\(\int \frac {(d+e x)^4 (f+g x)}{(a+b x+c x^2)^{7/2}} \, dx\) [36]

Optimal result

Integrand size = 27, antiderivative size = 214 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=-\frac {2 (d+e x)^4 (b f-2 a g+(2 c f-b g) x)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}+\frac {16 (2 c d f-b e f-b d g+2 a e g) (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{15 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}-\frac {128 \left (c d^2-b d e+a e^2\right ) (2 c d f-b e f-b d g+2 a e g) (b d-2 a e+(2 c d-b e) x)}{15 \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}} \] Output:

-2/5*(e*x+d)^4*(b*f-2*a*g+(-b*g+2*c*f)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(5/2) 
+16/15*(2*a*e*g-b*d*g-b*e*f+2*c*d*f)*(e*x+d)^2*(b*d-2*a*e+(-b*e+2*c*d)*x)/ 
(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(3/2)-128/15*(a*e^2-b*d*e+c*d^2)*(2*a*e*g-b*d 
*g-b*e*f+2*c*d*f)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)^3/(c*x^2+b*x+a)^ 
(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1051\) vs. \(2(214)=428\).

Time = 17.75 (sec) , antiderivative size = 1051, normalized size of antiderivative = 4.91 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {2 \left (b^5 \left (90 d^2 e^2 x^2 (-f+g x)+20 d e^3 x^3 (3 f+g x)-20 d^3 e x (f+3 g x)+e^4 x^4 (5 f+3 g x)-d^4 (3 f+5 g x)\right )+32 \left (8 a^5 e^4 g-8 c^5 d^4 f x^5-4 a c^4 d^2 x^3 \left (5 d^2 f+3 e^2 f x^2+2 d e g x^2\right )+4 a^4 c e^2 \left (2 d e f+3 d^2 g+5 e^2 g x^2\right )-a^2 c^3 x \left (15 d^4 f+30 d^2 e^2 f x^2+20 d^3 e g x^2+3 e^4 f x^4+12 d e^3 g x^4\right )+a^3 c^2 \left (12 d^3 e f+3 d^4 g+20 d e^3 f x^2+30 d^2 e^2 g x^2+15 e^4 g x^4\right )\right )+16 b^2 \left (2 c^3 d^2 x^3 \left (-9 e^2 f x^2+2 d e x (20 f-3 g x)-5 d^2 (3 f-2 g x)\right )-3 a c^2 x \left (e^4 f x^4+5 d^4 (f-2 g x)+30 d^2 e^2 x^2 (f-g x)+4 d e^3 x^3 (-5 f+g x)+20 d^3 e x (-2 f+g x)\right )+3 a^2 c \left (d^4 g+4 d^3 e (f-5 g x)+20 d e^3 x^2 (f-2 g x)+5 e^4 x^3 (-2 f+g x)+30 d^2 e^2 x (-f+g x)\right )+2 a^3 e^2 \left (9 d^2 g+d e (6 f-40 g x)+5 e^2 x (-2 f+3 g x)\right )\right )-8 b^3 \left (2 a^2 e \left (2 d^3 g+3 d^2 e (f-15 g x)-30 d e^2 x (f-2 g x)-5 e^3 x^2 (-3 f+g x)\right )-2 c^2 d x^2 \left (2 e^3 f x^3-5 d^3 (f-3 g x)+3 d e^2 x^2 (-15 f+g x)-30 d^2 e x (-2 f+g x)\right )-5 a c (d-e x)^2 \left (14 d e x (f-g x)-e^2 x^2 (3 f+g x)+d^2 (f+3 g x)\right )\right )-16 b \left (15 a^2 c^2 (d-e x)^4 (f-g x)-8 c^4 d^3 x^4 (-5 d f+4 e f x+d g x)+8 a^4 e^3 (4 d g+e (f-5 g x))+4 a^3 c e \left (6 d^3 g+3 d^2 e (3 f-5 g x)+5 e^3 x^2 (f-3 g x)-10 d e^2 x (f-2 g x)\right )-4 a c^3 d x^2 \left (6 e^3 f x^3-10 d^2 e x (-2 f+g x)+3 d e^2 x^2 (-5 f+3 g x)+d^3 (-15 f+5 g x)\right )\right )-2 b^4 \left (-c x \left (e^4 f x^4+20 d^3 e x (4 f-9 g x)+4 d e^3 x^3 (10 f+g x)+30 d^2 e^2 x^2 (-9 f+2 g x)+5 d^4 (f+4 g x)\right )+a \left (d^4 g+30 d^2 e^2 x (2 f-9 g x)+5 e^4 x^3 (4 f+g x)+20 d e^3 x^2 (-9 f+4 g x)+4 d^3 e (f+10 g x)\right )\right )\right )}{15 \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \] Input:

Integrate[((d + e*x)^4*(f + g*x))/(a + b*x + c*x^2)^(7/2),x]
 

Output:

(2*(b^5*(90*d^2*e^2*x^2*(-f + g*x) + 20*d*e^3*x^3*(3*f + g*x) - 20*d^3*e*x 
*(f + 3*g*x) + e^4*x^4*(5*f + 3*g*x) - d^4*(3*f + 5*g*x)) + 32*(8*a^5*e^4* 
g - 8*c^5*d^4*f*x^5 - 4*a*c^4*d^2*x^3*(5*d^2*f + 3*e^2*f*x^2 + 2*d*e*g*x^2 
) + 4*a^4*c*e^2*(2*d*e*f + 3*d^2*g + 5*e^2*g*x^2) - a^2*c^3*x*(15*d^4*f + 
30*d^2*e^2*f*x^2 + 20*d^3*e*g*x^2 + 3*e^4*f*x^4 + 12*d*e^3*g*x^4) + a^3*c^ 
2*(12*d^3*e*f + 3*d^4*g + 20*d*e^3*f*x^2 + 30*d^2*e^2*g*x^2 + 15*e^4*g*x^4 
)) + 16*b^2*(2*c^3*d^2*x^3*(-9*e^2*f*x^2 + 2*d*e*x*(20*f - 3*g*x) - 5*d^2* 
(3*f - 2*g*x)) - 3*a*c^2*x*(e^4*f*x^4 + 5*d^4*(f - 2*g*x) + 30*d^2*e^2*x^2 
*(f - g*x) + 4*d*e^3*x^3*(-5*f + g*x) + 20*d^3*e*x*(-2*f + g*x)) + 3*a^2*c 
*(d^4*g + 4*d^3*e*(f - 5*g*x) + 20*d*e^3*x^2*(f - 2*g*x) + 5*e^4*x^3*(-2*f 
 + g*x) + 30*d^2*e^2*x*(-f + g*x)) + 2*a^3*e^2*(9*d^2*g + d*e*(6*f - 40*g* 
x) + 5*e^2*x*(-2*f + 3*g*x))) - 8*b^3*(2*a^2*e*(2*d^3*g + 3*d^2*e*(f - 15* 
g*x) - 30*d*e^2*x*(f - 2*g*x) - 5*e^3*x^2*(-3*f + g*x)) - 2*c^2*d*x^2*(2*e 
^3*f*x^3 - 5*d^3*(f - 3*g*x) + 3*d*e^2*x^2*(-15*f + g*x) - 30*d^2*e*x*(-2* 
f + g*x)) - 5*a*c*(d - e*x)^2*(14*d*e*x*(f - g*x) - e^2*x^2*(3*f + g*x) + 
d^2*(f + 3*g*x))) - 16*b*(15*a^2*c^2*(d - e*x)^4*(f - g*x) - 8*c^4*d^3*x^4 
*(-5*d*f + 4*e*f*x + d*g*x) + 8*a^4*e^3*(4*d*g + e*(f - 5*g*x)) + 4*a^3*c* 
e*(6*d^3*g + 3*d^2*e*(3*f - 5*g*x) + 5*e^3*x^2*(f - 3*g*x) - 10*d*e^2*x*(f 
 - 2*g*x)) - 4*a*c^3*d*x^2*(6*e^3*f*x^3 - 10*d^2*e*x*(-2*f + g*x) + 3*d*e^ 
2*x^2*(-5*f + 3*g*x) + d^3*(-15*f + 5*g*x))) - 2*b^4*(-(c*x*(e^4*f*x^4 ...
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1227, 1153, 1158}

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.

\(\displaystyle \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 1227

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

\(\Big \downarrow \) 1153

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

\(\Big \downarrow \) 1158

\(\displaystyle -\frac {8 \left (\frac {16 \left (a e^2-b d e+c d^2\right ) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\right ) (2 a e g-b d g-b e f+2 c d f)}{5 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^4 (-2 a g+x (2 c f-b g)+b f)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\)

Input:

Int[((d + e*x)^4*(f + g*x))/(a + b*x + c*x^2)^(7/2),x]
 

Output:

(-2*(d + e*x)^4*(b*f - 2*a*g + (2*c*f - b*g)*x))/(5*(b^2 - 4*a*c)*(a + b*x 
 + c*x^2)^(5/2)) - (8*(2*c*d*f - b*e*f - b*d*g + 2*a*e*g)*((-2*(d + e*x)^2 
*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2) 
) + (16*(c*d^2 - b*d*e + a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(3*(b^2 - 
 4*a*c)^2*Sqrt[a + b*x + c*x^2])))/(5*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 1153

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*(2*p + 3)*((c*d^2 - 
b*d*e + a*e^2)/((p + 1)*(b^2 - 4*a*c)))   Int[(d + e*x)^(m - 2)*(a + b*x + 
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] 
&& LtQ[p, -1]
 

rule 1158

Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo 
l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x 
+ c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x]
 

rule 1227

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( 
(b*f - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[m*((b*( 
e*f + d*g) - 2*(c*d*f + a*e*g))/((p + 1)*(b^2 - 4*a*c)))   Int[(d + e*x)^(m 
 - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] 
&& EqQ[Simplify[m + 2*p + 3], 0] && LtQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1891\) vs. \(2(202)=404\).

Time = 2.14 (sec) , antiderivative size = 1892, normalized size of antiderivative = 8.84

method	result	size
trager	\(\text {Expression too large to display}\)	\(1892\)
gosper	\(\text {Expression too large to display}\)	\(1914\)
orering	\(\text {Expression too large to display}\)	\(1914\)
default	\(\text {Expression too large to display}\)	\(3082\)

Input:

int((e*x+d)^4*(g*x+f)/(c*x^2+b*x+a)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-2/15*(240*a^2*b*c^2*e^4*g*x^5-384*a^2*c^3*d*e^3*g*x^5-96*a^2*c^3*e^4*f*x^ 
5-40*a*b^3*c*e^4*g*x^5-192*a*b^2*c^2*d*e^3*g*x^5-48*a*b^2*c^2*e^4*f*x^5+57 
6*a*b*c^3*d^2*e^2*g*x^5+384*a*b*c^3*d*e^3*f*x^5-256*a*c^4*d^3*e*g*x^5-384* 
a*c^4*d^2*e^2*f*x^5+3*b^5*e^4*g*x^5+8*b^4*c*d*e^3*g*x^5+2*b^4*c*e^4*f*x^5+ 
48*b^3*c^2*d^2*e^2*g*x^5+32*b^3*c^2*d*e^3*f*x^5-192*b^2*c^3*d^3*e*g*x^5-28 
8*b^2*c^3*d^2*e^2*f*x^5+128*b*c^4*d^4*g*x^5+512*b*c^4*d^3*e*f*x^5-256*c^5* 
d^4*f*x^5+480*a^3*c^2*e^4*g*x^4+240*a^2*b^2*c*e^4*g*x^4-960*a^2*b*c^2*d*e^ 
3*g*x^4-240*a^2*b*c^2*e^4*f*x^4-10*a*b^4*e^4*g*x^4-480*a*b^3*c*d*e^3*g*x^4 
-120*a*b^3*c*e^4*f*x^4+1440*a*b^2*c^2*d^2*e^2*g*x^4+960*a*b^2*c^2*d*e^3*f* 
x^4-640*a*b*c^3*d^3*e*g*x^4-960*a*b*c^3*d^2*e^2*f*x^4+20*b^5*d*e^3*g*x^4+5 
*b^5*e^4*f*x^4+120*b^4*c*d^2*e^2*g*x^4+80*b^4*c*d*e^3*f*x^4-480*b^3*c^2*d^ 
3*e*g*x^4-720*b^3*c^2*d^2*e^2*f*x^4+320*b^2*c^3*d^4*g*x^4+1280*b^2*c^3*d^3 
*e*f*x^4-640*b*c^4*d^4*f*x^4+960*a^3*b*c*e^4*g*x^3+80*a^2*b^3*e^4*g*x^3-19 
20*a^2*b^2*c*d*e^3*g*x^3-480*a^2*b^2*c*e^4*f*x^3+1440*a^2*b*c^2*d^2*e^2*g* 
x^3+960*a^2*b*c^2*d*e^3*f*x^3-640*a^2*c^3*d^3*e*g*x^3-960*a^2*c^3*d^2*e^2* 
f*x^3-160*a*b^4*d*e^3*g*x^3-40*a*b^4*e^4*f*x^3+1200*a*b^3*c*d^2*e^2*g*x^3+ 
800*a*b^3*c*d*e^3*f*x^3-960*a*b^2*c^2*d^3*e*g*x^3-1440*a*b^2*c^2*d^2*e^2*f 
*x^3+320*a*b*c^3*d^4*g*x^3+1280*a*b*c^3*d^3*e*f*x^3-640*a*c^4*d^4*f*x^3+90 
*b^5*d^2*e^2*g*x^3+60*b^5*d*e^3*f*x^3-360*b^4*c*d^3*e*g*x^3-540*b^4*c*d^2* 
e^2*f*x^3+240*b^3*c^2*d^4*g*x^3+960*b^3*c^2*d^3*e*f*x^3-480*b^2*c^3*d^4...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1676 vs. \(2 (202) = 404\).

Time = 35.26 (sec) , antiderivative size = 1676, normalized size of antiderivative = 7.83 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^4*(g*x+f)/(c*x^2+b*x+a)^(7/2),x, algorithm="fricas")
 

Output:

-2/15*((2*(128*c^5*d^4 - 256*b*c^4*d^3*e + 48*(3*b^2*c^3 + 4*a*c^4)*d^2*e^ 
2 - 16*(b^3*c^2 + 12*a*b*c^3)*d*e^3 - (b^4*c - 24*a*b^2*c^2 - 48*a^2*c^3)* 
e^4)*f - (128*b*c^4*d^4 - 64*(3*b^2*c^3 + 4*a*c^4)*d^3*e + 48*(b^3*c^2 + 1 
2*a*b*c^3)*d^2*e^2 + 8*(b^4*c - 24*a*b^2*c^2 - 48*a^2*c^3)*d*e^3 + (3*b^5 
- 40*a*b^3*c + 240*a^2*b*c^2)*e^4)*g)*x^5 + 5*((128*b*c^4*d^4 - 256*b^2*c^ 
3*d^3*e + 48*(3*b^3*c^2 + 4*a*b*c^3)*d^2*e^2 - 16*(b^4*c + 12*a*b^2*c^2)*d 
*e^3 - (b^5 - 24*a*b^3*c - 48*a^2*b*c^2)*e^4)*f - 2*(32*b^2*c^3*d^4 - 16*( 
3*b^3*c^2 + 4*a*b*c^3)*d^3*e + 12*(b^4*c + 12*a*b^2*c^2)*d^2*e^2 + 2*(b^5 
- 24*a*b^3*c - 48*a^2*b*c^2)*d*e^3 - (a*b^4 - 24*a^2*b^2*c - 48*a^3*c^2)*e 
^4)*g)*x^4 + 10*(2*(8*(3*b^2*c^3 + 4*a*c^4)*d^4 - 16*(3*b^3*c^2 + 4*a*b*c^ 
3)*d^3*e + 3*(9*b^4*c + 24*a*b^2*c^2 + 16*a^2*c^3)*d^2*e^2 - (3*b^5 + 40*a 
*b^3*c + 48*a^2*b*c^2)*d*e^3 + 2*(a*b^4 + 12*a^2*b^2*c)*e^4)*f - (8*(3*b^3 
*c^2 + 4*a*b*c^3)*d^4 - 4*(9*b^4*c + 24*a*b^2*c^2 + 16*a^2*c^3)*d^3*e + 3* 
(3*b^5 + 40*a*b^3*c + 48*a^2*b*c^2)*d^2*e^2 - 16*(a*b^4 + 12*a^2*b^2*c)*d* 
e^3 + 8*(a^2*b^3 + 12*a^3*b*c)*e^4)*g)*x^3 + 10*((8*(b^3*c^2 + 12*a*b*c^3) 
*d^4 - 16*(b^4*c + 12*a*b^2*c^2)*d^3*e + 3*(3*b^5 + 40*a*b^3*c + 48*a^2*b* 
c^2)*d^2*e^2 - 4*(9*a*b^4 + 24*a^2*b^2*c + 16*a^3*c^2)*d*e^3 + 8*(3*a^2*b^ 
3 + 4*a^3*b*c)*e^4)*f - 2*(2*(b^4*c + 12*a*b^2*c^2)*d^4 - (3*b^5 + 40*a*b^ 
3*c + 48*a^2*b*c^2)*d^3*e + 3*(9*a*b^4 + 24*a^2*b^2*c + 16*a^3*c^2)*d^2*e^ 
2 - 16*(3*a^2*b^3 + 4*a^3*b*c)*d*e^3 + 8*(3*a^3*b^2 + 4*a^4*c)*e^4)*g)*...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**4*(g*x+f)/(c*x**2+b*x+a)**(7/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^4*(g*x+f)/(c*x^2+b*x+a)^(7/2),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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1816 vs. \(2 (202) = 404\).

Time = 0.34 (sec) , antiderivative size = 1816, normalized size of antiderivative = 8.49 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^4*(g*x+f)/(c*x^2+b*x+a)^(7/2),x, algorithm="giac")
 

Output:

-2/15*((((((256*c^5*d^4*f - 512*b*c^4*d^3*e*f + 288*b^2*c^3*d^2*e^2*f + 38 
4*a*c^4*d^2*e^2*f - 32*b^3*c^2*d*e^3*f - 384*a*b*c^3*d*e^3*f - 2*b^4*c*e^4 
*f + 48*a*b^2*c^2*e^4*f + 96*a^2*c^3*e^4*f - 128*b*c^4*d^4*g + 192*b^2*c^3 
*d^3*e*g + 256*a*c^4*d^3*e*g - 48*b^3*c^2*d^2*e^2*g - 576*a*b*c^3*d^2*e^2* 
g - 8*b^4*c*d*e^3*g + 192*a*b^2*c^2*d*e^3*g + 384*a^2*c^3*d*e^3*g - 3*b^5* 
e^4*g + 40*a*b^3*c*e^4*g - 240*a^2*b*c^2*e^4*g)*x/(b^6 - 12*a*b^4*c + 48*a 
^2*b^2*c^2 - 64*a^3*c^3) + 5*(128*b*c^4*d^4*f - 256*b^2*c^3*d^3*e*f + 144* 
b^3*c^2*d^2*e^2*f + 192*a*b*c^3*d^2*e^2*f - 16*b^4*c*d*e^3*f - 192*a*b^2*c 
^2*d*e^3*f - b^5*e^4*f + 24*a*b^3*c*e^4*f + 48*a^2*b*c^2*e^4*f - 64*b^2*c^ 
3*d^4*g + 96*b^3*c^2*d^3*e*g + 128*a*b*c^3*d^3*e*g - 24*b^4*c*d^2*e^2*g - 
288*a*b^2*c^2*d^2*e^2*g - 4*b^5*d*e^3*g + 96*a*b^3*c*d*e^3*g + 192*a^2*b*c 
^2*d*e^3*g + 2*a*b^4*e^4*g - 48*a^2*b^2*c*e^4*g - 96*a^3*c^2*e^4*g)/(b^6 - 
 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x + 10*(48*b^2*c^3*d^4*f + 64* 
a*c^4*d^4*f - 96*b^3*c^2*d^3*e*f - 128*a*b*c^3*d^3*e*f + 54*b^4*c*d^2*e^2* 
f + 144*a*b^2*c^2*d^2*e^2*f + 96*a^2*c^3*d^2*e^2*f - 6*b^5*d*e^3*f - 80*a* 
b^3*c*d*e^3*f - 96*a^2*b*c^2*d*e^3*f + 4*a*b^4*e^4*f + 48*a^2*b^2*c*e^4*f 
- 24*b^3*c^2*d^4*g - 32*a*b*c^3*d^4*g + 36*b^4*c*d^3*e*g + 96*a*b^2*c^2*d^ 
3*e*g + 64*a^2*c^3*d^3*e*g - 9*b^5*d^2*e^2*g - 120*a*b^3*c*d^2*e^2*g - 144 
*a^2*b*c^2*d^2*e^2*g + 16*a*b^4*d*e^3*g + 192*a^2*b^2*c*d*e^3*g - 8*a^2*b^ 
3*e^4*g - 96*a^3*b*c*e^4*g)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3...
 

Mupad [B] (verification not implemented)

Time = 19.84 (sec) , antiderivative size = 7970, normalized size of antiderivative = 37.24 \[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

int(((f + g*x)*(d + e*x)^4)/(a + b*x + c*x^2)^(7/2),x)
 

Output:

(x*((a*((16*c*e^3*(4*c*d*g - b*e*g + c*e*f))/(5*(4*a*c^2 - b^2*c)*(4*a*c - 
 b^2)) - (8*b*c*e^4*g)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (b*((b*(( 
16*c*e^3*(4*c*d*g - b*e*g + c*e*f))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - 
(8*b*c*e^4*g)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))))/c - (8*e^2*(5*b^2*e^2* 
g + 12*c^2*d^2*g - 14*a*c*e^2*g - 2*b*c*e^2*f + 8*c^2*d*e*f - 8*b*c*d*e*g) 
)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*e^3*(4*c*d*g - b*e*g + c*e*f) 
)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) + (16*a*c*e^4*g)/(5*(4*a*c^2 - b^2*c 
)*(4*a*c - b^2))))/c + (2*(256*c^5*d^4*f + 16*b^5*e^4*g + 160*a^2*c^3*e^4* 
f - 128*b*c^4*d^4*g - 4*b^4*c*e^4*f - 80*a*b^3*c*e^4*g + 128*a*c^4*d^3*e*g 
 - 512*b*c^4*d^3*e*f - 16*b^4*c*d*e^3*g + 40*a*b^2*c^2*e^4*f - 64*a^2*b*c^ 
2*e^4*g + 192*a*c^4*d^2*e^2*f + 640*a^2*c^3*d*e^3*g - 32*b^3*c^2*d*e^3*f + 
 224*b^2*c^3*d^3*e*g + 336*b^2*c^3*d^2*e^2*f - 48*b^3*c^2*d^2*e^2*g - 384* 
a*b*c^3*d*e^3*f - 576*a*b*c^3*d^2*e^2*g + 160*a*b^2*c^2*d*e^3*g))/(15*c*(4 
*a*c^2 - b^2*c)*(4*a*c - b^2)^2) - (4*b*e^2*(5*b^2*e^2*g + 12*c^2*d^2*g - 
14*a*c*e^2*g - 2*b*c*e^2*f + 8*c^2*d*e*f - 8*b*c*d*e*g))/(5*c*(4*a*c^2 - b 
^2*c)*(4*a*c - b^2))) - (a*((b*((16*c*e^3*(4*c*d*g - b*e*g + c*e*f))/(5*(4 
*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8*b*c*e^4*g)/(5*(4*a*c^2 - b^2*c)*(4*a*c 
 - b^2))))/c - (8*e^2*(5*b^2*e^2*g + 12*c^2*d^2*g - 14*a*c*e^2*g - 2*b*c*e 
^2*f + 8*c^2*d*e*f - 8*b*c*d*e*g))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - ( 
8*b*e^3*(4*c*d*g - b*e*g + c*e*f))/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) ...
 

Reduce [F]

\[ \int \frac {(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\int \frac {\left (e x +d \right )^{4} \left (g x +f \right )}{\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}d x \] Input:

int((e*x+d)^4*(g*x+f)/(c*x^2+b*x+a)^(7/2),x)
 

Output:

int((e*x+d)^4*(g*x+f)/(c*x^2+b*x+a)^(7/2),x)