\(\int \frac {(f+g x) (a+b x+c x^2)^{7/2}}{(d+e x)^7} \, dx\) [908]

Optimal result

Integrand size = 27, antiderivative size = 1117 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^7} \, dx=-\frac {7 c \left (5 b^2 e^2 g-8 c^2 d (e f-4 d g)+4 c e (b e f-7 b d g+a e g)\right ) \sqrt {a+b x+c x^2}}{8 e^8 (d+e x)}+\frac {7 \left (128 c^4 d^4 (e f-4 d g)-b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (44 a^2 e^2 g-b^2 d (2 e f-53 d g)+2 a b e (2 e f-49 d g)\right )+8 c^2 e^2 \left (b^2 d^2 (17 e f-114 d g)-2 a b d e (14 e f-83 d g)+2 a^2 e^2 (5 e f-27 d g)\right )+32 c^3 d^2 e (a e (7 e f-30 d g)-2 b d (4 e f-19 d g))\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{512 e^7 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {7 \left (5 b^2 e^2 g-8 c^2 d (e f-4 d g)+4 c e (b e f-7 b d g+a e g)\right ) \left (a+b x+c x^2\right )^{3/2}}{24 e^6 (d+e x)^3}+\frac {7 \left (24 c^2 d^2 (e f-4 d g)+b e^2 (b e f-37 b d g+36 a e g)+4 c e (a e (5 e f-23 d g)-3 b d (2 e f-11 d g))\right ) (b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{192 e^5 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}+\frac {7 (12 c d (e f-4 d g)-e (b e f-22 b d g+6 a e g)+5 e (2 c e f-8 c d g+3 b e g) x) \left (a+b x+c x^2\right )^{5/2}}{60 e^4 (d+e x)^5}-\frac {(e f-4 d g-3 e g x) \left (a+b x+c x^2\right )^{7/2}}{6 e^2 (d+e x)^6}+\frac {7 c^{3/2} \left (5 b^2 e^2 g-8 c^2 d (e f-4 d g)+4 c e (b e f-7 b d g+a e g)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 e^9}+\frac {7 \left (1024 c^6 d^6 (e f-4 d g)+b^5 e^6 (b e f+3 b d g-4 a e g)+40 b c^2 e^4 \left (24 a^3 e^3 g-4 a b^2 d e (2 e f-33 d g)+3 b^3 d^2 (e f-18 d g)+6 a^2 b e^2 (e f-17 d g)\right )+512 c^5 d^4 e (a e (5 e f-21 d g)-3 b d (2 e f-9 d g))+320 c^3 e^3 (b d-a e) \left (a b d e (5 e f-32 d g)-2 b^2 d^2 (2 e f-15 d g)-a^2 e^2 (e f-7 d g)\right )+640 c^4 d^2 e^2 \left (b^2 d^2 (5 e f-27 d g)+a^2 e^2 (3 e f-14 d g)-8 a b d e (e f-5 d g)\right )+4 b^3 c e^5 \left (40 a^2 e^2 g-5 a b e (e f+13 d g)+b^2 d (2 e f+27 d g)\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{1024 e^9 \left (c d^2-b d e+a e^2\right )^{5/2}} \] Output:

-7/8*c*(5*b^2*e^2*g-8*c^2*d*(-4*d*g+e*f)+4*c*e*(a*e*g-7*b*d*g+b*e*f))*(c*x 
^2+b*x+a)^(1/2)/e^8/(e*x+d)+7/512*(128*c^4*d^4*(-4*d*g+e*f)-b^3*e^4*(-4*a* 
e*g+3*b*d*g+b*e*f)+4*b*c*e^3*(44*a^2*e^2*g-b^2*d*(-53*d*g+2*e*f)+2*a*b*e*( 
-49*d*g+2*e*f))+8*c^2*e^2*(b^2*d^2*(-114*d*g+17*e*f)-2*a*b*d*e*(-83*d*g+14 
*e*f)+2*a^2*e^2*(-27*d*g+5*e*f))+32*c^3*d^2*e*(a*e*(-30*d*g+7*e*f)-2*b*d*( 
-19*d*g+4*e*f)))*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/e^7/(a*e^2 
-b*d*e+c*d^2)^2/(e*x+d)^2-7/24*(5*b^2*e^2*g-8*c^2*d*(-4*d*g+e*f)+4*c*e*(a* 
e*g-7*b*d*g+b*e*f))*(c*x^2+b*x+a)^(3/2)/e^6/(e*x+d)^3+7/192*(24*c^2*d^2*(- 
4*d*g+e*f)+b*e^2*(36*a*e*g-37*b*d*g+b*e*f)+4*c*e*(a*e*(-23*d*g+5*e*f)-3*b* 
d*(-11*d*g+2*e*f)))*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)/e^5/(a* 
e^2-b*d*e+c*d^2)/(e*x+d)^4+7/60*(12*c*d*(-4*d*g+e*f)-e*(6*a*e*g-22*b*d*g+b 
*e*f)+5*e*(3*b*e*g-8*c*d*g+2*c*e*f)*x)*(c*x^2+b*x+a)^(5/2)/e^4/(e*x+d)^5-1 
/6*(-3*e*g*x-4*d*g+e*f)*(c*x^2+b*x+a)^(7/2)/e^2/(e*x+d)^6+7/8*c^(3/2)*(5*b 
^2*e^2*g-8*c^2*d*(-4*d*g+e*f)+4*c*e*(a*e*g-7*b*d*g+b*e*f))*arctanh(1/2*(2* 
c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/e^9+7/1024*(1024*c^6*d^6*(-4*d*g+e*f)+ 
b^5*e^6*(-4*a*e*g+3*b*d*g+b*e*f)+40*b*c^2*e^4*(24*a^3*e^3*g-4*a*b^2*d*e*(- 
33*d*g+2*e*f)+3*b^3*d^2*(-18*d*g+e*f)+6*a^2*b*e^2*(-17*d*g+e*f))+512*c^5*d 
^4*e*(a*e*(-21*d*g+5*e*f)-3*b*d*(-9*d*g+2*e*f))+320*c^3*e^3*(-a*e+b*d)*(a* 
b*d*e*(-32*d*g+5*e*f)-2*b^2*d^2*(-15*d*g+2*e*f)-a^2*e^2*(-7*d*g+e*f))+640* 
c^4*d^2*e^2*(b^2*d^2*(-27*d*g+5*e*f)+a^2*e^2*(-14*d*g+3*e*f)-8*a*b*d*e*...
 

Mathematica [A] (verified)

Time = 22.94 (sec) , antiderivative size = 1673, normalized size of antiderivative = 1.50 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^7} \, dx =\text {Too large to display} \] Input:

Integrate[((f + g*x)*(a + b*x + c*x^2)^(7/2))/(d + e*x)^7,x]
 

Output:

(Sqrt[a + x*(b + c*x)]*(1920*c^2*(13*b*e*g + 4*c*(e*f - 7*d*g)) + 3840*c^3 
*e*g*x + (1280*(c*d^2 + e*(-(b*d) + a*e))^3*(-(e*f) + d*g))/(d + e*x)^6 - 
(128*(c*d^2 + e*(-(b*d) + a*e))^2*(2*c*d*(-37*e*f + 43*d*g) + e*(37*b*e*f 
- 49*b*d*g + 12*a*e*g)))/(d + e*x)^5 + (16*(c*d^2 + e*(-(b*d) + a*e))*(8*c 
^2*d^2*(-241*e*f + 334*d*g) - 3*b*e^2*(129*b*e*f - 253*b*d*g + 124*a*e*g) 
- 4*c*e*(a*e*(95*e*f - 281*d*g) + b*d*(-482*e*f + 761*d*g))))/(d + e*x)^4 
- (8*(48*c^3*d^3*(-153*e*f + 262*d*g) + b^2*e^3*(377*b*e*f - 1429*b*d*g + 
1052*a*e*g) - 8*c^2*d*e*(a*e*(541*e*f - 1323*d*g) + 3*b*d*(-459*e*f + 895* 
d*g)) + 2*c*e^2*(512*a^2*e^2*g + 2*a*b*e*(541*e*f - 2105*d*g) + b^2*d*(-22 
13*e*f + 5355*d*g))))/(d + e*x)^3 - (2*(-192*c^4*d^4*(-197*e*f + 438*d*g) 
+ 5*b^3*e^4*(7*b*e*f - 491*b*d*g + 484*a*e*g) + 32*c^3*d^2*e*(a*e*(1382*e* 
f - 3669*d*g) + 3*b*d*(-788*e*f + 1993*d*g)) + 4*b*c*e^3*(3364*a^2*e^2*g + 
 2*a*b*e*(947*e*f - 6126*d*g) + b^2*d*(-1964*e*f + 9563*d*g)) - 8*c^2*e^2* 
(2*a*b*d*e*(2764*e*f - 9625*d*g) + 58*a^2*e^2*(-15*e*f + 73*d*g) + b^2*d^2 
*(-5710*e*f + 17883*d*g))))/((c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + (-3 
84*c^5*d^5*(-223*e*f + 682*d*g) + 105*b^4*e^5*(b*e*f + 3*b*d*g - 4*a*e*g) 
- 10*b^2*c*e^4*(3496*a^2*e^2*g + 4*a*b*e*(49*e*f - 1881*d*g) + 13*b^2*d*(- 
7*e*f + 305*d*g)) + 64*c^4*d^3*e*(a*e*(2536*e*f - 8403*d*g) + 3*b*d*(-1115 
*e*f + 3869*d*g)) + 16*c^2*e^3*(-1856*a^3*e^3*g + a*b^2*d*e*(5317*e*f - 33 
365*d*g) + 2*b^3*d^2*(-1443*e*f + 9346*d*g) + a^2*b*e^2*(-2291*e*f + 16...
 

Rubi [A] (verified)

Time = 6.27 (sec) , antiderivative size = 1695, normalized size of antiderivative = 1.52, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {1230, 27, 1229, 27, 1229, 27, 1230, 25, 1269, 1092, 219, 1154, 219}

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.

\(\displaystyle \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^7} \, dx\)

\(\Big \downarrow \) 1230

\(\displaystyle -\frac {7 \int -\frac {2 (b e f-4 b d g+6 a e g+(2 c e f-8 c d g+3 b e g) x) \left (c x^2+b x+a\right )^{5/2}}{(d+e x)^6}dx}{24 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-4 d g+e f-3 e g x)}{6 e^2 (d+e x)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 \int \frac {(b e f-4 b d g+6 a e g+(2 c e f-8 c d g+3 b e g) x) \left (c x^2+b x+a\right )^{5/2}}{(d+e x)^6}dx}{12 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-4 d g+e f-3 e g x)}{6 e^2 (d+e x)^6}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {7 \left (-\frac {\int \frac {\left (3 e^2 (e f+3 d g) b^3-4 e \left (3 a e^2 g-c d (4 e f-25 d g)\right ) b^2-12 c \left (2 c (e f-4 d g) d^2+a e^2 (3 e f-17 d g)\right ) b-32 a c e \left (3 a e^2 g-c d (e f-4 d g)\right )-2 c \left (24 c^2 (e f-4 d g) d^2+b e^2 (b e f-37 b d g+36 a e g)+4 c e (a e (5 e f-23 d g)-3 b d (2 e f-11 d g))\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{2 (d+e x)^4}dx}{8 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (e^2 \left (48 a^2 e^2 g+4 a b e (2 e f-11 d g)-3 b^2 d (3 d g+e f)\right )+5 e x \left (-4 c e (b d (2 e f-11 d g)-a e (e f-7 d g))+b e^2 (12 a e g-13 b d g+b e f)+8 c^2 d^2 (e f-4 d g)\right )-4 c d e (b d (4 e f-25 d g)-a e (e f-7 d g))+24 c^2 d^3 (e f-4 d g)\right )}{40 e^2 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\right )}{12 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-4 d g+e f-3 e g x)}{6 e^2 (d+e x)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 \left (-\frac {\int \frac {\left (3 e^2 (e f+3 d g) b^3-4 \left (3 a e^3 g-c d e (4 e f-25 d g)\right ) b^2-12 c \left (2 c (e f-4 d g) d^2+a e^2 (3 e f-17 d g)\right ) b-32 a c e \left (3 a e^2 g-c d (e f-4 d g)\right )-2 c \left (24 c^2 (e f-4 d g) d^2+b e^2 (b e f-37 b d g+36 a e g)+4 c e (a e (5 e f-23 d g)-3 b d (2 e f-11 d g))\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{(d+e x)^4}dx}{16 e^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (a+b x+c x^2\right )^{5/2} \left (e^2 \left (48 a^2 e^2 g+4 a b e (2 e f-11 d g)-3 b^2 d (3 d g+e f)\right )+5 e x \left (-4 c e (b d (2 e f-11 d g)-a e (e f-7 d g))+b e^2 (12 a e g-13 b d g+b e f)+8 c^2 d^2 (e f-4 d g)\right )-4 c d e (b d (4 e f-25 d g)-a e (e f-7 d g))+24 c^2 d^3 (e f-4 d g)\right )}{40 e^2 (d+e x)^5 \left (a e^2-b d e+c d^2\right )}\right )}{12 e^2}-\frac {\left (a+b x+c x^2\right )^{7/2} (-4 d g+e f-3 e g x)}{6 e^2 (d+e x)^6}\)

\(\Big \downarrow \) 1229

\(\displaystyle \frac {7 \left (-\frac {\left (24 c^2 (e f-4 d g) d^3-4 c e (b d (4 e f-25 d g)-a e (e f-7 d g)) d+e^2 \left (-3 d (e f+3 d g) b^2+4 a e (2 e f-11 d g) b+48 a^2 e^2 g\right )+5 e \left (8 c^2 (e f-4 d g) d^2+b e^2 (b e f-13 b d g+12 a e g)-4 c e (b d (2 e f-11 d g)-a e (e f-7 d g))\right ) x\right ) \left (c x^2+b x+a\right )^{5/2}}{40 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^5}-\frac {\frac {\left (128 c^4 (e f-4 d g) d^5-32 c^3 e (b d (7 e f-34 d g)-a e (5 e f-22 d g)) d^3+8 c^2 e^2 \left (b^2 (11 e f-84 d g) d^2-10 a b e (e f-8 d g) d-2 a^2 e^2 (e f+d g)\right ) d+b^2 e^4 (b d-4 a e) (b e f+3 b d g-4 a e g)+4 c e^3 \left (d^2 (e f+24 d g) b^3-2 a d e (4 e f-3 d g) b^2+4 a^2 e^2 (3 e f-16 d g) b+32 a^3 e^3 g\right )+e \left (192 c^4 (e f-4 d g) d^4+32 c^3 e (a e (10 e f-43 d g)-3 b d (4 e f-19 d g)) d^2-3 b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (4 e f-85 d g) b^2+2 a e (5 e f-74 d g) b+60 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (26 e f-173 d g) d^2-2 a b e (20 e f-119 d g) d+10 a^2 e^2 (e f-7 d g)\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{4 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^3}-\frac {\int \frac {3 \left (e^4 (e f+3 d g) b^5-4 \left (a e^5 g-c d e^3 (e f+24 d g)\right ) b^4+8 c e^2 \left (c d^2 (11 e f-84 d g)-a e^2 (2 e f+29 d g)\right ) b^3+16 c e \left (9 a^2 g e^4-a c d (14 e f-99 d g) e^2-2 c^2 d^3 (7 e f-34 d g)\right ) b^2+16 c^2 \left (8 c^2 (e f-4 d g) d^4+2 a c e^2 (13 e f-60 d g) d^2+a^2 e^4 (11 e f-73 d g)\right ) b+64 a c^2 e \left (4 a^2 g e^4-a c d (3 e f-13 d g) e^2-2 c^2 d^3 (e f-4 d g)\right )+2 c \left (128 c^4 (e f-4 d g) d^4+32 c^3 e (a e (7 e f-30 d g)-2 b d (4 e f-19 d g)) d^2-b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (2 e f-53 d g) b^2+2 a e (2 e f-49 d g) b+44 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (17 e f-114 d g) d^2-2 a b e (14 e f-83 d g) d+2 a^2 e^2 (5 e f-27 d g)\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)^2}dx}{4 e^2 \left (c d^2-b e d+a e^2\right )}}{16 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{12 e^2}-\frac {(e f-4 d g-3 e g x) \left (c x^2+b x+a\right )^{7/2}}{6 e^2 (d+e x)^6}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 \left (-\frac {\left (24 c^2 (e f-4 d g) d^3-4 c e (b d (4 e f-25 d g)-a e (e f-7 d g)) d+e^2 \left (-3 d (e f+3 d g) b^2+4 a e (2 e f-11 d g) b+48 a^2 e^2 g\right )+5 e \left (8 c^2 (e f-4 d g) d^2+b e^2 (b e f-13 b d g+12 a e g)-4 c e (b d (2 e f-11 d g)-a e (e f-7 d g))\right ) x\right ) \left (c x^2+b x+a\right )^{5/2}}{40 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^5}-\frac {\frac {\left (128 c^4 (e f-4 d g) d^5-32 c^3 e (b d (7 e f-34 d g)-a e (5 e f-22 d g)) d^3+8 c^2 e^2 \left (b^2 (11 e f-84 d g) d^2-10 a b e (e f-8 d g) d-2 a^2 e^2 (e f+d g)\right ) d+b^2 e^4 (b d-4 a e) (b e f+3 b d g-4 a e g)+4 c e^3 \left (d^2 (e f+24 d g) b^3-2 a d e (4 e f-3 d g) b^2+4 a^2 e^2 (3 e f-16 d g) b+32 a^3 e^3 g\right )+e \left (192 c^4 (e f-4 d g) d^4+32 c^3 e (a e (10 e f-43 d g)-3 b d (4 e f-19 d g)) d^2-3 b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (4 e f-85 d g) b^2+2 a e (5 e f-74 d g) b+60 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (26 e f-173 d g) d^2-2 a b e (20 e f-119 d g) d+10 a^2 e^2 (e f-7 d g)\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{4 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^3}-\frac {3 \int \frac {\left (e^4 (e f+3 d g) b^5-4 \left (a e^5 g-c d e^3 (e f+24 d g)\right ) b^4+8 c e^2 \left (c d^2 (11 e f-84 d g)-a e^2 (2 e f+29 d g)\right ) b^3+16 c e \left (9 a^2 g e^4-a c d (14 e f-99 d g) e^2-2 c^2 d^3 (7 e f-34 d g)\right ) b^2+16 c^2 \left (8 c^2 (e f-4 d g) d^4+2 a c e^2 (13 e f-60 d g) d^2+a^2 e^4 (11 e f-73 d g)\right ) b+64 a c^2 e \left (4 a^2 g e^4-a c d (3 e f-13 d g) e^2-2 c^2 d^3 (e f-4 d g)\right )+2 c \left (128 c^4 (e f-4 d g) d^4+32 c^3 e (a e (7 e f-30 d g)-2 b d (4 e f-19 d g)) d^2-b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (2 e f-53 d g) b^2+2 a e (2 e f-49 d g) b+44 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (17 e f-114 d g) d^2-2 a b e (14 e f-83 d g) d+2 a^2 e^2 (5 e f-27 d g)\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{(d+e x)^2}dx}{8 e^2 \left (c d^2-b e d+a e^2\right )}}{16 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{12 e^2}-\frac {(e f-4 d g-3 e g x) \left (c x^2+b x+a\right )^{7/2}}{6 e^2 (d+e x)^6}\)

\(\Big \downarrow \) 1230

\(\displaystyle \frac {7 \left (-\frac {\left (24 c^2 (e f-4 d g) d^3-4 c e (b d (4 e f-25 d g)-a e (e f-7 d g)) d+e^2 \left (-3 d (e f+3 d g) b^2+4 a e (2 e f-11 d g) b+48 a^2 e^2 g\right )+5 e \left (8 c^2 (e f-4 d g) d^2+b e^2 (b e f-13 b d g+12 a e g)-4 c e (b d (2 e f-11 d g)-a e (e f-7 d g))\right ) x\right ) \left (c x^2+b x+a\right )^{5/2}}{40 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^5}-\frac {\frac {\left (128 c^4 (e f-4 d g) d^5-32 c^3 e (b d (7 e f-34 d g)-a e (5 e f-22 d g)) d^3+8 c^2 e^2 \left (b^2 (11 e f-84 d g) d^2-10 a b e (e f-8 d g) d-2 a^2 e^2 (e f+d g)\right ) d+b^2 e^4 (b d-4 a e) (b e f+3 b d g-4 a e g)+4 c e^3 \left (d^2 (e f+24 d g) b^3-2 a d e (4 e f-3 d g) b^2+4 a^2 e^2 (3 e f-16 d g) b+32 a^3 e^3 g\right )+e \left (192 c^4 (e f-4 d g) d^4+32 c^3 e (a e (10 e f-43 d g)-3 b d (4 e f-19 d g)) d^2-3 b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (4 e f-85 d g) b^2+2 a e (5 e f-74 d g) b+60 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (26 e f-173 d g) d^2-2 a b e (20 e f-119 d g) d+10 a^2 e^2 (e f-7 d g)\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{4 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^3}-\frac {3 \left (\frac {\left (512 c^5 (e f-4 d g) d^5+128 c^4 e (2 a e (4 e f-17 d g)-3 b d (3 e f-14 d g)) d^3-32 c^3 e^2 \left (-4 b^2 (6 e f-37 d g) d^2+a b e (41 e f-226 d g) d-16 a^2 e^2 (e f-5 d g)\right ) d-b^4 e^5 (b e f+3 b d g-4 a e g)-8 c^2 e^3 \left (5 d^2 (3 e f-38 d g) b^3-2 a d e (18 e f-197 d g) b^2+2 a^2 e^2 (11 e f-117 d g) b+32 a^3 e^3 g\right )-4 b^2 c e^4 \left (d (2 e f+27 d g) b^2-2 a e (2 e f+31 d g) b+36 a^2 e^2 g\right )+2 c e \left (128 c^4 (e f-4 d g) d^4+32 c^3 e (a e (7 e f-30 d g)-2 b d (4 e f-19 d g)) d^2-b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (2 e f-53 d g) b^2+2 a e (2 e f-49 d g) b+44 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (17 e f-114 d g) d^2-2 a b e (14 e f-83 d g) d+2 a^2 e^2 (5 e f-27 d g)\right )\right ) x\right ) \sqrt {c x^2+b x+a}}{e^2 (d+e x)}-\frac {\int -\frac {e^5 (e f+3 d g) b^6-4 \left (a e^6 g-c d e^4 (2 e f+27 d g)\right ) b^5+20 c e^3 \left (2 c d^2 (3 e f-38 d g)-a e^2 (e f+13 d g)\right ) b^4+32 c e^2 \left (5 a^2 g e^4-5 a c d (2 e f-25 d g) e^2-4 c^2 d^3 (6 e f-37 d g)\right ) b^3+16 c^2 e \left (24 c^2 (3 e f-14 d g) d^4+4 a c e^2 (29 e f-170 d g) d^2+5 a^2 e^4 (3 e f-43 d g)\right ) b^2+64 c^2 \left (15 a^3 g e^6-a^2 c d (22 e f-123 d g) e^4-16 a c^2 d^3 (2 e f-9 d g) e^2-8 c^3 d^5 (e f-4 d g)\right ) b+64 a c^3 e \left (8 c^2 (e f-4 d g) d^4+2 a c e^2 (7 e f-30 d g) d^2+a^2 e^4 (5 e f-27 d g)\right )+128 c^2 \left (c d^2-b e d+a e^2\right )^2 \left (-8 d (e f-4 d g) c^2+4 e (b e f-7 b d g+a e g) c+5 b^2 e^2 g\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}\right )}{8 e^2 \left (c d^2-b e d+a e^2\right )}}{16 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{12 e^2}-\frac {(e f-4 d g-3 e g x) \left (c x^2+b x+a\right )^{7/2}}{6 e^2 (d+e x)^6}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {7 \left (-\frac {\left (24 c^2 (e f-4 d g) d^3-4 c e (b d (4 e f-25 d g)-a e (e f-7 d g)) d+e^2 \left (-3 d (e f+3 d g) b^2+4 a e (2 e f-11 d g) b+48 a^2 e^2 g\right )+5 e \left (8 c^2 (e f-4 d g) d^2+b e^2 (b e f-13 b d g+12 a e g)-4 c e (b d (2 e f-11 d g)-a e (e f-7 d g))\right ) x\right ) \left (c x^2+b x+a\right )^{5/2}}{40 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^5}-\frac {\frac {\left (128 c^4 (e f-4 d g) d^5-32 c^3 e (b d (7 e f-34 d g)-a e (5 e f-22 d g)) d^3+8 c^2 e^2 \left (b^2 (11 e f-84 d g) d^2-10 a b e (e f-8 d g) d-2 a^2 e^2 (e f+d g)\right ) d+b^2 e^4 (b d-4 a e) (b e f+3 b d g-4 a e g)+4 c e^3 \left (d^2 (e f+24 d g) b^3-2 a d e (4 e f-3 d g) b^2+4 a^2 e^2 (3 e f-16 d g) b+32 a^3 e^3 g\right )+e \left (192 c^4 (e f-4 d g) d^4+32 c^3 e (a e (10 e f-43 d g)-3 b d (4 e f-19 d g)) d^2-3 b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (4 e f-85 d g) b^2+2 a e (5 e f-74 d g) b+60 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (26 e f-173 d g) d^2-2 a b e (20 e f-119 d g) d+10 a^2 e^2 (e f-7 d g)\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{4 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^3}-\frac {3 \left (\frac {\sqrt {c x^2+b x+a} \left (512 c^5 (e f-4 d g) d^5+128 c^4 e (2 a e (4 e f-17 d g)-3 b d (3 e f-14 d g)) d^3-32 c^3 e^2 \left (-4 b^2 (6 e f-37 d g) d^2+a b e (41 e f-226 d g) d-16 a^2 e^2 (e f-5 d g)\right ) d-b^4 e^5 (b e f+3 b d g-4 a e g)-8 c^2 e^3 \left (5 d^2 (3 e f-38 d g) b^3-2 a d e (18 e f-197 d g) b^2+2 a^2 e^2 (11 e f-117 d g) b+32 a^3 e^3 g\right )-4 b^2 c e^4 \left (d (2 e f+27 d g) b^2-2 a e (2 e f+31 d g) b+36 a^2 e^2 g\right )+2 c e \left (128 c^4 (e f-4 d g) d^4+32 c^3 e (a e (7 e f-30 d g)-2 b d (4 e f-19 d g)) d^2-b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (2 e f-53 d g) b^2+2 a e (2 e f-49 d g) b+44 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (17 e f-114 d g) d^2-2 a b e (14 e f-83 d g) d+2 a^2 e^2 (5 e f-27 d g)\right )\right ) x\right )}{e^2 (d+e x)}+\frac {\int \frac {e^5 (e f+3 d g) b^6-4 \left (a e^6 g-c d e^4 (2 e f+27 d g)\right ) b^5+20 c e^3 \left (2 c d^2 (3 e f-38 d g)-a e^2 (e f+13 d g)\right ) b^4+32 c e^2 \left (5 a^2 g e^4-5 a c d (2 e f-25 d g) e^2-4 c^2 d^3 (6 e f-37 d g)\right ) b^3+16 c^2 e \left (24 c^2 (3 e f-14 d g) d^4+4 a c e^2 (29 e f-170 d g) d^2+5 a^2 e^4 (3 e f-43 d g)\right ) b^2+64 c^2 \left (15 a^3 g e^6-a^2 c d (22 e f-123 d g) e^4-16 a c^2 d^3 (2 e f-9 d g) e^2-8 c^3 d^5 (e f-4 d g)\right ) b+64 a c^3 e \left (8 c^2 (e f-4 d g) d^4+2 a c e^2 (7 e f-30 d g) d^2+a^2 e^4 (5 e f-27 d g)\right )+128 c^2 \left (c d^2-b e d+a e^2\right )^2 \left (-8 d (e f-4 d g) c^2+4 e (b e f-7 b d g+a e g) c+5 b^2 e^2 g\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 e^2}\right )}{8 e^2 \left (c d^2-b e d+a e^2\right )}}{16 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{12 e^2}-\frac {(e f-4 d g-3 e g x) \left (c x^2+b x+a\right )^{7/2}}{6 e^2 (d+e x)^6}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {7 \left (-\frac {\left (24 c^2 (e f-4 d g) d^3-4 c e (b d (4 e f-25 d g)-a e (e f-7 d g)) d+e^2 \left (-3 d (e f+3 d g) b^2+4 a e (2 e f-11 d g) b+48 a^2 e^2 g\right )+5 e \left (8 c^2 (e f-4 d g) d^2+b e^2 (b e f-13 b d g+12 a e g)-4 c e (b d (2 e f-11 d g)-a e (e f-7 d g))\right ) x\right ) \left (c x^2+b x+a\right )^{5/2}}{40 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^5}-\frac {\frac {\left (128 c^4 (e f-4 d g) d^5-32 c^3 e (b d (7 e f-34 d g)-a e (5 e f-22 d g)) d^3+8 c^2 e^2 \left (b^2 (11 e f-84 d g) d^2-10 a b e (e f-8 d g) d-2 a^2 e^2 (e f+d g)\right ) d+b^2 e^4 (b d-4 a e) (b e f+3 b d g-4 a e g)+4 c e^3 \left (d^2 (e f+24 d g) b^3-2 a d e (4 e f-3 d g) b^2+4 a^2 e^2 (3 e f-16 d g) b+32 a^3 e^3 g\right )+e \left (192 c^4 (e f-4 d g) d^4+32 c^3 e (a e (10 e f-43 d g)-3 b d (4 e f-19 d g)) d^2-3 b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (4 e f-85 d g) b^2+2 a e (5 e f-74 d g) b+60 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (26 e f-173 d g) d^2-2 a b e (20 e f-119 d g) d+10 a^2 e^2 (e f-7 d g)\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{4 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^3}-\frac {3 \left (\frac {\sqrt {c x^2+b x+a} \left (512 c^5 (e f-4 d g) d^5+128 c^4 e (2 a e (4 e f-17 d g)-3 b d (3 e f-14 d g)) d^3-32 c^3 e^2 \left (-4 b^2 (6 e f-37 d g) d^2+a b e (41 e f-226 d g) d-16 a^2 e^2 (e f-5 d g)\right ) d-b^4 e^5 (b e f+3 b d g-4 a e g)-8 c^2 e^3 \left (5 d^2 (3 e f-38 d g) b^3-2 a d e (18 e f-197 d g) b^2+2 a^2 e^2 (11 e f-117 d g) b+32 a^3 e^3 g\right )-4 b^2 c e^4 \left (d (2 e f+27 d g) b^2-2 a e (2 e f+31 d g) b+36 a^2 e^2 g\right )+2 c e \left (128 c^4 (e f-4 d g) d^4+32 c^3 e (a e (7 e f-30 d g)-2 b d (4 e f-19 d g)) d^2-b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (2 e f-53 d g) b^2+2 a e (2 e f-49 d g) b+44 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (17 e f-114 d g) d^2-2 a b e (14 e f-83 d g) d+2 a^2 e^2 (5 e f-27 d g)\right )\right ) x\right )}{e^2 (d+e x)}+\frac {\frac {128 c^2 \left (-8 d (e f-4 d g) c^2+4 e (b e f-7 b d g+a e g) c+5 b^2 e^2 g\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx \left (c d^2-b e d+a e^2\right )^2}{e}+\frac {\left (1024 c^6 (e f-4 d g) d^6+512 c^5 e (a e (5 e f-21 d g)-3 b d (2 e f-9 d g)) d^4+640 c^4 e^2 \left (b^2 (5 e f-27 d g) d^2-8 a b e (e f-5 d g) d+a^2 e^2 (3 e f-14 d g)\right ) d^2+b^5 e^6 (b e f+3 b d g-4 a e g)+40 b c^2 e^4 \left (3 d^2 (e f-18 d g) b^3-4 a d e (2 e f-33 d g) b^2+6 a^2 e^2 (e f-17 d g) b+24 a^3 e^3 g\right )+320 c^3 e^3 (b d-a e) \left (-2 b^2 (2 e f-15 d g) d^2+a b e (5 e f-32 d g) d-a^2 e^2 (e f-7 d g)\right )+4 b^3 c e^5 \left (d (2 e f+27 d g) b^2-5 a e (e f+13 d g) b+40 a^2 e^2 g\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}\right )}{8 e^2 \left (c d^2-b e d+a e^2\right )}}{16 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{12 e^2}-\frac {(e f-4 d g-3 e g x) \left (c x^2+b x+a\right )^{7/2}}{6 e^2 (d+e x)^6}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {7 \left (-\frac {\left (24 c^2 (e f-4 d g) d^3-4 c e (b d (4 e f-25 d g)-a e (e f-7 d g)) d+e^2 \left (-3 d (e f+3 d g) b^2+4 a e (2 e f-11 d g) b+48 a^2 e^2 g\right )+5 e \left (8 c^2 (e f-4 d g) d^2+b e^2 (b e f-13 b d g+12 a e g)-4 c e (b d (2 e f-11 d g)-a e (e f-7 d g))\right ) x\right ) \left (c x^2+b x+a\right )^{5/2}}{40 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^5}-\frac {\frac {\left (128 c^4 (e f-4 d g) d^5-32 c^3 e (b d (7 e f-34 d g)-a e (5 e f-22 d g)) d^3+8 c^2 e^2 \left (b^2 (11 e f-84 d g) d^2-10 a b e (e f-8 d g) d-2 a^2 e^2 (e f+d g)\right ) d+b^2 e^4 (b d-4 a e) (b e f+3 b d g-4 a e g)+4 c e^3 \left (d^2 (e f+24 d g) b^3-2 a d e (4 e f-3 d g) b^2+4 a^2 e^2 (3 e f-16 d g) b+32 a^3 e^3 g\right )+e \left (192 c^4 (e f-4 d g) d^4+32 c^3 e (a e (10 e f-43 d g)-3 b d (4 e f-19 d g)) d^2-3 b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (4 e f-85 d g) b^2+2 a e (5 e f-74 d g) b+60 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (26 e f-173 d g) d^2-2 a b e (20 e f-119 d g) d+10 a^2 e^2 (e f-7 d g)\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{4 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^3}-\frac {3 \left (\frac {\sqrt {c x^2+b x+a} \left (512 c^5 (e f-4 d g) d^5+128 c^4 e (2 a e (4 e f-17 d g)-3 b d (3 e f-14 d g)) d^3-32 c^3 e^2 \left (-4 b^2 (6 e f-37 d g) d^2+a b e (41 e f-226 d g) d-16 a^2 e^2 (e f-5 d g)\right ) d-b^4 e^5 (b e f+3 b d g-4 a e g)-8 c^2 e^3 \left (5 d^2 (3 e f-38 d g) b^3-2 a d e (18 e f-197 d g) b^2+2 a^2 e^2 (11 e f-117 d g) b+32 a^3 e^3 g\right )-4 b^2 c e^4 \left (d (2 e f+27 d g) b^2-2 a e (2 e f+31 d g) b+36 a^2 e^2 g\right )+2 c e \left (128 c^4 (e f-4 d g) d^4+32 c^3 e (a e (7 e f-30 d g)-2 b d (4 e f-19 d g)) d^2-b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (2 e f-53 d g) b^2+2 a e (2 e f-49 d g) b+44 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (17 e f-114 d g) d^2-2 a b e (14 e f-83 d g) d+2 a^2 e^2 (5 e f-27 d g)\right )\right ) x\right )}{e^2 (d+e x)}+\frac {\frac {256 c^2 \left (-8 d (e f-4 d g) c^2+4 e (b e f-7 b d g+a e g) c+5 b^2 e^2 g\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}} \left (c d^2-b e d+a e^2\right )^2}{e}+\frac {\left (1024 c^6 (e f-4 d g) d^6+512 c^5 e (a e (5 e f-21 d g)-3 b d (2 e f-9 d g)) d^4+640 c^4 e^2 \left (b^2 (5 e f-27 d g) d^2-8 a b e (e f-5 d g) d+a^2 e^2 (3 e f-14 d g)\right ) d^2+b^5 e^6 (b e f+3 b d g-4 a e g)+40 b c^2 e^4 \left (3 d^2 (e f-18 d g) b^3-4 a d e (2 e f-33 d g) b^2+6 a^2 e^2 (e f-17 d g) b+24 a^3 e^3 g\right )+320 c^3 e^3 (b d-a e) \left (-2 b^2 (2 e f-15 d g) d^2+a b e (5 e f-32 d g) d-a^2 e^2 (e f-7 d g)\right )+4 b^3 c e^5 \left (d (2 e f+27 d g) b^2-5 a e (e f+13 d g) b+40 a^2 e^2 g\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}\right )}{8 e^2 \left (c d^2-b e d+a e^2\right )}}{16 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{12 e^2}-\frac {(e f-4 d g-3 e g x) \left (c x^2+b x+a\right )^{7/2}}{6 e^2 (d+e x)^6}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {7 \left (-\frac {\left (24 c^2 (e f-4 d g) d^3-4 c e (b d (4 e f-25 d g)-a e (e f-7 d g)) d+e^2 \left (-3 d (e f+3 d g) b^2+4 a e (2 e f-11 d g) b+48 a^2 e^2 g\right )+5 e \left (8 c^2 (e f-4 d g) d^2+b e^2 (b e f-13 b d g+12 a e g)-4 c e (b d (2 e f-11 d g)-a e (e f-7 d g))\right ) x\right ) \left (c x^2+b x+a\right )^{5/2}}{40 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^5}-\frac {\frac {\left (128 c^4 (e f-4 d g) d^5-32 c^3 e (b d (7 e f-34 d g)-a e (5 e f-22 d g)) d^3+8 c^2 e^2 \left (b^2 (11 e f-84 d g) d^2-10 a b e (e f-8 d g) d-2 a^2 e^2 (e f+d g)\right ) d+b^2 e^4 (b d-4 a e) (b e f+3 b d g-4 a e g)+4 c e^3 \left (d^2 (e f+24 d g) b^3-2 a d e (4 e f-3 d g) b^2+4 a^2 e^2 (3 e f-16 d g) b+32 a^3 e^3 g\right )+e \left (192 c^4 (e f-4 d g) d^4+32 c^3 e (a e (10 e f-43 d g)-3 b d (4 e f-19 d g)) d^2-3 b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (4 e f-85 d g) b^2+2 a e (5 e f-74 d g) b+60 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (26 e f-173 d g) d^2-2 a b e (20 e f-119 d g) d+10 a^2 e^2 (e f-7 d g)\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{4 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^3}-\frac {3 \left (\frac {\sqrt {c x^2+b x+a} \left (512 c^5 (e f-4 d g) d^5+128 c^4 e (2 a e (4 e f-17 d g)-3 b d (3 e f-14 d g)) d^3-32 c^3 e^2 \left (-4 b^2 (6 e f-37 d g) d^2+a b e (41 e f-226 d g) d-16 a^2 e^2 (e f-5 d g)\right ) d-b^4 e^5 (b e f+3 b d g-4 a e g)-8 c^2 e^3 \left (5 d^2 (3 e f-38 d g) b^3-2 a d e (18 e f-197 d g) b^2+2 a^2 e^2 (11 e f-117 d g) b+32 a^3 e^3 g\right )-4 b^2 c e^4 \left (d (2 e f+27 d g) b^2-2 a e (2 e f+31 d g) b+36 a^2 e^2 g\right )+2 c e \left (128 c^4 (e f-4 d g) d^4+32 c^3 e (a e (7 e f-30 d g)-2 b d (4 e f-19 d g)) d^2-b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (2 e f-53 d g) b^2+2 a e (2 e f-49 d g) b+44 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (17 e f-114 d g) d^2-2 a b e (14 e f-83 d g) d+2 a^2 e^2 (5 e f-27 d g)\right )\right ) x\right )}{e^2 (d+e x)}+\frac {\frac {128 c^{3/2} \left (-8 d (e f-4 d g) c^2+4 e (b e f-7 b d g+a e g) c+5 b^2 e^2 g\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )^2}{e}+\frac {\left (1024 c^6 (e f-4 d g) d^6+512 c^5 e (a e (5 e f-21 d g)-3 b d (2 e f-9 d g)) d^4+640 c^4 e^2 \left (b^2 (5 e f-27 d g) d^2-8 a b e (e f-5 d g) d+a^2 e^2 (3 e f-14 d g)\right ) d^2+b^5 e^6 (b e f+3 b d g-4 a e g)+40 b c^2 e^4 \left (3 d^2 (e f-18 d g) b^3-4 a d e (2 e f-33 d g) b^2+6 a^2 e^2 (e f-17 d g) b+24 a^3 e^3 g\right )+320 c^3 e^3 (b d-a e) \left (-2 b^2 (2 e f-15 d g) d^2+a b e (5 e f-32 d g) d-a^2 e^2 (e f-7 d g)\right )+4 b^3 c e^5 \left (d (2 e f+27 d g) b^2-5 a e (e f+13 d g) b+40 a^2 e^2 g\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{2 e^2}\right )}{8 e^2 \left (c d^2-b e d+a e^2\right )}}{16 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{12 e^2}-\frac {(e f-4 d g-3 e g x) \left (c x^2+b x+a\right )^{7/2}}{6 e^2 (d+e x)^6}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {7 \left (-\frac {\left (24 c^2 (e f-4 d g) d^3-4 c e (b d (4 e f-25 d g)-a e (e f-7 d g)) d+e^2 \left (-3 d (e f+3 d g) b^2+4 a e (2 e f-11 d g) b+48 a^2 e^2 g\right )+5 e \left (8 c^2 (e f-4 d g) d^2+b e^2 (b e f-13 b d g+12 a e g)-4 c e (b d (2 e f-11 d g)-a e (e f-7 d g))\right ) x\right ) \left (c x^2+b x+a\right )^{5/2}}{40 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^5}-\frac {\frac {\left (128 c^4 (e f-4 d g) d^5-32 c^3 e (b d (7 e f-34 d g)-a e (5 e f-22 d g)) d^3+8 c^2 e^2 \left (b^2 (11 e f-84 d g) d^2-10 a b e (e f-8 d g) d-2 a^2 e^2 (e f+d g)\right ) d+b^2 e^4 (b d-4 a e) (b e f+3 b d g-4 a e g)+4 c e^3 \left (d^2 (e f+24 d g) b^3-2 a d e (4 e f-3 d g) b^2+4 a^2 e^2 (3 e f-16 d g) b+32 a^3 e^3 g\right )+e \left (192 c^4 (e f-4 d g) d^4+32 c^3 e (a e (10 e f-43 d g)-3 b d (4 e f-19 d g)) d^2-3 b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (4 e f-85 d g) b^2+2 a e (5 e f-74 d g) b+60 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (26 e f-173 d g) d^2-2 a b e (20 e f-119 d g) d+10 a^2 e^2 (e f-7 d g)\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{4 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^3}-\frac {3 \left (\frac {\sqrt {c x^2+b x+a} \left (512 c^5 (e f-4 d g) d^5+128 c^4 e (2 a e (4 e f-17 d g)-3 b d (3 e f-14 d g)) d^3-32 c^3 e^2 \left (-4 b^2 (6 e f-37 d g) d^2+a b e (41 e f-226 d g) d-16 a^2 e^2 (e f-5 d g)\right ) d-b^4 e^5 (b e f+3 b d g-4 a e g)-8 c^2 e^3 \left (5 d^2 (3 e f-38 d g) b^3-2 a d e (18 e f-197 d g) b^2+2 a^2 e^2 (11 e f-117 d g) b+32 a^3 e^3 g\right )-4 b^2 c e^4 \left (d (2 e f+27 d g) b^2-2 a e (2 e f+31 d g) b+36 a^2 e^2 g\right )+2 c e \left (128 c^4 (e f-4 d g) d^4+32 c^3 e (a e (7 e f-30 d g)-2 b d (4 e f-19 d g)) d^2-b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (2 e f-53 d g) b^2+2 a e (2 e f-49 d g) b+44 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (17 e f-114 d g) d^2-2 a b e (14 e f-83 d g) d+2 a^2 e^2 (5 e f-27 d g)\right )\right ) x\right )}{e^2 (d+e x)}+\frac {\frac {128 c^{3/2} \left (c d^2-b e d+a e^2\right )^2 \left (-8 d (e f-4 d g) c^2+4 e (b e f-7 b d g+a e g) c+5 b^2 e^2 g\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right )}{e}-\frac {2 \left (1024 c^6 (e f-4 d g) d^6+512 c^5 e (a e (5 e f-21 d g)-3 b d (2 e f-9 d g)) d^4+640 c^4 e^2 \left (b^2 (5 e f-27 d g) d^2-8 a b e (e f-5 d g) d+a^2 e^2 (3 e f-14 d g)\right ) d^2+b^5 e^6 (b e f+3 b d g-4 a e g)+40 b c^2 e^4 \left (3 d^2 (e f-18 d g) b^3-4 a d e (2 e f-33 d g) b^2+6 a^2 e^2 (e f-17 d g) b+24 a^3 e^3 g\right )+320 c^3 e^3 (b d-a e) \left (-2 b^2 (2 e f-15 d g) d^2+a b e (5 e f-32 d g) d-a^2 e^2 (e f-7 d g)\right )+4 b^3 c e^5 \left (d (2 e f+27 d g) b^2-5 a e (e f+13 d g) b+40 a^2 e^2 g\right )\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}}{2 e^2}\right )}{8 e^2 \left (c d^2-b e d+a e^2\right )}}{16 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{12 e^2}-\frac {(e f-4 d g-3 e g x) \left (c x^2+b x+a\right )^{7/2}}{6 e^2 (d+e x)^6}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {7 \left (-\frac {\left (24 c^2 (e f-4 d g) d^3-4 c e (b d (4 e f-25 d g)-a e (e f-7 d g)) d+e^2 \left (-3 d (e f+3 d g) b^2+4 a e (2 e f-11 d g) b+48 a^2 e^2 g\right )+5 e \left (8 c^2 (e f-4 d g) d^2+b e^2 (b e f-13 b d g+12 a e g)-4 c e (b d (2 e f-11 d g)-a e (e f-7 d g))\right ) x\right ) \left (c x^2+b x+a\right )^{5/2}}{40 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^5}-\frac {\frac {\left (128 c^4 (e f-4 d g) d^5-32 c^3 e (b d (7 e f-34 d g)-a e (5 e f-22 d g)) d^3+8 c^2 e^2 \left (b^2 (11 e f-84 d g) d^2-10 a b e (e f-8 d g) d-2 a^2 e^2 (e f+d g)\right ) d+b^2 e^4 (b d-4 a e) (b e f+3 b d g-4 a e g)+4 c e^3 \left (d^2 (e f+24 d g) b^3-2 a d e (4 e f-3 d g) b^2+4 a^2 e^2 (3 e f-16 d g) b+32 a^3 e^3 g\right )+e \left (192 c^4 (e f-4 d g) d^4+32 c^3 e (a e (10 e f-43 d g)-3 b d (4 e f-19 d g)) d^2-3 b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (4 e f-85 d g) b^2+2 a e (5 e f-74 d g) b+60 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (26 e f-173 d g) d^2-2 a b e (20 e f-119 d g) d+10 a^2 e^2 (e f-7 d g)\right )\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{4 e^2 \left (c d^2-b e d+a e^2\right ) (d+e x)^3}-\frac {3 \left (\frac {\sqrt {c x^2+b x+a} \left (512 c^5 (e f-4 d g) d^5+128 c^4 e (2 a e (4 e f-17 d g)-3 b d (3 e f-14 d g)) d^3-32 c^3 e^2 \left (-4 b^2 (6 e f-37 d g) d^2+a b e (41 e f-226 d g) d-16 a^2 e^2 (e f-5 d g)\right ) d-b^4 e^5 (b e f+3 b d g-4 a e g)-8 c^2 e^3 \left (5 d^2 (3 e f-38 d g) b^3-2 a d e (18 e f-197 d g) b^2+2 a^2 e^2 (11 e f-117 d g) b+32 a^3 e^3 g\right )-4 b^2 c e^4 \left (d (2 e f+27 d g) b^2-2 a e (2 e f+31 d g) b+36 a^2 e^2 g\right )+2 c e \left (128 c^4 (e f-4 d g) d^4+32 c^3 e (a e (7 e f-30 d g)-2 b d (4 e f-19 d g)) d^2-b^3 e^4 (b e f+3 b d g-4 a e g)+4 b c e^3 \left (-d (2 e f-53 d g) b^2+2 a e (2 e f-49 d g) b+44 a^2 e^2 g\right )+8 c^2 e^2 \left (b^2 (17 e f-114 d g) d^2-2 a b e (14 e f-83 d g) d+2 a^2 e^2 (5 e f-27 d g)\right )\right ) x\right )}{e^2 (d+e x)}+\frac {\frac {128 c^{3/2} \left (-8 d (e f-4 d g) c^2+4 e (b e f-7 b d g+a e g) c+5 b^2 e^2 g\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {c x^2+b x+a}}\right ) \left (c d^2-b e d+a e^2\right )^2}{e}+\frac {\left (1024 c^6 (e f-4 d g) d^6+512 c^5 e (a e (5 e f-21 d g)-3 b d (2 e f-9 d g)) d^4+640 c^4 e^2 \left (b^2 (5 e f-27 d g) d^2-8 a b e (e f-5 d g) d+a^2 e^2 (3 e f-14 d g)\right ) d^2+b^5 e^6 (b e f+3 b d g-4 a e g)+40 b c^2 e^4 \left (3 d^2 (e f-18 d g) b^3-4 a d e (2 e f-33 d g) b^2+6 a^2 e^2 (e f-17 d g) b+24 a^3 e^3 g\right )+320 c^3 e^3 (b d-a e) \left (-2 b^2 (2 e f-15 d g) d^2+a b e (5 e f-32 d g) d-a^2 e^2 (e f-7 d g)\right )+4 b^3 c e^5 \left (d (2 e f+27 d g) b^2-5 a e (e f+13 d g) b+40 a^2 e^2 g\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b e d+a e^2} \sqrt {c x^2+b x+a}}\right )}{e \sqrt {c d^2-b e d+a e^2}}}{2 e^2}\right )}{8 e^2 \left (c d^2-b e d+a e^2\right )}}{16 e^2 \left (c d^2-b e d+a e^2\right )}\right )}{12 e^2}-\frac {(e f-4 d g-3 e g x) \left (c x^2+b x+a\right )^{7/2}}{6 e^2 (d+e x)^6}\)

Input:

Int[((f + g*x)*(a + b*x + c*x^2)^(7/2))/(d + e*x)^7,x]
 

Output:

-1/6*((e*f - 4*d*g - 3*e*g*x)*(a + b*x + c*x^2)^(7/2))/(e^2*(d + e*x)^6) + 
 (7*(-1/40*((24*c^2*d^3*(e*f - 4*d*g) - 4*c*d*e*(b*d*(4*e*f - 25*d*g) - a* 
e*(e*f - 7*d*g)) + e^2*(48*a^2*e^2*g + 4*a*b*e*(2*e*f - 11*d*g) - 3*b^2*d* 
(e*f + 3*d*g)) + 5*e*(8*c^2*d^2*(e*f - 4*d*g) + b*e^2*(b*e*f - 13*b*d*g + 
12*a*e*g) - 4*c*e*(b*d*(2*e*f - 11*d*g) - a*e*(e*f - 7*d*g)))*x)*(a + b*x 
+ c*x^2)^(5/2))/(e^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^5) - (((128*c^4*d^5 
*(e*f - 4*d*g) + b^2*e^4*(b*d - 4*a*e)*(b*e*f + 3*b*d*g - 4*a*e*g) - 32*c^ 
3*d^3*e*(b*d*(7*e*f - 34*d*g) - a*e*(5*e*f - 22*d*g)) + 8*c^2*d*e^2*(b^2*d 
^2*(11*e*f - 84*d*g) - 10*a*b*d*e*(e*f - 8*d*g) - 2*a^2*e^2*(e*f + d*g)) + 
 4*c*e^3*(32*a^3*e^3*g + 4*a^2*b*e^2*(3*e*f - 16*d*g) - 2*a*b^2*d*e*(4*e*f 
 - 3*d*g) + b^3*d^2*(e*f + 24*d*g)) + e*(192*c^4*d^4*(e*f - 4*d*g) - 3*b^3 
*e^4*(b*e*f + 3*b*d*g - 4*a*e*g) + 4*b*c*e^3*(60*a^2*e^2*g - b^2*d*(4*e*f 
- 85*d*g) + 2*a*b*e*(5*e*f - 74*d*g)) + 32*c^3*d^2*e*(a*e*(10*e*f - 43*d*g 
) - 3*b*d*(4*e*f - 19*d*g)) + 8*c^2*e^2*(b^2*d^2*(26*e*f - 173*d*g) - 2*a* 
b*d*e*(20*e*f - 119*d*g) + 10*a^2*e^2*(e*f - 7*d*g)))*x)*(a + b*x + c*x^2) 
^(3/2))/(4*e^2*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3) - (3*(((512*c^5*d^5*(e 
*f - 4*d*g) - b^4*e^5*(b*e*f + 3*b*d*g - 4*a*e*g) - 8*c^2*e^3*(32*a^3*e^3* 
g - 2*a*b^2*d*e*(18*e*f - 197*d*g) + 2*a^2*b*e^2*(11*e*f - 117*d*g) + 5*b^ 
3*d^2*(3*e*f - 38*d*g)) + 128*c^4*d^3*e*(2*a*e*(4*e*f - 17*d*g) - 3*b*d*(3 
*e*f - 14*d*g)) - 32*c^3*d*e^2*(a*b*d*e*(41*e*f - 226*d*g) - 4*b^2*d^2*...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1229

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + b*x + c*x^2 
)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* 
d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 
- b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 
)*(m + 2)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 
)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + 
p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c 
*(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( 
m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g 
}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 
0]
 

rule 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(12970\) vs. \(2(1073)=2146\).

Time = 5.60 (sec) , antiderivative size = 12971, normalized size of antiderivative = 11.61

method	result	size
risch	\(\text {Expression too large to display}\)	\(12971\)
default	\(\text {Expression too large to display}\)	\(45361\)

Input:

int((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^7,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^7} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^7,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^7} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((g*x+f)*(c*x**2+b*x+a)**(7/2)/(e*x+d)**7,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^7,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?` f 
or more de
 

Giac [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^7} \, dx=\text {Timed out} \] Input:

integrate((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^7,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^7} \, dx=\text {Hanged} \] Input:

int(((f + g*x)*(a + b*x + c*x^2)^(7/2))/(d + e*x)^7,x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{7/2}}{(d+e x)^7} \, dx=\int \frac {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{\left (e x +d \right )^{7}}d x \] Input:

int((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^7,x)
 

Output:

int((g*x+f)*(c*x^2+b*x+a)^(7/2)/(e*x+d)^7,x)