \left (g x +f \right )^{\frac {5}{2}}}{5}-48 a c d \,e^{2} f^{3} g^{3} \sqrt {g x +f}+\frac {12 a c d \,e^{2} g^{3} \left (g x +f \right )^{\frac {7}{2}}}{7}-4 c^{2} d \,e^{2} f g \left (g x +f \right )^{\frac {9}{2}}+\frac {4 a c \,e^{3} g^{2} \left (g x +f \right )^{\frac {9}{2}}}{9}-\frac {20 a c \,e^{3} f \,g^{2} \left (g x +f \right )^{\frac {7}{2}}}{7}-\frac {8 c^{2} d^{3} f \,g^{3} \left (g x +f \right )^{\frac {5}{2}}}{5}+6 a^{2} d^{2} e \,g^{6} \sqrt {g x +f}-\frac {30 c^{2} d^{2} e f \,g^{2} \left (g x +f \right )^{\frac {7}{2}}}{7}-20 c^{2} d^{2} e \,f^{3} g^{2} \left (g x +f \right )^{\frac {3}{2}}+6 a^{2} e^{3} f^{2} g^{4} \sqrt {g x +f}+\frac {2 c^{2} d^{3} g^{3} \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {6 c^{2} d \,e^{2} g \left (g x +f \right )^{\frac {11}{2}}}{11}+4 c^{2} d^{3} f^{2} g^{3} \left (g x +f \right )^{\frac {3}{2}}+\frac {2 c^{2} d^{2} e \,g^{2} \left (g x +f \right )^{\frac {9}{2}}}{3}-36 c^{2} d \,e^{2} f^{5} g \sqrt {g x +f}+30 c^{2} d \,e^{2} f^{4} g \left (g x +f \right )^{\frac {3}{2}}+\frac {12 a c \,d^{2} e \,g^{4} \left (g x +f \right )^{\frac {5}{2}}}{5}+20 a c \,e^{3} f^{4} g^{2} \sqrt {g x +f}+8 a c \,e^{3} f^{2} g^{2} \left (g x +f \right )^{\frac {5}{2}}+12 c^{2} d^{2} e \,f^{2} g^{2} \left (g x +f \right )^{\frac {5}{2}}+\frac {90 c^{2} d \,e^{2} f^{2} g \left (g x +f \right )^{\frac {7}{2}}}{7}+14 c^{2} e^{3} f^{6} \sqrt {g x +f}-14 c^{2} e^{3} f^{5} \left (g x +f \right )^{\frac {3}{2}}-\frac {14 c^{2} e^{3} f \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {14 c^{2} e^{3} f^{2} \left (g x +f \right )^{\frac {9}{2}}}{3}+14 c^{2} e^{3} f^{4} \left (g x +f \right )^{\frac {5}{2}}+\frac {2 a^{2} e^{3} g^{4} \left (g x +f \right )^{\frac {5}{2}}}{5}-10 c^{2} e^{3} f^{3} \left (g x +f \right )^{\frac {7}{2}}+\frac {2 c^{2} e^{3} \left (g x +f \right )^{\frac {13}{2}}}{13}-\frac {2 \left (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}\right )}{\sqrt {g x +f}}}{g^{8}}\)	\(977\)
default	\(\frac {-\frac {40 a c \,e^{3} f^{3} g^{2} \left (g x +f \right )^{\frac {3}{2}}}{3}+30 c^{2} d^{2} e \,f^{4} g^{2} \sqrt {g x +f}-24 c^{2} d \,e^{2} f^{3} g \left (g x +f \right )^{\frac {5}{2}}-12 a c \,d^{2} e f \,g^{4} \left (g x +f \right )^{\frac {3}{2}}-2 a^{2} e^{3} f \,g^{4} \left (g x +f \right )^{\frac {3}{2}}-12 a^{2} d \,e^{2} f \,g^{5} \sqrt {g x +f}-8 c^{2} d^{3} f^{3} g^{3} \sqrt {g x +f}-8 a c \,d^{3} f \,g^{5} \sqrt {g x +f}+\frac {4 a c \,d^{3} g^{5} \left (g x +f \right )^{\frac {3}{2}}}{3}+2 a^{2} d \,e^{2} g^{5} \left (g x +f \right )^{\frac {3}{2}}+24 a c d \,e^{2} f^{2} g^{3} \left (g x +f \right )^{\frac {3}{2}}+36 a c \,d^{2} e \,f^{2} g^{4} \sqrt {g x +f}-\frac {48 a c d \,e^{2} f \,g^{3} \left (g x +f \right )^{\frac {5}{2}}}{5}-48 a c d \,e^{2} f^{3} g^{3} \sqrt {g x +f}+\frac {12 a c d \,e^{2} g^{3} \left (g x +f \right )^{\frac {7}{2}}}{7}-4 c^{2} d \,e^{2} f g \left (g x +f \right )^{\frac {9}{2}}+\frac {4 a c \,e^{3} g^{2} \left (g x +f \right )^{\frac {9}{2}}}{9}-\frac {20 a c \,e^{3} f \,g^{2} \left (g x +f \right )^{\frac {7}{2}}}{7}-\frac {8 c^{2} d^{3} f \,g^{3} \left (g x +f \right )^{\frac {5}{2}}}{5}+6 a^{2} d^{2} e \,g^{6} \sqrt {g x +f}-\frac {30 c^{2} d^{2} e f \,g^{2} \left (g x +f \right )^{\frac {7}{2}}}{7}-20 c^{2} d^{2} e \,f^{3} g^{2} \left (g x +f \right )^{\frac {3}{2}}+6 a^{2} e^{3} f^{2} g^{4} \sqrt {g x +f}+\frac {2 c^{2} d^{3} g^{3} \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {6 c^{2} d \,e^{2} g \left (g x +f \right )^{\frac {11}{2}}}{11}+4 c^{2} d^{3} f^{2} g^{3} \left (g x +f \right )^{\frac {3}{2}}+\frac {2 c^{2} d^{2} e \,g^{2} \left (g x +f \right )^{\frac {9}{2}}}{3}-36 c^{2} d \,e^{2} f^{5} g \sqrt {g x +f}+30 c^{2} d \,e^{2} f^{4} g \left (g x +f \right )^{\frac {3}{2}}+\frac {12 a c \,d^{2} e \,g^{4} \left (g x +f \right )^{\frac {5}{2}}}{5}+20 a c \,e^{3} f^{4} g^{2} \sqrt {g x +f}+8 a c \,e^{3} f^{2} g^{2} \left (g x +f \right )^{\frac {5}{2}}+12 c^{2} d^{2} e \,f^{2} g^{2} \left (g x +f \right )^{\frac {5}{2}}+\frac {90 c^{2} d \,e^{2} f^{2} g \left (g x +f \right )^{\frac {7}{2}}}{7}+14 c^{2} e^{3} f^{6} \sqrt {g x +f}-14 c^{2} e^{3} f^{5} \left (g x +f \right )^{\frac {3}{2}}-\frac {14 c^{2} e^{3} f \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {14 c^{2} e^{3} f^{2} \left (g x +f \right )^{\frac {9}{2}}}{3}+14 c^{2} e^{3} f^{4} \left (g x +f \right )^{\frac {5}{2}}+\frac {2 a^{2} e^{3} g^{4} \left (g x +f \right )^{\frac {5}{2}}}{5}-10 c^{2} e^{3} f^{3} \left (g x +f \right )^{\frac {7}{2}}+\frac {2 c^{2} e^{3} \left (g x +f \right )^{\frac {13}{2}}}{13}-\frac {2 \left (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}\right )}{\sqrt {g x +f}}}{g^{8}}\)	\(977\)

Input:

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

Output:

-2/(g*x+f)^(1/2)*(((-1/13*c^2*x^7-1/5*a^2*x^3-2/9*a*c*x^5)*e^3-x^2*d*(6/7* 
a*c*x^2+3/11*c^2*x^4+a^2)*e^2-3*(2/5*a*c*x^2+a^2+1/9*c^2*x^4)*x*d^2*e+d^3* 
(-1/7*c^2*x^4-2/3*a*c*x^2+a^2))*g^7-6*f*((-7/429*c^2*x^6-10/189*a*c*x^4-1/ 
15*a^2*x^2)*e^3-2/3*x*(1/11*c^2*x^4+12/35*a*c*x^2+a^2)*d*e^2+d^2*(-5/63*c^ 
2*x^4-2/5*a*c*x^2+a^2)*e-4/9*c*x*d^3*(3/35*c*x^2+a))*g^6+8*f^2*((-7/429*c^ 
2*x^5-4/63*a*c*x^3-1/5*a^2*x)*e^3+d*(-5/77*c^2*x^4-12/35*a*c*x^2+a^2)*e^2- 
6/5*c*x*d^2*(5/63*c*x^2+a)*e+2/3*(-3/35*c*x^2+a)*c*d^3)*g^5-16/5*f^3*((-25 
/429*c^2*x^4-20/63*a*c*x^2+a^2)*e^3-24/7*c*(5/66*c*x^2+a)*x*d*e^2+6*c*(-5/ 
63*c*x^2+a)*d^2*e-4/7*c^2*d^3*x)*g^4+768/35*(-5/27*(21/286*c*x^2+a)*x*e^3+ 
d*(-5/66*c*x^2+a)*e^2-5/18*c*d^2*e*x+1/6*c*d^3)*f^4*c*g^3-512/63*e*((-21/2 
86*c*x^2+a)*e^2-9/11*c*d*x*e+3/2*c*d^2)*f^5*c*g^2+1024/77*e^2*f^6*c^2*(-7/ 
39*e*x+d)*g-2048/429*c^2*e^3*f^7)/g^8
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 715, normalized size of antiderivative = 1.52 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )^2}{(f+g x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/45045*(3465*c^2*e^3*g^7*x^7 + 215040*c^2*e^3*f^7 - 599040*c^2*d*e^2*f^6* 
g + 270270*a^2*d^2*e*f*g^6 - 45045*a^2*d^3*g^7 + 183040*(3*c^2*d^2*e + 2*a 
*c*e^3)*f^5*g^2 - 164736*(c^2*d^3 + 6*a*c*d*e^2)*f^4*g^3 + 144144*(6*a*c*d 
^2*e + a^2*e^3)*f^3*g^4 - 120120*(2*a*c*d^3 + 3*a^2*d*e^2)*f^2*g^5 - 315*( 
14*c^2*e^3*f*g^6 - 39*c^2*d*e^2*g^7)*x^6 + 35*(168*c^2*e^3*f^2*g^5 - 468*c 
^2*d*e^2*f*g^6 + 143*(3*c^2*d^2*e + 2*a*c*e^3)*g^7)*x^5 - 5*(1680*c^2*e^3* 
f^3*g^4 - 4680*c^2*d*e^2*f^2*g^5 + 1430*(3*c^2*d^2*e + 2*a*c*e^3)*f*g^6 - 
1287*(c^2*d^3 + 6*a*c*d*e^2)*g^7)*x^4 + (13440*c^2*e^3*f^4*g^3 - 37440*c^2 
*d*e^2*f^3*g^4 + 11440*(3*c^2*d^2*e + 2*a*c*e^3)*f^2*g^5 - 10296*(c^2*d^3 
+ 6*a*c*d*e^2)*f*g^6 + 9009*(6*a*c*d^2*e + a^2*e^3)*g^7)*x^3 - (26880*c^2* 
e^3*f^5*g^2 - 74880*c^2*d*e^2*f^4*g^3 + 22880*(3*c^2*d^2*e + 2*a*c*e^3)*f^ 
3*g^4 - 20592*(c^2*d^3 + 6*a*c*d*e^2)*f^2*g^5 + 18018*(6*a*c*d^2*e + a^2*e 
^3)*f*g^6 - 15015*(2*a*c*d^3 + 3*a^2*d*e^2)*g^7)*x^2 + (107520*c^2*e^3*f^6 
*g - 299520*c^2*d*e^2*f^5*g^2 + 135135*a^2*d^2*e*g^7 + 91520*(3*c^2*d^2*e 
+ 2*a*c*e^3)*f^4*g^3 - 82368*(c^2*d^3 + 6*a*c*d*e^2)*f^3*g^4 + 72072*(6*a* 
c*d^2*e + a^2*e^3)*f^2*g^5 - 60060*(2*a*c*d^3 + 3*a^2*d*e^2)*f*g^6)*x)*sqr 
t(g*x + f)/(g^9*x + f*g^8)
 

Sympy [A] (verification not implemented)

Time = 53.67 (sec) , antiderivative size = 872, normalized size of antiderivative = 1.86 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )^2}{(f+g x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((2*(c**2*e**3*(f + g*x)**(13/2)/(13*g**7) + (f + g*x)**(11/2)*(3 
*c**2*d*e**2*g - 7*c**2*e**3*f)/(11*g**7) + (f + g*x)**(9/2)*(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)/(9*g**7 
) + (f + g*x)**(7/2)*(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)/( 
7*g**7) + (f + g*x)**(5/2)*(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)/(5*g**7) + (f + g*x 
)**(3/2)*(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)/(3*g**7) + sqrt(f + g*x)*(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 
)/g**7 - (a*g**2 + c*f**2)**2*(d*g - e*f)**3/(g**7*sqrt(f + g*x)))/g, Ne(g 
, 0)), ((a**2*d**3*x + 3*a**2*d**2*e*x**2/2 + 3*c**2*d*e**2*x**7/7 + c**2* 
e**3*x**8/8 + x**6*(2*a*c*e**3 + 3*c**2*d**2*e)/6 + x**5*(6*a*c*d*e**2 + c 
**2*d**3)/5 + x**4*(a**2*e**3 + 6*a*c*d**2*e)/4 + x**3*(3*a**2*d*e**2 + 2* 
a*c*d**3)/3)/f**(3/2), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 712, normalized size of antiderivative = 1.51 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )^2}{(f+g x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/45045*((3465*(g*x + f)^(13/2)*c^2*e^3 - 4095*(7*c^2*e^3*f - 3*c^2*d*e^2* 
g)*(g*x + f)^(11/2) + 5005*(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)^(9/2) - 6435*(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)^(7/2) + 9009*(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)^(5/2) - 15015*(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)^(3/2) + 45045*(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)* 
sqrt(g*x + f))/g^7 + 45045*(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)/(sqrt(g*x + f)*g^7))/g
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1014 vs. \(2 (442) = 884\).

Time = 0.14 (sec) , antiderivative size = 1014, normalized size of antiderivative = 2.16 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )^2}{(f+g x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

2*(c^2*e^3*f^7 - 3*c^2*d*e^2*f^6*g + 3*c^2*d^2*e*f^5*g^2 + 2*a*c*e^3*f^5*g 
^2 - c^2*d^3*f^4*g^3 - 6*a*c*d*e^2*f^4*g^3 + 6*a*c*d^2*e*f^3*g^4 + a^2*e^3 
*f^3*g^4 - 2*a*c*d^3*f^2*g^5 - 3*a^2*d*e^2*f^2*g^5 + 3*a^2*d^2*e*f*g^6 - a 
^2*d^3*g^7)/(sqrt(g*x + f)*g^8) + 2/45045*(3465*(g*x + f)^(13/2)*c^2*e^3*g 
^96 - 28665*(g*x + f)^(11/2)*c^2*e^3*f*g^96 + 105105*(g*x + f)^(9/2)*c^2*e 
^3*f^2*g^96 - 225225*(g*x + f)^(7/2)*c^2*e^3*f^3*g^96 + 315315*(g*x + f)^( 
5/2)*c^2*e^3*f^4*g^96 - 315315*(g*x + f)^(3/2)*c^2*e^3*f^5*g^96 + 315315*s 
qrt(g*x + f)*c^2*e^3*f^6*g^96 + 12285*(g*x + f)^(11/2)*c^2*d*e^2*g^97 - 90 
090*(g*x + f)^(9/2)*c^2*d*e^2*f*g^97 + 289575*(g*x + f)^(7/2)*c^2*d*e^2*f^ 
2*g^97 - 540540*(g*x + f)^(5/2)*c^2*d*e^2*f^3*g^97 + 675675*(g*x + f)^(3/2 
)*c^2*d*e^2*f^4*g^97 - 810810*sqrt(g*x + f)*c^2*d*e^2*f^5*g^97 + 15015*(g* 
x + f)^(9/2)*c^2*d^2*e*g^98 + 10010*(g*x + f)^(9/2)*a*c*e^3*g^98 - 96525*( 
g*x + f)^(7/2)*c^2*d^2*e*f*g^98 - 64350*(g*x + f)^(7/2)*a*c*e^3*f*g^98 + 2 
70270*(g*x + f)^(5/2)*c^2*d^2*e*f^2*g^98 + 180180*(g*x + f)^(5/2)*a*c*e^3* 
f^2*g^98 - 450450*(g*x + f)^(3/2)*c^2*d^2*e*f^3*g^98 - 300300*(g*x + f)^(3 
/2)*a*c*e^3*f^3*g^98 + 675675*sqrt(g*x + f)*c^2*d^2*e*f^4*g^98 + 450450*sq 
rt(g*x + f)*a*c*e^3*f^4*g^98 + 6435*(g*x + f)^(7/2)*c^2*d^3*g^99 + 38610*( 
g*x + f)^(7/2)*a*c*d*e^2*g^99 - 36036*(g*x + f)^(5/2)*c^2*d^3*f*g^99 - 216 
216*(g*x + f)^(5/2)*a*c*d*e^2*f*g^99 + 90090*(g*x + f)^(3/2)*c^2*d^3*f^2*g 
^99 + 540540*(g*x + f)^(3/2)*a*c*d*e^2*f^2*g^99 - 180180*sqrt(g*x + f)*...
 

Mupad [B] (verification not implemented)

Time = 5.80 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.31 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )^2}{(f+g x)^{3/2}} \, dx=\frac {{\left (f+g\,x\right )}^{7/2}\,\left (2\,c^2\,d^3\,g^3-30\,c^2\,d^2\,e\,f\,g^2+90\,c^2\,d\,e^2\,f^2\,g-70\,c^2\,e^3\,f^3+12\,a\,c\,d\,e^2\,g^3-20\,a\,c\,e^3\,f\,g^2\right )}{7\,g^8}-\frac {2\,a^2\,d^3\,g^7-6\,a^2\,d^2\,e\,f\,g^6+6\,a^2\,d\,e^2\,f^2\,g^5-2\,a^2\,e^3\,f^3\,g^4+4\,a\,c\,d^3\,f^2\,g^5-12\,a\,c\,d^2\,e\,f^3\,g^4+12\,a\,c\,d\,e^2\,f^4\,g^3-4\,a\,c\,e^3\,f^5\,g^2+2\,c^2\,d^3\,f^4\,g^3-6\,c^2\,d^2\,e\,f^5\,g^2+6\,c^2\,d\,e^2\,f^6\,g-2\,c^2\,e^3\,f^7}{g^8\,\sqrt {f+g\,x}}+\frac {{\left (f+g\,x\right )}^{5/2}\,\left (2\,a^2\,e^3\,g^4+12\,a\,c\,d^2\,e\,g^4-48\,a\,c\,d\,e^2\,f\,g^3+40\,a\,c\,e^3\,f^2\,g^2-8\,c^2\,d^3\,f\,g^3+60\,c^2\,d^2\,e\,f^2\,g^2-120\,c^2\,d\,e^2\,f^3\,g+70\,c^2\,e^3\,f^4\right )}{5\,g^8}+\frac {2\,c^2\,e^3\,{\left (f+g\,x\right )}^{13/2}}{13\,g^8}+\frac {2\,{\left (f+g\,x\right )}^{3/2}\,\left (d\,g-e\,f\right )\,\left (3\,a^2\,e^2\,g^4+2\,a\,c\,d^2\,g^4-16\,a\,c\,d\,e\,f\,g^3+20\,a\,c\,e^2\,f^2\,g^2+6\,c^2\,d^2\,f^2\,g^2-24\,c^2\,d\,e\,f^3\,g+21\,c^2\,e^2\,f^4\right )}{3\,g^8}+\frac {2\,\sqrt {f+g\,x}\,\left (c\,f^2+a\,g^2\right )\,{\left (d\,g-e\,f\right )}^2\,\left (7\,c\,e\,f^2-4\,c\,d\,f\,g+3\,a\,e\,g^2\right )}{g^8}+\frac {2\,c^2\,e^2\,{\left (f+g\,x\right )}^{11/2}\,\left (3\,d\,g-7\,e\,f\right )}{11\,g^8}+\frac {2\,c\,e\,{\left (f+g\,x\right )}^{9/2}\,\left (3\,c\,d^2\,g^2-18\,c\,d\,e\,f\,g+21\,c\,e^2\,f^2+2\,a\,e^2\,g^2\right )}{9\,g^8} \] Input:

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

Output:

((f + g*x)^(7/2)*(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))/(7*g^8) - (2*a^2* 
d^3*g^7 - 2*c^2*e^3*f^7 - 2*a^2*e^3*f^3*g^4 + 2*c^2*d^3*f^4*g^3 + 4*a*c*d^ 
3*f^2*g^5 - 4*a*c*e^3*f^5*g^2 - 6*a^2*d^2*e*f*g^6 + 6*c^2*d*e^2*f^6*g + 6* 
a^2*d*e^2*f^2*g^5 - 6*c^2*d^2*e*f^5*g^2 + 12*a*c*d*e^2*f^4*g^3 - 12*a*c*d^ 
2*e*f^3*g^4)/(g^8*(f + g*x)^(1/2)) + ((f + g*x)^(5/2)*(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 - 12 
0*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))/(5*g^8) + 
(2*c^2*e^3*(f + g*x)^(13/2))/(13*g^8) + (2*(f + g*x)^(3/2)*(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))/(3*g^8) + (2*(f + g*x) 
^(1/2)*(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))/ 
g^8 + (2*c^2*e^2*(f + g*x)^(11/2)*(3*d*g - 7*e*f))/(11*g^8) + (2*c*e*(f + 
g*x)^(9/2)*(2*a*e^2*g^2 + 3*c*d^2*g^2 + 21*c*e^2*f^2 - 18*c*d*e*f*g))/(9*g 
^8)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 818, normalized size of antiderivative = 1.74 \[ \int \frac {(d+e x)^3 \left (a+c x^2\right )^2}{(f+g x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(2*( - 45045*a**2*d**3*g**7 + 270270*a**2*d**2*e*f*g**6 + 135135*a**2*d**2 
*e*g**7*x - 360360*a**2*d*e**2*f**2*g**5 - 180180*a**2*d*e**2*f*g**6*x + 4 
5045*a**2*d*e**2*g**7*x**2 + 144144*a**2*e**3*f**3*g**4 + 72072*a**2*e**3* 
f**2*g**5*x - 18018*a**2*e**3*f*g**6*x**2 + 9009*a**2*e**3*g**7*x**3 - 240 
240*a*c*d**3*f**2*g**5 - 120120*a*c*d**3*f*g**6*x + 30030*a*c*d**3*g**7*x* 
*2 + 864864*a*c*d**2*e*f**3*g**4 + 432432*a*c*d**2*e*f**2*g**5*x - 108108* 
a*c*d**2*e*f*g**6*x**2 + 54054*a*c*d**2*e*g**7*x**3 - 988416*a*c*d*e**2*f* 
*4*g**3 - 494208*a*c*d*e**2*f**3*g**4*x + 123552*a*c*d*e**2*f**2*g**5*x**2 
 - 61776*a*c*d*e**2*f*g**6*x**3 + 38610*a*c*d*e**2*g**7*x**4 + 366080*a*c* 
e**3*f**5*g**2 + 183040*a*c*e**3*f**4*g**3*x - 45760*a*c*e**3*f**3*g**4*x* 
*2 + 22880*a*c*e**3*f**2*g**5*x**3 - 14300*a*c*e**3*f*g**6*x**4 + 10010*a* 
c*e**3*g**7*x**5 - 164736*c**2*d**3*f**4*g**3 - 82368*c**2*d**3*f**3*g**4* 
x + 20592*c**2*d**3*f**2*g**5*x**2 - 10296*c**2*d**3*f*g**6*x**3 + 6435*c* 
*2*d**3*g**7*x**4 + 549120*c**2*d**2*e*f**5*g**2 + 274560*c**2*d**2*e*f**4 
*g**3*x - 68640*c**2*d**2*e*f**3*g**4*x**2 + 34320*c**2*d**2*e*f**2*g**5*x 
**3 - 21450*c**2*d**2*e*f*g**6*x**4 + 15015*c**2*d**2*e*g**7*x**5 - 599040 
*c**2*d*e**2*f**6*g - 299520*c**2*d*e**2*f**5*g**2*x + 74880*c**2*d*e**2*f 
**4*g**3*x**2 - 37440*c**2*d*e**2*f**3*g**4*x**3 + 23400*c**2*d*e**2*f**2* 
g**5*x**4 - 16380*c**2*d*e**2*f*g**6*x**5 + 12285*c**2*d*e**2*g**7*x**6 + 
215040*c**2*e**3*f**7 + 107520*c**2*e**3*f**6*g*x - 26880*c**2*e**3*f**...