\(\int \frac {(d+e x)^2 (a+c x^2)^2}{\sqrt {f+g x}} \, dx\) [78]

Optimal result

Integrand size = 26, antiderivative size = 338 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 (e f-d g)^2 \left (c f^2+a g^2\right )^2 \sqrt {f+g x}}{g^7}-\frac {4 (e f-d g) \left (c f^2+a g^2\right ) \left (a e g^2+c f (3 e f-2 d g)\right ) (f+g x)^{3/2}}{3 g^7}+\frac {2 \left (a^2 e^2 g^4+2 a c g^2 \left (6 e^2 f^2-6 d e f g+d^2 g^2\right )+c^2 f^2 \left (15 e^2 f^2-20 d e f g+6 d^2 g^2\right )\right ) (f+g x)^{5/2}}{5 g^7}-\frac {8 c \left (a e g^2 (2 e f-d g)+c f \left (5 e^2 f^2-5 d e f g+d^2 g^2\right )\right ) (f+g x)^{7/2}}{7 g^7}+\frac {2 c \left (2 a e^2 g^2+c \left (15 e^2 f^2-10 d e f g+d^2 g^2\right )\right ) (f+g x)^{9/2}}{9 g^7}-\frac {4 c^2 e (3 e f-d g) (f+g x)^{11/2}}{11 g^7}+\frac {2 c^2 e^2 (f+g x)^{13/2}}{13 g^7} \] Output:

2*(-d*g+e*f)^2*(a*g^2+c*f^2)^2*(g*x+f)^(1/2)/g^7-4/3*(-d*g+e*f)*(a*g^2+c*f 
^2)*(a*e*g^2+c*f*(-2*d*g+3*e*f))*(g*x+f)^(3/2)/g^7+2/5*(a^2*e^2*g^4+2*a*c* 
g^2*(d^2*g^2-6*d*e*f*g+6*e^2*f^2)+c^2*f^2*(6*d^2*g^2-20*d*e*f*g+15*e^2*f^2 
))*(g*x+f)^(5/2)/g^7-8/7*c*(a*e*g^2*(-d*g+2*e*f)+c*f*(d^2*g^2-5*d*e*f*g+5* 
e^2*f^2))*(g*x+f)^(7/2)/g^7+2/9*c*(2*a*e^2*g^2+c*(d^2*g^2-10*d*e*f*g+15*e^ 
2*f^2))*(g*x+f)^(9/2)/g^7-4/11*c^2*e*(-d*g+3*e*f)*(g*x+f)^(11/2)/g^7+2/13* 
c^2*e^2*(g*x+f)^(13/2)/g^7
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.07 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (3003 a^2 g^4 \left (15 d^2 g^2+10 d e g (-2 f+g x)+e^2 \left (8 f^2-4 f g x+3 g^2 x^2\right )\right )+286 a c g^2 \left (21 d^2 g^2 \left (8 f^2-4 f g x+3 g^2 x^2\right )+18 d e g \left (-16 f^3+8 f^2 g x-6 f g^2 x^2+5 g^3 x^3\right )+e^2 \left (128 f^4-64 f^3 g x+48 f^2 g^2 x^2-40 f g^3 x^3+35 g^4 x^4\right )\right )+c^2 \left (143 d^2 g^2 \left (128 f^4-64 f^3 g x+48 f^2 g^2 x^2-40 f g^3 x^3+35 g^4 x^4\right )+130 d e g \left (-256 f^5+128 f^4 g x-96 f^3 g^2 x^2+80 f^2 g^3 x^3-70 f g^4 x^4+63 g^5 x^5\right )+15 e^2 \left (1024 f^6-512 f^5 g x+384 f^4 g^2 x^2-320 f^3 g^3 x^3+280 f^2 g^4 x^4-252 f g^5 x^5+231 g^6 x^6\right )\right )\right )}{45045 g^7} \] Input:

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

Output:

(2*Sqrt[f + g*x]*(3003*a^2*g^4*(15*d^2*g^2 + 10*d*e*g*(-2*f + g*x) + e^2*( 
8*f^2 - 4*f*g*x + 3*g^2*x^2)) + 286*a*c*g^2*(21*d^2*g^2*(8*f^2 - 4*f*g*x + 
 3*g^2*x^2) + 18*d*e*g*(-16*f^3 + 8*f^2*g*x - 6*f*g^2*x^2 + 5*g^3*x^3) + e 
^2*(128*f^4 - 64*f^3*g*x + 48*f^2*g^2*x^2 - 40*f*g^3*x^3 + 35*g^4*x^4)) + 
c^2*(143*d^2*g^2*(128*f^4 - 64*f^3*g*x + 48*f^2*g^2*x^2 - 40*f*g^3*x^3 + 3 
5*g^4*x^4) + 130*d*e*g*(-256*f^5 + 128*f^4*g*x - 96*f^3*g^2*x^2 + 80*f^2*g 
^3*x^3 - 70*f*g^4*x^4 + 63*g^5*x^5) + 15*e^2*(1024*f^6 - 512*f^5*g*x + 384 
*f^4*g^2*x^2 - 320*f^3*g^3*x^3 + 280*f^2*g^4*x^4 - 252*f*g^5*x^5 + 231*g^6 
*x^6))))/(45045*g^7)
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 338, 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.

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

\(\Big \downarrow \) 652

\(\displaystyle \int \left (\frac {(f+g x)^{3/2} \left (a^2 e^2 g^4+2 a c g^2 \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )+c^2 f^2 \left (6 d^2 g^2-20 d e f g+15 e^2 f^2\right )\right )}{g^6}+\frac {c (f+g x)^{7/2} \left (2 a e^2 g^2+c \left (d^2 g^2-10 d e f g+15 e^2 f^2\right )\right )}{g^6}+\frac {4 c (f+g x)^{5/2} \left (-a e g^2 (2 e f-d g)-c f \left (d^2 g^2-5 d e f g+5 e^2 f^2\right )\right )}{g^6}+\frac {2 \sqrt {f+g x} \left (a g^2+c f^2\right ) (e f-d g) \left (-a e g^2-c f (3 e f-2 d g)\right )}{g^6}+\frac {\left (a g^2+c f^2\right )^2 (d g-e f)^2}{g^6 \sqrt {f+g x}}-\frac {2 c^2 e (f+g x)^{9/2} (3 e f-d g)}{g^6}+\frac {c^2 e^2 (f+g x)^{11/2}}{g^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 (f+g x)^{5/2} \left (a^2 e^2 g^4+2 a c g^2 \left (d^2 g^2-6 d e f g+6 e^2 f^2\right )+c^2 f^2 \left (6 d^2 g^2-20 d e f g+15 e^2 f^2\right )\right )}{5 g^7}+\frac {2 c (f+g x)^{9/2} \left (2 a e^2 g^2+c \left (d^2 g^2-10 d e f g+15 e^2 f^2\right )\right )}{9 g^7}-\frac {8 c (f+g x)^{7/2} \left (a e g^2 (2 e f-d g)+c f \left (d^2 g^2-5 d e f g+5 e^2 f^2\right )\right )}{7 g^7}-\frac {4 (f+g x)^{3/2} \left (a g^2+c f^2\right ) (e f-d g) \left (a e g^2+c f (3 e f-2 d g)\right )}{3 g^7}+\frac {2 \sqrt {f+g x} \left (a g^2+c f^2\right )^2 (e f-d g)^2}{g^7}-\frac {4 c^2 e (f+g x)^{11/2} (3 e f-d g)}{11 g^7}+\frac {2 c^2 e^2 (f+g x)^{13/2}}{13 g^7}\)

Input:

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

Output:

(2*(e*f - d*g)^2*(c*f^2 + a*g^2)^2*Sqrt[f + g*x])/g^7 - (4*(e*f - d*g)*(c* 
f^2 + a*g^2)*(a*e*g^2 + c*f*(3*e*f - 2*d*g))*(f + g*x)^(3/2))/(3*g^7) + (2 
*(a^2*e^2*g^4 + 2*a*c*g^2*(6*e^2*f^2 - 6*d*e*f*g + d^2*g^2) + c^2*f^2*(15* 
e^2*f^2 - 20*d*e*f*g + 6*d^2*g^2))*(f + g*x)^(5/2))/(5*g^7) - (8*c*(a*e*g^ 
2*(2*e*f - d*g) + c*f*(5*e^2*f^2 - 5*d*e*f*g + d^2*g^2))*(f + g*x)^(7/2))/ 
(7*g^7) + (2*c*(2*a*e^2*g^2 + c*(15*e^2*f^2 - 10*d*e*f*g + d^2*g^2))*(f + 
g*x)^(9/2))/(9*g^7) - (4*c^2*e*(3*e*f - d*g)*(f + g*x)^(11/2))/(11*g^7) + 
(2*c^2*e^2*(f + g*x)^(13/2))/(13*g^7)
 

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 = 1.65 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.97

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {g x +f}\, \left (\left (\left (\frac {1}{13} e^{2} x^{6}+\frac {1}{9} d^{2} x^{4}+\frac {2}{11} d e \,x^{5}\right ) c^{2}+\frac {2 x^{2} a \left (d^{2}+\frac {5}{9} e^{2} x^{2}+\frac {10}{7} d e x \right ) c}{5}+a^{2} \left (\frac {1}{5} e^{2} x^{2}+d^{2}+\frac {2}{3} d e x \right )\right ) g^{6}-\frac {4 \left (\left (\frac {9}{143} e^{2} x^{5}+\frac {5}{33} d e \,x^{4}+\frac {2}{21} d^{2} x^{3}\right ) c^{2}+\frac {2 \left (\frac {10}{21} e^{2} x^{2}+\frac {9}{7} d e x +d^{2}\right ) x a c}{5}+a^{2} e \left (\frac {e x}{5}+d \right )\right ) f \,g^{5}}{3}+\frac {8 f^{2} \left (\frac {2 \left (\frac {175}{286} e^{2} x^{2}+\frac {50}{33} d e x +d^{2}\right ) x^{2} c^{2}}{7}+2 \left (\frac {2}{7} e^{2} x^{2}+\frac {6}{7} d e x +d^{2}\right ) a c +e^{2} a^{2}\right ) g^{4}}{15}-\frac {64 f^{3} \left (\left (\frac {25}{429} e^{2} x^{3}+\frac {5}{33} d e \,x^{2}+\frac {1}{9} d^{2} x \right ) c +e a \left (\frac {2 e x}{9}+d \right )\right ) c \,g^{3}}{35}+\frac {256 f^{4} c \left (\left (\frac {45}{286} e^{2} x^{2}+\frac {5}{11} d e x +\frac {1}{2} d^{2}\right ) c +a \,e^{2}\right ) g^{2}}{315}-\frac {512 e \,f^{5} c^{2} \left (\frac {3 e x}{13}+d \right ) g}{693}+\frac {1024 c^{2} e^{2} f^{6}}{3003}\right )}{g^{7}}\)	\(328\)
derivativedivides	\(\frac {\frac {2 c^{2} e^{2} \left (g x +f \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 e \left (d g -e f \right ) c^{2}-4 f \,c^{2} e^{2}\right ) \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (d g -e f \right )^{2} c^{2}-8 e \left (d g -e f \right ) c^{2} f +e^{2} \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )\right ) \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-4 \left (d g -e f \right )^{2} c^{2} f +2 e \left (d g -e f \right ) \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-4 e^{2} \left (a \,g^{2}+c \,f^{2}\right ) c f \right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (d g -e f \right )^{2} \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-8 e \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right ) c f +e^{2} \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-4 \left (d g -e f \right )^{2} \left (a \,g^{2}+c \,f^{2}\right ) c f +2 e \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {3}{2}}}{3}+2 \left (d g -e f \right )^{2} \left (a \,g^{2}+c \,f^{2}\right )^{2} \sqrt {g x +f}}{g^{7}}\)	\(374\)
default	\(\frac {\frac {2 c^{2} e^{2} \left (g x +f \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 e \left (d g -e f \right ) c^{2}-4 f \,c^{2} e^{2}\right ) \left (g x +f \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (d g -e f \right )^{2} c^{2}-8 e \left (d g -e f \right ) c^{2} f +e^{2} \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )\right ) \left (g x +f \right )^{\frac {9}{2}}}{9}+\frac {2 \left (-4 \left (d g -e f \right )^{2} c^{2} f +2 e \left (d g -e f \right ) \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-4 e^{2} \left (a \,g^{2}+c \,f^{2}\right ) c f \right ) \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (d g -e f \right )^{2} \left (2 \left (a \,g^{2}+c \,f^{2}\right ) c +4 c^{2} f^{2}\right )-8 e \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right ) c f +e^{2} \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-4 \left (d g -e f \right )^{2} \left (a \,g^{2}+c \,f^{2}\right ) c f +2 e \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right )^{2}\right ) \left (g x +f \right )^{\frac {3}{2}}}{3}+2 \left (d g -e f \right )^{2} \left (a \,g^{2}+c \,f^{2}\right )^{2} \sqrt {g x +f}}{g^{7}}\)	\(374\)
gosper	\(\frac {2 \sqrt {g x +f}\, \left (3465 c^{2} e^{2} x^{6} g^{6}+8190 c^{2} d e \,g^{6} x^{5}-3780 c^{2} e^{2} f \,g^{5} x^{5}+10010 a c \,e^{2} g^{6} x^{4}+5005 c^{2} d^{2} g^{6} x^{4}-9100 c^{2} d e f \,g^{5} x^{4}+4200 c^{2} e^{2} f^{2} g^{4} x^{4}+25740 a c d e \,g^{6} x^{3}-11440 a c \,e^{2} f \,g^{5} x^{3}-5720 c^{2} d^{2} f \,g^{5} x^{3}+10400 c^{2} d e \,f^{2} g^{4} x^{3}-4800 c^{2} e^{2} f^{3} g^{3} x^{3}+9009 a^{2} e^{2} g^{6} x^{2}+18018 a c \,d^{2} g^{6} x^{2}-30888 a c d e f \,g^{5} x^{2}+13728 a c \,e^{2} f^{2} g^{4} x^{2}+6864 c^{2} d^{2} f^{2} g^{4} x^{2}-12480 c^{2} d e \,f^{3} g^{3} x^{2}+5760 c^{2} e^{2} f^{4} g^{2} x^{2}+30030 a^{2} d e \,g^{6} x -12012 a^{2} e^{2} f \,g^{5} x -24024 a c \,d^{2} f \,g^{5} x +41184 a c d e \,f^{2} g^{4} x -18304 a c \,e^{2} f^{3} g^{3} x -9152 c^{2} d^{2} f^{3} g^{3} x +16640 c^{2} d e \,f^{4} g^{2} x -7680 c^{2} e^{2} f^{5} g x +45045 a^{2} d^{2} g^{6}-60060 a^{2} d e f \,g^{5}+24024 a^{2} e^{2} f^{2} g^{4}+48048 a c \,d^{2} f^{2} g^{4}-82368 a c d e \,f^{3} g^{3}+36608 a c \,e^{2} f^{4} g^{2}+18304 c^{2} d^{2} f^{4} g^{2}-33280 c^{2} d e \,f^{5} g +15360 c^{2} e^{2} f^{6}\right )}{45045 g^{7}}\)	\(509\)
trager	\(\frac {2 \sqrt {g x +f}\, \left (3465 c^{2} e^{2} x^{6} g^{6}+8190 c^{2} d e \,g^{6} x^{5}-3780 c^{2} e^{2} f \,g^{5} x^{5}+10010 a c \,e^{2} g^{6} x^{4}+5005 c^{2} d^{2} g^{6} x^{4}-9100 c^{2} d e f \,g^{5} x^{4}+4200 c^{2} e^{2} f^{2} g^{4} x^{4}+25740 a c d e \,g^{6} x^{3}-11440 a c \,e^{2} f \,g^{5} x^{3}-5720 c^{2} d^{2} f \,g^{5} x^{3}+10400 c^{2} d e \,f^{2} g^{4} x^{3}-4800 c^{2} e^{2} f^{3} g^{3} x^{3}+9009 a^{2} e^{2} g^{6} x^{2}+18018 a c \,d^{2} g^{6} x^{2}-30888 a c d e f \,g^{5} x^{2}+13728 a c \,e^{2} f^{2} g^{4} x^{2}+6864 c^{2} d^{2} f^{2} g^{4} x^{2}-12480 c^{2} d e \,f^{3} g^{3} x^{2}+5760 c^{2} e^{2} f^{4} g^{2} x^{2}+30030 a^{2} d e \,g^{6} x -12012 a^{2} e^{2} f \,g^{5} x -24024 a c \,d^{2} f \,g^{5} x +41184 a c d e \,f^{2} g^{4} x -18304 a c \,e^{2} f^{3} g^{3} x -9152 c^{2} d^{2} f^{3} g^{3} x +16640 c^{2} d e \,f^{4} g^{2} x -7680 c^{2} e^{2} f^{5} g x +45045 a^{2} d^{2} g^{6}-60060 a^{2} d e f \,g^{5}+24024 a^{2} e^{2} f^{2} g^{4}+48048 a c \,d^{2} f^{2} g^{4}-82368 a c d e \,f^{3} g^{3}+36608 a c \,e^{2} f^{4} g^{2}+18304 c^{2} d^{2} f^{4} g^{2}-33280 c^{2} d e \,f^{5} g +15360 c^{2} e^{2} f^{6}\right )}{45045 g^{7}}\)	\(509\)
risch	\(\frac {2 \sqrt {g x +f}\, \left (3465 c^{2} e^{2} x^{6} g^{6}+8190 c^{2} d e \,g^{6} x^{5}-3780 c^{2} e^{2} f \,g^{5} x^{5}+10010 a c \,e^{2} g^{6} x^{4}+5005 c^{2} d^{2} g^{6} x^{4}-9100 c^{2} d e f \,g^{5} x^{4}+4200 c^{2} e^{2} f^{2} g^{4} x^{4}+25740 a c d e \,g^{6} x^{3}-11440 a c \,e^{2} f \,g^{5} x^{3}-5720 c^{2} d^{2} f \,g^{5} x^{3}+10400 c^{2} d e \,f^{2} g^{4} x^{3}-4800 c^{2} e^{2} f^{3} g^{3} x^{3}+9009 a^{2} e^{2} g^{6} x^{2}+18018 a c \,d^{2} g^{6} x^{2}-30888 a c d e f \,g^{5} x^{2}+13728 a c \,e^{2} f^{2} g^{4} x^{2}+6864 c^{2} d^{2} f^{2} g^{4} x^{2}-12480 c^{2} d e \,f^{3} g^{3} x^{2}+5760 c^{2} e^{2} f^{4} g^{2} x^{2}+30030 a^{2} d e \,g^{6} x -12012 a^{2} e^{2} f \,g^{5} x -24024 a c \,d^{2} f \,g^{5} x +41184 a c d e \,f^{2} g^{4} x -18304 a c \,e^{2} f^{3} g^{3} x -9152 c^{2} d^{2} f^{3} g^{3} x +16640 c^{2} d e \,f^{4} g^{2} x -7680 c^{2} e^{2} f^{5} g x +45045 a^{2} d^{2} g^{6}-60060 a^{2} d e f \,g^{5}+24024 a^{2} e^{2} f^{2} g^{4}+48048 a c \,d^{2} f^{2} g^{4}-82368 a c d e \,f^{3} g^{3}+36608 a c \,e^{2} f^{4} g^{2}+18304 c^{2} d^{2} f^{4} g^{2}-33280 c^{2} d e \,f^{5} g +15360 c^{2} e^{2} f^{6}\right )}{45045 g^{7}}\)	\(509\)
orering	\(\frac {2 \sqrt {g x +f}\, \left (3465 c^{2} e^{2} x^{6} g^{6}+8190 c^{2} d e \,g^{6} x^{5}-3780 c^{2} e^{2} f \,g^{5} x^{5}+10010 a c \,e^{2} g^{6} x^{4}+5005 c^{2} d^{2} g^{6} x^{4}-9100 c^{2} d e f \,g^{5} x^{4}+4200 c^{2} e^{2} f^{2} g^{4} x^{4}+25740 a c d e \,g^{6} x^{3}-11440 a c \,e^{2} f \,g^{5} x^{3}-5720 c^{2} d^{2} f \,g^{5} x^{3}+10400 c^{2} d e \,f^{2} g^{4} x^{3}-4800 c^{2} e^{2} f^{3} g^{3} x^{3}+9009 a^{2} e^{2} g^{6} x^{2}+18018 a c \,d^{2} g^{6} x^{2}-30888 a c d e f \,g^{5} x^{2}+13728 a c \,e^{2} f^{2} g^{4} x^{2}+6864 c^{2} d^{2} f^{2} g^{4} x^{2}-12480 c^{2} d e \,f^{3} g^{3} x^{2}+5760 c^{2} e^{2} f^{4} g^{2} x^{2}+30030 a^{2} d e \,g^{6} x -12012 a^{2} e^{2} f \,g^{5} x -24024 a c \,d^{2} f \,g^{5} x +41184 a c d e \,f^{2} g^{4} x -18304 a c \,e^{2} f^{3} g^{3} x -9152 c^{2} d^{2} f^{3} g^{3} x +16640 c^{2} d e \,f^{4} g^{2} x -7680 c^{2} e^{2} f^{5} g x +45045 a^{2} d^{2} g^{6}-60060 a^{2} d e f \,g^{5}+24024 a^{2} e^{2} f^{2} g^{4}+48048 a c \,d^{2} f^{2} g^{4}-82368 a c d e \,f^{3} g^{3}+36608 a c \,e^{2} f^{4} g^{2}+18304 c^{2} d^{2} f^{4} g^{2}-33280 c^{2} d e \,f^{5} g +15360 c^{2} e^{2} f^{6}\right )}{45045 g^{7}}\)	\(509\)

Input:

int((e*x+d)^2*(c*x^2+a)^2/(g*x+f)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2*(g*x+f)^(1/2)*(((1/13*e^2*x^6+1/9*d^2*x^4+2/11*d*e*x^5)*c^2+2/5*x^2*a*(d 
^2+5/9*e^2*x^2+10/7*d*e*x)*c+a^2*(1/5*e^2*x^2+d^2+2/3*d*e*x))*g^6-4/3*((9/ 
143*e^2*x^5+5/33*d*e*x^4+2/21*d^2*x^3)*c^2+2/5*(10/21*e^2*x^2+9/7*d*e*x+d^ 
2)*x*a*c+a^2*e*(1/5*e*x+d))*f*g^5+8/15*f^2*(2/7*(175/286*e^2*x^2+50/33*d*e 
*x+d^2)*x^2*c^2+2*(2/7*e^2*x^2+6/7*d*e*x+d^2)*a*c+e^2*a^2)*g^4-64/35*f^3*( 
(25/429*e^2*x^3+5/33*d*e*x^2+1/9*d^2*x)*c+e*a*(2/9*e*x+d))*c*g^3+256/315*f 
^4*c*((45/286*e^2*x^2+5/11*d*e*x+1/2*d^2)*c+a*e^2)*g^2-512/693*e*f^5*c^2*( 
3/13*e*x+d)*g+1024/3003*c^2*e^2*f^6)/g^7
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.34 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (3465 \, c^{2} e^{2} g^{6} x^{6} + 15360 \, c^{2} e^{2} f^{6} - 33280 \, c^{2} d e f^{5} g - 82368 \, a c d e f^{3} g^{3} - 60060 \, a^{2} d e f g^{5} + 45045 \, a^{2} d^{2} g^{6} + 18304 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f^{4} g^{2} + 24024 \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} f^{2} g^{4} - 630 \, {\left (6 \, c^{2} e^{2} f g^{5} - 13 \, c^{2} d e g^{6}\right )} x^{5} + 35 \, {\left (120 \, c^{2} e^{2} f^{2} g^{4} - 260 \, c^{2} d e f g^{5} + 143 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} g^{6}\right )} x^{4} - 20 \, {\left (240 \, c^{2} e^{2} f^{3} g^{3} - 520 \, c^{2} d e f^{2} g^{4} - 1287 \, a c d e g^{6} + 286 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f g^{5}\right )} x^{3} + 3 \, {\left (1920 \, c^{2} e^{2} f^{4} g^{2} - 4160 \, c^{2} d e f^{3} g^{3} - 10296 \, a c d e f g^{5} + 2288 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f^{2} g^{4} + 3003 \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} g^{6}\right )} x^{2} - 2 \, {\left (3840 \, c^{2} e^{2} f^{5} g - 8320 \, c^{2} d e f^{4} g^{2} - 20592 \, a c d e f^{2} g^{4} - 15015 \, a^{2} d e g^{6} + 4576 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f^{3} g^{3} + 6006 \, {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} f g^{5}\right )} x\right )} \sqrt {g x + f}}{45045 \, g^{7}} \] Input:

integrate((e*x+d)^2*(c*x^2+a)^2/(g*x+f)^(1/2),x, algorithm="fricas")
 

Output:

2/45045*(3465*c^2*e^2*g^6*x^6 + 15360*c^2*e^2*f^6 - 33280*c^2*d*e*f^5*g - 
82368*a*c*d*e*f^3*g^3 - 60060*a^2*d*e*f*g^5 + 45045*a^2*d^2*g^6 + 18304*(c 
^2*d^2 + 2*a*c*e^2)*f^4*g^2 + 24024*(2*a*c*d^2 + a^2*e^2)*f^2*g^4 - 630*(6 
*c^2*e^2*f*g^5 - 13*c^2*d*e*g^6)*x^5 + 35*(120*c^2*e^2*f^2*g^4 - 260*c^2*d 
*e*f*g^5 + 143*(c^2*d^2 + 2*a*c*e^2)*g^6)*x^4 - 20*(240*c^2*e^2*f^3*g^3 - 
520*c^2*d*e*f^2*g^4 - 1287*a*c*d*e*g^6 + 286*(c^2*d^2 + 2*a*c*e^2)*f*g^5)* 
x^3 + 3*(1920*c^2*e^2*f^4*g^2 - 4160*c^2*d*e*f^3*g^3 - 10296*a*c*d*e*f*g^5 
 + 2288*(c^2*d^2 + 2*a*c*e^2)*f^2*g^4 + 3003*(2*a*c*d^2 + a^2*e^2)*g^6)*x^ 
2 - 2*(3840*c^2*e^2*f^5*g - 8320*c^2*d*e*f^4*g^2 - 20592*a*c*d*e*f^2*g^4 - 
 15015*a^2*d*e*g^6 + 4576*(c^2*d^2 + 2*a*c*e^2)*f^3*g^3 + 6006*(2*a*c*d^2 
+ a^2*e^2)*f*g^5)*x)*sqrt(g*x + f)/g^7
 

Sympy [A] (verification not implemented)

Time = 1.16 (sec) , antiderivative size = 665, normalized size of antiderivative = 1.97 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} e^{2} \left (f + g x\right )^{\frac {13}{2}}}{13 g^{6}} + \frac {\left (f + g x\right )^{\frac {11}{2}} \cdot \left (2 c^{2} d e g - 6 c^{2} e^{2} f\right )}{11 g^{6}} + \frac {\left (f + g x\right )^{\frac {9}{2}} \cdot \left (2 a c e^{2} g^{2} + c^{2} d^{2} g^{2} - 10 c^{2} d e f g + 15 c^{2} e^{2} f^{2}\right )}{9 g^{6}} + \frac {\left (f + g x\right )^{\frac {7}{2}} \cdot \left (4 a c d e g^{3} - 8 a c e^{2} f g^{2} - 4 c^{2} d^{2} f g^{2} + 20 c^{2} d e f^{2} g - 20 c^{2} e^{2} f^{3}\right )}{7 g^{6}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \left (a^{2} e^{2} g^{4} + 2 a c d^{2} g^{4} - 12 a c d e f g^{3} + 12 a c e^{2} f^{2} g^{2} + 6 c^{2} d^{2} f^{2} g^{2} - 20 c^{2} d e f^{3} g + 15 c^{2} e^{2} f^{4}\right )}{5 g^{6}} + \frac {\left (f + g x\right )^{\frac {3}{2}} \cdot \left (2 a^{2} d e g^{5} - 2 a^{2} e^{2} f g^{4} - 4 a c d^{2} f g^{4} + 12 a c d e f^{2} g^{3} - 8 a c e^{2} f^{3} g^{2} - 4 c^{2} d^{2} f^{3} g^{2} + 10 c^{2} d e f^{4} g - 6 c^{2} e^{2} f^{5}\right )}{3 g^{6}} + \frac {\sqrt {f + g x} \left (a^{2} d^{2} g^{6} - 2 a^{2} d e f g^{5} + a^{2} e^{2} f^{2} g^{4} + 2 a c d^{2} f^{2} g^{4} - 4 a c d e f^{3} g^{3} + 2 a c e^{2} f^{4} g^{2} + c^{2} d^{2} f^{4} g^{2} - 2 c^{2} d e f^{5} g + c^{2} e^{2} f^{6}\right )}{g^{6}}\right )}{g} & \text {for}\: g \neq 0 \\\frac {a^{2} d^{2} x + a^{2} d e x^{2} + a c d e x^{4} + \frac {c^{2} d e x^{6}}{3} + \frac {c^{2} e^{2} x^{7}}{7} + \frac {x^{5} \cdot \left (2 a c e^{2} + c^{2} d^{2}\right )}{5} + \frac {x^{3} \left (a^{2} e^{2} + 2 a c d^{2}\right )}{3}}{\sqrt {f}} & \text {otherwise} \end {cases} \] Input:

integrate((e*x+d)**2*(c*x**2+a)**2/(g*x+f)**(1/2),x)
 

Output:

Piecewise((2*(c**2*e**2*(f + g*x)**(13/2)/(13*g**6) + (f + g*x)**(11/2)*(2 
*c**2*d*e*g - 6*c**2*e**2*f)/(11*g**6) + (f + g*x)**(9/2)*(2*a*c*e**2*g**2 
 + c**2*d**2*g**2 - 10*c**2*d*e*f*g + 15*c**2*e**2*f**2)/(9*g**6) + (f + g 
*x)**(7/2)*(4*a*c*d*e*g**3 - 8*a*c*e**2*f*g**2 - 4*c**2*d**2*f*g**2 + 20*c 
**2*d*e*f**2*g - 20*c**2*e**2*f**3)/(7*g**6) + (f + g*x)**(5/2)*(a**2*e**2 
*g**4 + 2*a*c*d**2*g**4 - 12*a*c*d*e*f*g**3 + 12*a*c*e**2*f**2*g**2 + 6*c* 
*2*d**2*f**2*g**2 - 20*c**2*d*e*f**3*g + 15*c**2*e**2*f**4)/(5*g**6) + (f 
+ g*x)**(3/2)*(2*a**2*d*e*g**5 - 2*a**2*e**2*f*g**4 - 4*a*c*d**2*f*g**4 + 
12*a*c*d*e*f**2*g**3 - 8*a*c*e**2*f**3*g**2 - 4*c**2*d**2*f**3*g**2 + 10*c 
**2*d*e*f**4*g - 6*c**2*e**2*f**5)/(3*g**6) + sqrt(f + g*x)*(a**2*d**2*g** 
6 - 2*a**2*d*e*f*g**5 + a**2*e**2*f**2*g**4 + 2*a*c*d**2*f**2*g**4 - 4*a*c 
*d*e*f**3*g**3 + 2*a*c*e**2*f**4*g**2 + c**2*d**2*f**4*g**2 - 2*c**2*d*e*f 
**5*g + c**2*e**2*f**6)/g**6)/g, Ne(g, 0)), ((a**2*d**2*x + a**2*d*e*x**2 
+ a*c*d*e*x**4 + c**2*d*e*x**6/3 + c**2*e**2*x**7/7 + x**5*(2*a*c*e**2 + c 
**2*d**2)/5 + x**3*(a**2*e**2 + 2*a*c*d**2)/3)/sqrt(f), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.32 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (3465 \, {\left (g x + f\right )}^{\frac {13}{2}} c^{2} e^{2} - 8190 \, {\left (3 \, c^{2} e^{2} f - c^{2} d e g\right )} {\left (g x + f\right )}^{\frac {11}{2}} + 5005 \, {\left (15 \, c^{2} e^{2} f^{2} - 10 \, c^{2} d e f g + {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} g^{2}\right )} {\left (g x + f\right )}^{\frac {9}{2}} - 25740 \, {\left (5 \, c^{2} e^{2} f^{3} - 5 \, c^{2} d e f^{2} g - a c d e g^{3} + {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f g^{2}\right )} {\left (g x + f\right )}^{\frac {7}{2}} + 9009 \, {\left (15 \, c^{2} e^{2} f^{4} - 20 \, c^{2} d e f^{3} g - 12 \, a c d e f g^{3} + 6 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f^{2} g^{2} + {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} g^{4}\right )} {\left (g x + f\right )}^{\frac {5}{2}} - 30030 \, {\left (3 \, c^{2} e^{2} f^{5} - 5 \, c^{2} d e f^{4} g - 6 \, a c d e f^{2} g^{3} - a^{2} d e g^{5} + 2 \, {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f^{3} g^{2} + {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} f g^{4}\right )} {\left (g x + f\right )}^{\frac {3}{2}} + 45045 \, {\left (c^{2} e^{2} f^{6} - 2 \, c^{2} d e f^{5} g - 4 \, a c d e f^{3} g^{3} - 2 \, a^{2} d e f g^{5} + a^{2} d^{2} g^{6} + {\left (c^{2} d^{2} + 2 \, a c e^{2}\right )} f^{4} g^{2} + {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} f^{2} g^{4}\right )} \sqrt {g x + f}\right )}}{45045 \, g^{7}} \] Input:

integrate((e*x+d)^2*(c*x^2+a)^2/(g*x+f)^(1/2),x, algorithm="maxima")
 

Output:

2/45045*(3465*(g*x + f)^(13/2)*c^2*e^2 - 8190*(3*c^2*e^2*f - c^2*d*e*g)*(g 
*x + f)^(11/2) + 5005*(15*c^2*e^2*f^2 - 10*c^2*d*e*f*g + (c^2*d^2 + 2*a*c* 
e^2)*g^2)*(g*x + f)^(9/2) - 25740*(5*c^2*e^2*f^3 - 5*c^2*d*e*f^2*g - a*c*d 
*e*g^3 + (c^2*d^2 + 2*a*c*e^2)*f*g^2)*(g*x + f)^(7/2) + 9009*(15*c^2*e^2*f 
^4 - 20*c^2*d*e*f^3*g - 12*a*c*d*e*f*g^3 + 6*(c^2*d^2 + 2*a*c*e^2)*f^2*g^2 
 + (2*a*c*d^2 + a^2*e^2)*g^4)*(g*x + f)^(5/2) - 30030*(3*c^2*e^2*f^5 - 5*c 
^2*d*e*f^4*g - 6*a*c*d*e*f^2*g^3 - a^2*d*e*g^5 + 2*(c^2*d^2 + 2*a*c*e^2)*f 
^3*g^2 + (2*a*c*d^2 + a^2*e^2)*f*g^4)*(g*x + f)^(3/2) + 45045*(c^2*e^2*f^6 
 - 2*c^2*d*e*f^5*g - 4*a*c*d*e*f^3*g^3 - 2*a^2*d*e*f*g^5 + a^2*d^2*g^6 + ( 
c^2*d^2 + 2*a*c*e^2)*f^4*g^2 + (2*a*c*d^2 + a^2*e^2)*f^2*g^4)*sqrt(g*x + f 
))/g^7
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.45 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (45045 \, \sqrt {g x + f} a^{2} d^{2} + \frac {30030 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {g x + f} f\right )} a^{2} d e}{g} + \frac {6006 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} a c d^{2}}{g^{2}} + \frac {3003 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} a^{2} e^{2}}{g^{2}} + \frac {5148 \, {\left (5 \, {\left (g x + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x + f} f^{3}\right )} a c d e}{g^{3}} + \frac {143 \, {\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} - 180 \, {\left (g x + f\right )}^{\frac {7}{2}} f + 378 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{2} - 420 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{3} + 315 \, \sqrt {g x + f} f^{4}\right )} c^{2} d^{2}}{g^{4}} + \frac {286 \, {\left (35 \, {\left (g x + f\right )}^{\frac {9}{2}} - 180 \, {\left (g x + f\right )}^{\frac {7}{2}} f + 378 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{2} - 420 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{3} + 315 \, \sqrt {g x + f} f^{4}\right )} a c e^{2}}{g^{4}} + \frac {130 \, {\left (63 \, {\left (g x + f\right )}^{\frac {11}{2}} - 385 \, {\left (g x + f\right )}^{\frac {9}{2}} f + 990 \, {\left (g x + f\right )}^{\frac {7}{2}} f^{2} - 1386 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{3} + 1155 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{4} - 693 \, \sqrt {g x + f} f^{5}\right )} c^{2} d e}{g^{5}} + \frac {15 \, {\left (231 \, {\left (g x + f\right )}^{\frac {13}{2}} - 1638 \, {\left (g x + f\right )}^{\frac {11}{2}} f + 5005 \, {\left (g x + f\right )}^{\frac {9}{2}} f^{2} - 8580 \, {\left (g x + f\right )}^{\frac {7}{2}} f^{3} + 9009 \, {\left (g x + f\right )}^{\frac {5}{2}} f^{4} - 6006 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{5} + 3003 \, \sqrt {g x + f} f^{6}\right )} c^{2} e^{2}}{g^{6}}\right )}}{45045 \, g} \] Input:

integrate((e*x+d)^2*(c*x^2+a)^2/(g*x+f)^(1/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

2/45045*(45045*sqrt(g*x + f)*a^2*d^2 + 30030*((g*x + f)^(3/2) - 3*sqrt(g*x 
 + f)*f)*a^2*d*e/g + 6006*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/2)*f + 15*s 
qrt(g*x + f)*f^2)*a*c*d^2/g^2 + 3003*(3*(g*x + f)^(5/2) - 10*(g*x + f)^(3/ 
2)*f + 15*sqrt(g*x + f)*f^2)*a^2*e^2/g^2 + 5148*(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*e/g 
^3 + 143*(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^2/g^4 + 286* 
(35*(g*x + f)^(9/2) - 180*(g*x + f)^(7/2)*f + 378*(g*x + f)^(5/2)*f^2 - 42 
0*(g*x + f)^(3/2)*f^3 + 315*sqrt(g*x + f)*f^4)*a*c*e^2/g^4 + 130*(63*(g*x 
+ f)^(11/2) - 385*(g*x + f)^(9/2)*f + 990*(g*x + f)^(7/2)*f^2 - 1386*(g*x 
+ f)^(5/2)*f^3 + 1155*(g*x + f)^(3/2)*f^4 - 693*sqrt(g*x + f)*f^5)*c^2*d*e 
/g^5 + 15*(231*(g*x + f)^(13/2) - 1638*(g*x + f)^(11/2)*f + 5005*(g*x + f) 
^(9/2)*f^2 - 8580*(g*x + f)^(7/2)*f^3 + 9009*(g*x + f)^(5/2)*f^4 - 6006*(g 
*x + f)^(3/2)*f^5 + 3003*sqrt(g*x + f)*f^6)*c^2*e^2/g^6)/g
 

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {{\left (f+g\,x\right )}^{5/2}\,\left (2\,a^2\,e^2\,g^4+4\,a\,c\,d^2\,g^4-24\,a\,c\,d\,e\,f\,g^3+24\,a\,c\,e^2\,f^2\,g^2+12\,c^2\,d^2\,f^2\,g^2-40\,c^2\,d\,e\,f^3\,g+30\,c^2\,e^2\,f^4\right )}{5\,g^7}-\frac {{\left (f+g\,x\right )}^{7/2}\,\left (8\,c^2\,d^2\,f\,g^2-40\,c^2\,d\,e\,f^2\,g+40\,c^2\,e^2\,f^3-8\,a\,c\,d\,e\,g^3+16\,a\,c\,e^2\,f\,g^2\right )}{7\,g^7}+\frac {{\left (f+g\,x\right )}^{9/2}\,\left (2\,c^2\,d^2\,g^2-20\,c^2\,d\,e\,f\,g+30\,c^2\,e^2\,f^2+4\,a\,c\,e^2\,g^2\right )}{9\,g^7}+\frac {2\,c^2\,e^2\,{\left (f+g\,x\right )}^{13/2}}{13\,g^7}+\frac {2\,\sqrt {f+g\,x}\,{\left (c\,f^2+a\,g^2\right )}^2\,{\left (d\,g-e\,f\right )}^2}{g^7}+\frac {4\,c^2\,e\,{\left (f+g\,x\right )}^{11/2}\,\left (d\,g-3\,e\,f\right )}{11\,g^7}+\frac {4\,{\left (f+g\,x\right )}^{3/2}\,\left (c\,f^2+a\,g^2\right )\,\left (d\,g-e\,f\right )\,\left (3\,c\,e\,f^2-2\,c\,d\,f\,g+a\,e\,g^2\right )}{3\,g^7} \] Input:

int(((a + c*x^2)^2*(d + e*x)^2)/(f + g*x)^(1/2),x)
 

Output:

((f + g*x)^(5/2)*(2*a^2*e^2*g^4 + 30*c^2*e^2*f^4 + 12*c^2*d^2*f^2*g^2 + 4* 
a*c*d^2*g^4 - 40*c^2*d*e*f^3*g + 24*a*c*e^2*f^2*g^2 - 24*a*c*d*e*f*g^3))/( 
5*g^7) - ((f + g*x)^(7/2)*(40*c^2*e^2*f^3 + 8*c^2*d^2*f*g^2 + 16*a*c*e^2*f 
*g^2 - 40*c^2*d*e*f^2*g - 8*a*c*d*e*g^3))/(7*g^7) + ((f + g*x)^(9/2)*(2*c^ 
2*d^2*g^2 + 30*c^2*e^2*f^2 + 4*a*c*e^2*g^2 - 20*c^2*d*e*f*g))/(9*g^7) + (2 
*c^2*e^2*(f + g*x)^(13/2))/(13*g^7) + (2*(f + g*x)^(1/2)*(a*g^2 + c*f^2)^2 
*(d*g - e*f)^2)/g^7 + (4*c^2*e*(f + g*x)^(11/2)*(d*g - 3*e*f))/(11*g^7) + 
(4*(f + g*x)^(3/2)*(a*g^2 + c*f^2)*(d*g - e*f)*(a*e*g^2 + 3*c*e*f^2 - 2*c* 
d*f*g))/(3*g^7)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.50 \[ \int \frac {(d+e x)^2 \left (a+c x^2\right )^2}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {g x +f}\, \left (3465 c^{2} e^{2} g^{6} x^{6}+8190 c^{2} d e \,g^{6} x^{5}-3780 c^{2} e^{2} f \,g^{5} x^{5}+10010 a c \,e^{2} g^{6} x^{4}+5005 c^{2} d^{2} g^{6} x^{4}-9100 c^{2} d e f \,g^{5} x^{4}+4200 c^{2} e^{2} f^{2} g^{4} x^{4}+25740 a c d e \,g^{6} x^{3}-11440 a c \,e^{2} f \,g^{5} x^{3}-5720 c^{2} d^{2} f \,g^{5} x^{3}+10400 c^{2} d e \,f^{2} g^{4} x^{3}-4800 c^{2} e^{2} f^{3} g^{3} x^{3}+9009 a^{2} e^{2} g^{6} x^{2}+18018 a c \,d^{2} g^{6} x^{2}-30888 a c d e f \,g^{5} x^{2}+13728 a c \,e^{2} f^{2} g^{4} x^{2}+6864 c^{2} d^{2} f^{2} g^{4} x^{2}-12480 c^{2} d e \,f^{3} g^{3} x^{2}+5760 c^{2} e^{2} f^{4} g^{2} x^{2}+30030 a^{2} d e \,g^{6} x -12012 a^{2} e^{2} f \,g^{5} x -24024 a c \,d^{2} f \,g^{5} x +41184 a c d e \,f^{2} g^{4} x -18304 a c \,e^{2} f^{3} g^{3} x -9152 c^{2} d^{2} f^{3} g^{3} x +16640 c^{2} d e \,f^{4} g^{2} x -7680 c^{2} e^{2} f^{5} g x +45045 a^{2} d^{2} g^{6}-60060 a^{2} d e f \,g^{5}+24024 a^{2} e^{2} f^{2} g^{4}+48048 a c \,d^{2} f^{2} g^{4}-82368 a c d e \,f^{3} g^{3}+36608 a c \,e^{2} f^{4} g^{2}+18304 c^{2} d^{2} f^{4} g^{2}-33280 c^{2} d e \,f^{5} g +15360 c^{2} e^{2} f^{6}\right )}{45045 g^{7}} \] Input:

int((e*x+d)^2*(c*x^2+a)^2/(g*x+f)^(1/2),x)
 

Output:

(2*sqrt(f + g*x)*(45045*a**2*d**2*g**6 - 60060*a**2*d*e*f*g**5 + 30030*a** 
2*d*e*g**6*x + 24024*a**2*e**2*f**2*g**4 - 12012*a**2*e**2*f*g**5*x + 9009 
*a**2*e**2*g**6*x**2 + 48048*a*c*d**2*f**2*g**4 - 24024*a*c*d**2*f*g**5*x 
+ 18018*a*c*d**2*g**6*x**2 - 82368*a*c*d*e*f**3*g**3 + 41184*a*c*d*e*f**2* 
g**4*x - 30888*a*c*d*e*f*g**5*x**2 + 25740*a*c*d*e*g**6*x**3 + 36608*a*c*e 
**2*f**4*g**2 - 18304*a*c*e**2*f**3*g**3*x + 13728*a*c*e**2*f**2*g**4*x**2 
 - 11440*a*c*e**2*f*g**5*x**3 + 10010*a*c*e**2*g**6*x**4 + 18304*c**2*d**2 
*f**4*g**2 - 9152*c**2*d**2*f**3*g**3*x + 6864*c**2*d**2*f**2*g**4*x**2 - 
5720*c**2*d**2*f*g**5*x**3 + 5005*c**2*d**2*g**6*x**4 - 33280*c**2*d*e*f** 
5*g + 16640*c**2*d*e*f**4*g**2*x - 12480*c**2*d*e*f**3*g**3*x**2 + 10400*c 
**2*d*e*f**2*g**4*x**3 - 9100*c**2*d*e*f*g**5*x**4 + 8190*c**2*d*e*g**6*x* 
*5 + 15360*c**2*e**2*f**6 - 7680*c**2*e**2*f**5*g*x + 5760*c**2*e**2*f**4* 
g**2*x**2 - 4800*c**2*e**2*f**3*g**3*x**3 + 4200*c**2*e**2*f**2*g**4*x**4 
- 3780*c**2*e**2*f*g**5*x**5 + 3465*c**2*e**2*g**6*x**6))/(45045*g**7)