\(\int \frac {(d+e x)^3 (f+g x)}{(a+b x+c x^2)^{5/2}} \, dx\) [988]

Optimal result

Integrand size = 27, antiderivative size = 324 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^3 (b f-2 a g+(2 c f-b g) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {4 (d+e x)^2 \left (4 a c e f-b^2 (3 e f+2 d g)+4 b (c d f+a e g)+\left (8 c^2 d f-b^2 e g-4 c (b e f+b d g-3 a e g)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 e \left (32 c^3 d^2 f+3 b^3 e^2 g-20 a b c e^2 g-8 c^2 (b d (3 e f+2 d g)-2 a e (e f+3 d g))+2 c e \left (8 c^2 d f-b^2 e g-4 c (b e f+b d g-3 a e g)\right ) x\right ) \sqrt {a+b x+c x^2}}{3 c^2 \left (b^2-4 a c\right )^2}+\frac {e^3 g \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{5/2}} \] Output:

-2/3*(e*x+d)^3*(b*f-2*a*g+(-b*g+2*c*f)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(3/2) 
+4/3*(e*x+d)^2*(4*a*c*e*f-b^2*(2*d*g+3*e*f)+4*b*(a*e*g+c*d*f)+(8*c^2*d*f-b 
^2*e*g-4*c*(-3*a*e*g+b*d*g+b*e*f))*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(1/2)-2 
/3*e*(32*c^3*d^2*f+3*b^3*e^2*g-20*a*b*c*e^2*g-8*c^2*(b*d*(2*d*g+3*e*f)-2*a 
*e*(3*d*g+e*f))+2*c*e*(8*c^2*d*f-b^2*e*g-4*c*(-3*a*e*g+b*d*g+b*e*f))*x)*(c 
*x^2+b*x+a)^(1/2)/c^2/(-4*a*c+b^2)^2+e^3*g*arctanh(1/2*(2*c*x+b)/c^(1/2)/( 
c*x^2+b*x+a)^(1/2))/c^(5/2)
 

Mathematica [A] (verified)

Time = 5.28 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.61 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (3 b^5 e^3 g x^2+2 b^4 e^3 g x \left (3 a+2 c x^2\right )-2 b^2 c \left (21 a^2 e^3 g x+3 c^2 d x \left (e^2 f x^2+d^2 (f-2 g x)+d e x (-6 f+g x)\right )-a c \left (d^3 g+e^3 x^2 (3 f-14 g x)+3 d^2 e (f-6 g x)+9 d e^2 x (-2 f+g x)\right )\right )+4 b c \left (-5 a^3 e^3 g+2 c^3 d^2 x^2 (-3 d f+3 e f x+d g x)-6 a^2 c e \left (d^2 g-e^2 f x+d e (f-3 g x)\right )+3 a c^2 \left (e^3 f x^3+3 d^2 e x (f-g x)+d^3 (-f+g x)+3 d e^2 x^2 (-f+g x)\right )\right )+b^3 \left (3 a^2 e^3 g-18 a c e^3 g x^2+c^2 \left (-e^3 f x^3+9 d^2 e x (f-g x)-3 d e^2 x^2 (3 f+g x)+d^3 (f+3 g x)\right )\right )+8 c^2 \left (-2 c^3 d^3 f x^3+a^3 e^2 (2 e f+6 d g+3 e g x)-3 a c^2 d x \left (d^2 f+e^2 f x^2+d e g x^2\right )+a^2 c \left (3 d^2 e f+d^3 g+9 d e^2 g x^2+e^3 x^2 (3 f+4 g x)\right )\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}-\frac {e^3 g \log \left (c^2 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{c^{5/2}} \] Input:

Integrate[((d + e*x)^3*(f + g*x))/(a + b*x + c*x^2)^(5/2),x]
 

Output:

(-2*(3*b^5*e^3*g*x^2 + 2*b^4*e^3*g*x*(3*a + 2*c*x^2) - 2*b^2*c*(21*a^2*e^3 
*g*x + 3*c^2*d*x*(e^2*f*x^2 + d^2*(f - 2*g*x) + d*e*x*(-6*f + g*x)) - a*c* 
(d^3*g + e^3*x^2*(3*f - 14*g*x) + 3*d^2*e*(f - 6*g*x) + 9*d*e^2*x*(-2*f + 
g*x))) + 4*b*c*(-5*a^3*e^3*g + 2*c^3*d^2*x^2*(-3*d*f + 3*e*f*x + d*g*x) - 
6*a^2*c*e*(d^2*g - e^2*f*x + d*e*(f - 3*g*x)) + 3*a*c^2*(e^3*f*x^3 + 3*d^2 
*e*x*(f - g*x) + d^3*(-f + g*x) + 3*d*e^2*x^2*(-f + g*x))) + b^3*(3*a^2*e^ 
3*g - 18*a*c*e^3*g*x^2 + c^2*(-(e^3*f*x^3) + 9*d^2*e*x*(f - g*x) - 3*d*e^2 
*x^2*(3*f + g*x) + d^3*(f + 3*g*x))) + 8*c^2*(-2*c^3*d^3*f*x^3 + a^3*e^2*( 
2*e*f + 6*d*g + 3*e*g*x) - 3*a*c^2*d*x*(d^2*f + e^2*f*x^2 + d*e*g*x^2) + a 
^2*c*(3*d^2*e*f + d^3*g + 9*d*e^2*g*x^2 + e^3*x^2*(3*f + 4*g*x)))))/(3*c^2 
*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2)) - (e^3*g*Log[c^2*(b + 2*c*x - 2* 
Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(5/2)
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1233, 27, 1224, 1092, 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.

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

\(\Big \downarrow \) 1233

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1224

\(\displaystyle \frac {2 (d+e x)^2 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {-\frac {3 e^3 g \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{c}-\frac {2 \left (x \left (-2 c^2 e \left (12 a^2 e^2 g+4 a b e (4 d g+e f)+b^2 (-d) (3 d g+4 e f)\right )+2 b^2 c e^2 g (11 a e+b d)-8 c^3 d (b d (d g+3 e f)-a e (3 d g+2 e f))-3 b^4 e^3 g+16 c^4 d^3 f\right )+4 b c \left (5 a^2 e^3 g+a c d e (5 d g+6 e f)+2 c^2 d^3 f\right )+b^3 e g \left (c d^2-3 a e^2\right )-16 a c^2 e \left (a e (3 d g+e f)+c d^2 f\right )-4 b^2 c^2 d^2 (d g+2 e f)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}}{3 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {2 (d+e x)^2 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {-\frac {6 e^3 g \left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c}-\frac {2 \left (x \left (-2 c^2 e \left (12 a^2 e^2 g+4 a b e (4 d g+e f)+b^2 (-d) (3 d g+4 e f)\right )+2 b^2 c e^2 g (11 a e+b d)-8 c^3 d (b d (d g+3 e f)-a e (3 d g+2 e f))-3 b^4 e^3 g+16 c^4 d^3 f\right )+4 b c \left (5 a^2 e^3 g+a c d e (5 d g+6 e f)+2 c^2 d^3 f\right )+b^3 e g \left (c d^2-3 a e^2\right )-16 a c^2 e \left (a e (3 d g+e f)+c d^2 f\right )-4 b^2 c^2 d^2 (d g+2 e f)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}}{3 c \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 (d+e x)^2 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {-\frac {2 \left (x \left (-2 c^2 e \left (12 a^2 e^2 g+4 a b e (4 d g+e f)+b^2 (-d) (3 d g+4 e f)\right )+2 b^2 c e^2 g (11 a e+b d)-8 c^3 d (b d (d g+3 e f)-a e (3 d g+2 e f))-3 b^4 e^3 g+16 c^4 d^3 f\right )+4 b c \left (5 a^2 e^3 g+a c d e (5 d g+6 e f)+2 c^2 d^3 f\right )+b^3 e g \left (c d^2-3 a e^2\right )-16 a c^2 e \left (a e (3 d g+e f)+c d^2 f\right )-4 b^2 c^2 d^2 (d g+2 e f)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {3 e^3 g \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}}}{3 c \left (b^2-4 a c\right )}\)

Input:

Int[((d + e*x)^3*(f + g*x))/(a + b*x + c*x^2)^(5/2),x]
 

Output:

(2*(d + e*x)^2*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e 
*g - c*(b*e*f + b*d*g + 2*a*e*g))*x))/(3*c*(b^2 - 4*a*c)*(a + b*x + c*x^2) 
^(3/2)) - ((-2*(b^3*e*(c*d^2 - 3*a*e^2)*g - 4*b^2*c^2*d^2*(2*e*f + d*g) - 
16*a*c^2*e*(c*d^2*f + a*e*(e*f + 3*d*g)) + 4*b*c*(2*c^2*d^3*f + 5*a^2*e^3* 
g + a*c*d*e*(6*e*f + 5*d*g)) + (16*c^4*d^3*f - 3*b^4*e^3*g + 2*b^2*c*e^2*( 
b*d + 11*a*e)*g - 8*c^3*d*(b*d*(3*e*f + d*g) - a*e*(2*e*f + 3*d*g)) - 2*c^ 
2*e*(12*a^2*e^2*g - b^2*d*(4*e*f + 3*d*g) + 4*a*b*e*(e*f + 4*d*g)))*x))/(c 
*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) - (3*(b^2 - 4*a*c)*e^3*g*ArcTanh[(b 
+ 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(3/2))/(3*c*(b^2 - 4*a*c))
 

Defintions of rubi rules used

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 1224

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

rule 1233

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 + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c 
*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( 
p + 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim 
p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f 
*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( 
m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* 
p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && 
GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | 
|  !ILtQ[m + 2*p + 3, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1148\) vs. \(2(304)=608\).

Time = 2.32 (sec) , antiderivative size = 1149, normalized size of antiderivative = 3.55

method	result	size
default	\(\text {Expression too large to display}\)	\(1149\)

Input:

int((e*x+d)^3*(g*x+f)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 972 vs. \(2 (304) = 608\).

Time = 6.15 (sec) , antiderivative size = 1947, normalized size of antiderivative = 6.01 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^3*(g*x+f)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/6*(3*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^3*g*x^4 + 2*(b^5*c - 8*a*b 
^3*c^2 + 16*a^2*b*c^3)*e^3*g*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*e^3*g*x^ 
2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e^3*g*x + (a^2*b^4 - 8*a^3*b^2* 
c + 16*a^4*c^2)*e^3*g)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x 
^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(((16*c^6*d^3 - 24*b*c^5*d^ 
2*e + 6*(b^2*c^4 + 4*a*c^5)*d*e^2 + (b^3*c^3 - 12*a*b*c^4)*e^3)*f - (8*b*c 
^5*d^3 - 6*(b^2*c^4 + 4*a*c^5)*d^2*e - 3*(b^3*c^3 - 12*a*b*c^4)*d*e^2 + 4* 
(b^4*c^2 - 7*a*b^2*c^3 + 8*a^2*c^4)*e^3)*g)*x^3 + 3*((8*b*c^5*d^3 - 12*b^2 
*c^4*d^2*e + 3*(b^3*c^3 + 4*a*b*c^4)*d*e^2 - 2*(a*b^2*c^3 + 4*a^2*c^4)*e^3 
)*f - (4*b^2*c^4*d^3 - 3*(b^3*c^3 + 4*a*b*c^4)*d^2*e + 6*(a*b^2*c^3 + 4*a^ 
2*c^4)*d*e^2 + (b^5*c - 6*a*b^3*c^2)*e^3)*g)*x^2 + (24*a^2*b*c^3*d*e^2 - 1 
6*a^3*c^3*e^3 - (b^3*c^3 - 12*a*b*c^4)*d^3 - 6*(a*b^2*c^3 + 4*a^2*c^4)*d^2 
*e)*f + (24*a^2*b*c^3*d^2*e - 48*a^3*c^3*d*e^2 - 2*(a*b^2*c^3 + 4*a^2*c^4) 
*d^3 - (3*a^2*b^3*c - 20*a^3*b*c^2)*e^3)*g + 3*((12*a*b^2*c^3*d*e^2 - 8*a^ 
2*b*c^3*e^3 + 2*(b^2*c^4 + 4*a*c^5)*d^3 - 3*(b^3*c^3 + 4*a*b*c^4)*d^2*e)*f 
 + (12*a*b^2*c^3*d^2*e - 24*a^2*b*c^3*d*e^2 - (b^3*c^3 + 4*a*b*c^4)*d^3 - 
2*(a*b^4*c - 7*a^2*b^2*c^2 + 4*a^3*c^3)*e^3)*g)*x)*sqrt(c*x^2 + b*x + a))/ 
(a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^ 
2*c^7)*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^3 + (b^6*c^3 - 6*a 
*b^4*c^4 + 32*a^3*c^6)*x^2 + 2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**3*(g*x+f)/(c*x**2+b*x+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x)^3 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)^3*(g*x+f)/(c*x^2+b*x+a)^(5/2),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. 807 vs. \(2 (304) = 608\).

Time = 0.42 (sec) , antiderivative size = 807, normalized size of antiderivative = 2.49 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {e^{3} g \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {5}{2}}} + \frac {2 \, {\left ({\left ({\left (\frac {{\left (16 \, c^{5} d^{3} f - 24 \, b c^{4} d^{2} e f + 6 \, b^{2} c^{3} d e^{2} f + 24 \, a c^{4} d e^{2} f + b^{3} c^{2} e^{3} f - 12 \, a b c^{3} e^{3} f - 8 \, b c^{4} d^{3} g + 6 \, b^{2} c^{3} d^{2} e g + 24 \, a c^{4} d^{2} e g + 3 \, b^{3} c^{2} d e^{2} g - 36 \, a b c^{3} d e^{2} g - 4 \, b^{4} c e^{3} g + 28 \, a b^{2} c^{2} e^{3} g - 32 \, a^{2} c^{3} e^{3} g\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac {3 \, {\left (8 \, b c^{4} d^{3} f - 12 \, b^{2} c^{3} d^{2} e f + 3 \, b^{3} c^{2} d e^{2} f + 12 \, a b c^{3} d e^{2} f - 2 \, a b^{2} c^{2} e^{3} f - 8 \, a^{2} c^{3} e^{3} f - 4 \, b^{2} c^{3} d^{3} g + 3 \, b^{3} c^{2} d^{2} e g + 12 \, a b c^{3} d^{2} e g - 6 \, a b^{2} c^{2} d e^{2} g - 24 \, a^{2} c^{3} d e^{2} g - b^{5} e^{3} g + 6 \, a b^{3} c e^{3} g\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac {3 \, {\left (2 \, b^{2} c^{3} d^{3} f + 8 \, a c^{4} d^{3} f - 3 \, b^{3} c^{2} d^{2} e f - 12 \, a b c^{3} d^{2} e f + 12 \, a b^{2} c^{2} d e^{2} f - 8 \, a^{2} b c^{2} e^{3} f - b^{3} c^{2} d^{3} g - 4 \, a b c^{3} d^{3} g + 12 \, a b^{2} c^{2} d^{2} e g - 24 \, a^{2} b c^{2} d e^{2} g - 2 \, a b^{4} e^{3} g + 14 \, a^{2} b^{2} c e^{3} g - 8 \, a^{3} c^{2} e^{3} g\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac {b^{3} c^{2} d^{3} f - 12 \, a b c^{3} d^{3} f + 6 \, a b^{2} c^{2} d^{2} e f + 24 \, a^{2} c^{3} d^{2} e f - 24 \, a^{2} b c^{2} d e^{2} f + 16 \, a^{3} c^{2} e^{3} f + 2 \, a b^{2} c^{2} d^{3} g + 8 \, a^{2} c^{3} d^{3} g - 24 \, a^{2} b c^{2} d^{2} e g + 48 \, a^{3} c^{2} d e^{2} g + 3 \, a^{2} b^{3} e^{3} g - 20 \, a^{3} b c e^{3} g}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \] Input:

integrate((e*x+d)^3*(g*x+f)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

-e^3*g*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(5/2) 
 + 2/3*((((16*c^5*d^3*f - 24*b*c^4*d^2*e*f + 6*b^2*c^3*d*e^2*f + 24*a*c^4* 
d*e^2*f + b^3*c^2*e^3*f - 12*a*b*c^3*e^3*f - 8*b*c^4*d^3*g + 6*b^2*c^3*d^2 
*e*g + 24*a*c^4*d^2*e*g + 3*b^3*c^2*d*e^2*g - 36*a*b*c^3*d*e^2*g - 4*b^4*c 
*e^3*g + 28*a*b^2*c^2*e^3*g - 32*a^2*c^3*e^3*g)*x/(b^4*c^2 - 8*a*b^2*c^3 + 
 16*a^2*c^4) + 3*(8*b*c^4*d^3*f - 12*b^2*c^3*d^2*e*f + 3*b^3*c^2*d*e^2*f + 
 12*a*b*c^3*d*e^2*f - 2*a*b^2*c^2*e^3*f - 8*a^2*c^3*e^3*f - 4*b^2*c^3*d^3* 
g + 3*b^3*c^2*d^2*e*g + 12*a*b*c^3*d^2*e*g - 6*a*b^2*c^2*d*e^2*g - 24*a^2* 
c^3*d*e^2*g - b^5*e^3*g + 6*a*b^3*c*e^3*g)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2 
*c^4))*x + 3*(2*b^2*c^3*d^3*f + 8*a*c^4*d^3*f - 3*b^3*c^2*d^2*e*f - 12*a*b 
*c^3*d^2*e*f + 12*a*b^2*c^2*d*e^2*f - 8*a^2*b*c^2*e^3*f - b^3*c^2*d^3*g - 
4*a*b*c^3*d^3*g + 12*a*b^2*c^2*d^2*e*g - 24*a^2*b*c^2*d*e^2*g - 2*a*b^4*e^ 
3*g + 14*a^2*b^2*c*e^3*g - 8*a^3*c^2*e^3*g)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^ 
2*c^4))*x - (b^3*c^2*d^3*f - 12*a*b*c^3*d^3*f + 6*a*b^2*c^2*d^2*e*f + 24*a 
^2*c^3*d^2*e*f - 24*a^2*b*c^2*d*e^2*f + 16*a^3*c^2*e^3*f + 2*a*b^2*c^2*d^3 
*g + 8*a^2*c^3*d^3*g - 24*a^2*b*c^2*d^2*e*g + 48*a^3*c^2*d*e^2*g + 3*a^2*b 
^3*e^3*g - 20*a^3*b*c*e^3*g)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 
+ b*x + a)^(3/2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (d+e\,x\right )}^3}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \] Input:

int(((f + g*x)*(d + e*x)^3)/(a + b*x + c*x^2)^(5/2),x)
 

Output:

int(((f + g*x)*(d + e*x)^3)/(a + b*x + c*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 3456, normalized size of antiderivative = 10.67 \[ \int \frac {(d+e x)^3 (f+g x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int((e*x+d)^3*(g*x+f)/(c*x^2+b*x+a)^(5/2),x)
 

Output:

(40*sqrt(a + b*x + c*x**2)*a**3*b*c**2*e**3*g - 96*sqrt(a + b*x + c*x**2)* 
a**3*c**3*d*e**2*g - 32*sqrt(a + b*x + c*x**2)*a**3*c**3*e**3*f - 48*sqrt( 
a + b*x + c*x**2)*a**3*c**3*e**3*g*x - 6*sqrt(a + b*x + c*x**2)*a**2*b**3* 
c*e**3*g + 84*sqrt(a + b*x + c*x**2)*a**2*b**2*c**2*e**3*g*x + 48*sqrt(a + 
 b*x + c*x**2)*a**2*b*c**3*d**2*e*g + 48*sqrt(a + b*x + c*x**2)*a**2*b*c** 
3*d*e**2*f - 144*sqrt(a + b*x + c*x**2)*a**2*b*c**3*d*e**2*g*x - 48*sqrt(a 
 + b*x + c*x**2)*a**2*b*c**3*e**3*f*x - 16*sqrt(a + b*x + c*x**2)*a**2*c** 
4*d**3*g - 48*sqrt(a + b*x + c*x**2)*a**2*c**4*d**2*e*f - 144*sqrt(a + b*x 
 + c*x**2)*a**2*c**4*d*e**2*g*x**2 - 48*sqrt(a + b*x + c*x**2)*a**2*c**4*e 
**3*f*x**2 - 64*sqrt(a + b*x + c*x**2)*a**2*c**4*e**3*g*x**3 - 12*sqrt(a + 
 b*x + c*x**2)*a*b**4*c*e**3*g*x + 36*sqrt(a + b*x + c*x**2)*a*b**3*c**2*e 
**3*g*x**2 - 4*sqrt(a + b*x + c*x**2)*a*b**2*c**3*d**3*g - 12*sqrt(a + b*x 
 + c*x**2)*a*b**2*c**3*d**2*e*f + 72*sqrt(a + b*x + c*x**2)*a*b**2*c**3*d* 
*2*e*g*x + 72*sqrt(a + b*x + c*x**2)*a*b**2*c**3*d*e**2*f*x - 36*sqrt(a + 
b*x + c*x**2)*a*b**2*c**3*d*e**2*g*x**2 - 12*sqrt(a + b*x + c*x**2)*a*b**2 
*c**3*e**3*f*x**2 + 56*sqrt(a + b*x + c*x**2)*a*b**2*c**3*e**3*g*x**3 + 24 
*sqrt(a + b*x + c*x**2)*a*b*c**4*d**3*f - 24*sqrt(a + b*x + c*x**2)*a*b*c* 
*4*d**3*g*x - 72*sqrt(a + b*x + c*x**2)*a*b*c**4*d**2*e*f*x + 72*sqrt(a + 
b*x + c*x**2)*a*b*c**4*d**2*e*g*x**2 + 72*sqrt(a + b*x + c*x**2)*a*b*c**4* 
d*e**2*f*x**2 - 72*sqrt(a + b*x + c*x**2)*a*b*c**4*d*e**2*g*x**3 - 24*s...