\(\int (d+e x)^3 (f+g x) \sqrt {a+b x+c x^2} \, dx\) [870]

Optimal result

Integrand size = 27, antiderivative size = 530 \[ \int (d+e x)^3 (f+g x) \sqrt {a+b x+c x^2} \, dx=\frac {\left (128 c^4 d^3 f+21 b^4 e^3 g-28 b^2 c e^2 (b e f+3 b d g+2 a e g)-32 c^3 d (3 a e (e f+d g)+2 b d (3 e f+d g))+8 c^2 e \left (2 a^2 e^2 g+15 b^2 d (e f+d g)+6 a b e (e f+3 d g)\right )\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^5}+\frac {(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{20 c^2}+\frac {g (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}-\frac {\left (105 b^3 e^3 g-64 c^3 d^2 (12 e f+d g)-28 b c e^2 (7 a e g+5 b (e f+3 d g))+8 c^2 e (16 a e (e f+3 d g)+3 b d (25 e f+19 d g))-6 c e \left (21 b^2 e^2 g+8 c^2 d (7 e f+d g)-4 c e (7 b e f+9 b d g+5 a e g)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{960 c^4}-\frac {\left (b^2-4 a c\right ) \left (128 c^4 d^3 f+21 b^4 e^3 g-28 b^2 c e^2 (b e f+3 b d g+2 a e g)-32 c^3 d (3 a e (e f+d g)+2 b d (3 e f+d g))+8 c^2 e \left (2 a^2 e^2 g+15 b^2 d (e f+d g)+6 a b e (e f+3 d g)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{11/2}} \] Output:

1/512*(128*c^4*d^3*f+21*b^4*e^3*g-28*b^2*c*e^2*(2*a*e*g+3*b*d*g+b*e*f)-32* 
c^3*d*(3*a*e*(d*g+e*f)+2*b*d*(d*g+3*e*f))+8*c^2*e*(2*a^2*e^2*g+15*b^2*d*(d 
*g+e*f)+6*a*b*e*(3*d*g+e*f)))*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^5+1/20*(-3*b 
*e*g+2*c*d*g+4*c*e*f)*(e*x+d)^2*(c*x^2+b*x+a)^(3/2)/c^2+1/6*g*(e*x+d)^3*(c 
*x^2+b*x+a)^(3/2)/c-1/960*(105*b^3*e^3*g-64*c^3*d^2*(d*g+12*e*f)-28*b*c*e^ 
2*(7*a*e*g+5*b*(3*d*g+e*f))+8*c^2*e*(16*a*e*(3*d*g+e*f)+3*b*d*(19*d*g+25*e 
*f))-6*c*e*(21*b^2*e^2*g+8*c^2*d*(d*g+7*e*f)-4*c*e*(5*a*e*g+9*b*d*g+7*b*e* 
f))*x)*(c*x^2+b*x+a)^(3/2)/c^4-1/1024*(-4*a*c+b^2)*(128*c^4*d^3*f+21*b^4*e 
^3*g-28*b^2*c*e^2*(2*a*e*g+3*b*d*g+b*e*f)-32*c^3*d*(3*a*e*(d*g+e*f)+2*b*d* 
(d*g+3*e*f))+8*c^2*e*(2*a^2*e^2*g+15*b^2*d*(d*g+e*f)+6*a*b*e*(3*d*g+e*f))) 
*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(11/2)
 

Mathematica [A] (verified)

Time = 5.06 (sec) , antiderivative size = 613, normalized size of antiderivative = 1.16 \[ \int (d+e x)^3 (f+g x) \sqrt {a+b x+c x^2} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (315 b^5 e^3 g-210 b^4 c e^2 (2 e f+6 d g+e g x)+8 b^3 c e \left (-210 a e^2 g+c \left (225 d^2 g+7 e^2 x (5 f+3 g x)+15 d e (15 f+7 g x)\right )\right )-32 c^3 \left (a^2 e^2 (32 e f+96 d g+15 e g x)-4 c^2 x \left (10 d^3 (3 f+2 g x)+15 d^2 e x (4 f+3 g x)+9 d e^2 x^2 (5 f+4 g x)+2 e^3 x^3 (6 f+5 g x)\right )-2 a c \left (40 d^3 g+15 d^2 e (8 f+3 g x)+e^3 x^2 (8 f+5 g x)+3 d e^2 x (15 f+8 g x)\right )\right )-16 b^2 c^2 \left (-a e^2 (115 e f+345 d g+56 e g x)+c \left (60 d^3 g+15 d^2 e (12 f+5 g x)+e^3 x^2 (14 f+9 g x)+3 d e^2 x (25 f+14 g x)\right )\right )+16 b c^2 \left (113 a^2 e^3 g+4 c^2 \left (15 d^2 e x (2 f+g x)+10 d^3 (3 f+g x)+e^3 x^3 (3 f+2 g x)+3 d e^2 x^2 (5 f+3 g x)\right )-2 a c e \left (195 d^2 g+e^2 x (29 f+17 g x)+3 d e (65 f+29 g x)\right )\right )\right )+15 \left (b^2-4 a c\right ) \left (128 c^4 d^3 f+21 b^4 e^3 g-28 b^2 c e^2 (b e f+3 b d g+2 a e g)-32 c^3 d (3 a e (e f+d g)+2 b d (3 e f+d g))+8 c^2 e \left (2 a^2 e^2 g+15 b^2 d (e f+d g)+6 a b e (e f+3 d g)\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{15360 c^{11/2}} \] Input:

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

Output:

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(315*b^5*e^3*g - 210*b^4*c*e^2*(2*e*f + 6 
*d*g + e*g*x) + 8*b^3*c*e*(-210*a*e^2*g + c*(225*d^2*g + 7*e^2*x*(5*f + 3* 
g*x) + 15*d*e*(15*f + 7*g*x))) - 32*c^3*(a^2*e^2*(32*e*f + 96*d*g + 15*e*g 
*x) - 4*c^2*x*(10*d^3*(3*f + 2*g*x) + 15*d^2*e*x*(4*f + 3*g*x) + 9*d*e^2*x 
^2*(5*f + 4*g*x) + 2*e^3*x^3*(6*f + 5*g*x)) - 2*a*c*(40*d^3*g + 15*d^2*e*( 
8*f + 3*g*x) + e^3*x^2*(8*f + 5*g*x) + 3*d*e^2*x*(15*f + 8*g*x))) - 16*b^2 
*c^2*(-(a*e^2*(115*e*f + 345*d*g + 56*e*g*x)) + c*(60*d^3*g + 15*d^2*e*(12 
*f + 5*g*x) + e^3*x^2*(14*f + 9*g*x) + 3*d*e^2*x*(25*f + 14*g*x))) + 16*b* 
c^2*(113*a^2*e^3*g + 4*c^2*(15*d^2*e*x*(2*f + g*x) + 10*d^3*(3*f + g*x) + 
e^3*x^3*(3*f + 2*g*x) + 3*d*e^2*x^2*(5*f + 3*g*x)) - 2*a*c*e*(195*d^2*g + 
e^2*x*(29*f + 17*g*x) + 3*d*e*(65*f + 29*g*x)))) + 15*(b^2 - 4*a*c)*(128*c 
^4*d^3*f + 21*b^4*e^3*g - 28*b^2*c*e^2*(b*e*f + 3*b*d*g + 2*a*e*g) - 32*c^ 
3*d*(3*a*e*(e*f + d*g) + 2*b*d*(3*e*f + d*g)) + 8*c^2*e*(2*a^2*e^2*g + 15* 
b^2*d*(e*f + d*g) + 6*a*b*e*(e*f + 3*d*g)))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt 
[a + x*(b + c*x)]])/(15360*c^(11/2))
 

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 439, normalized size of antiderivative = 0.83, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {1236, 27, 1236, 27, 1225, 1087, 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.

\(\displaystyle \int (d+e x)^3 (f+g x) \sqrt {a+b x+c x^2} \, dx\)

\(\Big \downarrow \) 1236

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\frac {\int \frac {1}{2} (d+e x) \left (40 c^2 f d^2+3 b e (3 b d+4 a e) g-4 c (b d (3 e f+4 d g)+a e (4 e f+7 d g))+\left (8 d (7 e f+d g) c^2-4 e (7 b e f+9 b d g+5 a e g) c+21 b^2 e^2 g\right ) x\right ) \sqrt {c x^2+b x+a}dx}{5 c}+\frac {(d+e x)^2 \left (a+b x+c x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{5 c}}{4 c}+\frac {g (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int (d+e x) \left (40 c^2 f d^2+3 b e (3 b d+4 a e) g-4 c (b d (3 e f+4 d g)+a e (4 e f+7 d g))+\left (8 d (7 e f+d g) c^2-4 e (7 b e f+9 b d g+5 a e g) c+21 b^2 e^2 g\right ) x\right ) \sqrt {c x^2+b x+a}dx}{10 c}+\frac {(d+e x)^2 \left (a+b x+c x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{5 c}}{4 c}+\frac {g (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {\frac {5 \left (8 c^2 e \left (2 a^2 e^2 g+6 a b e (3 d g+e f)+15 b^2 d (d g+e f)\right )-28 b^2 c e^2 (2 a e g+3 b d g+b e f)-32 c^3 d (3 a e (d g+e f)+2 b d (d g+3 e f))+21 b^4 e^3 g+128 c^4 d^3 f\right ) \int \sqrt {c x^2+b x+a}dx}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c e x \left (-4 c e (5 a e g+9 b d g+7 b e f)+21 b^2 e^2 g+8 c^2 d (d g+7 e f)\right )+8 c^2 e (16 a e (3 d g+e f)+3 b d (19 d g+25 e f))-28 b c e^2 (7 a e g+5 b (3 d g+e f))+105 b^3 e^3 g-64 c^3 d^2 (d g+12 e f)\right )}{24 c^2}}{10 c}+\frac {(d+e x)^2 \left (a+b x+c x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{5 c}}{4 c}+\frac {g (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\frac {5 \left (8 c^2 e \left (2 a^2 e^2 g+6 a b e (3 d g+e f)+15 b^2 d (d g+e f)\right )-28 b^2 c e^2 (2 a e g+3 b d g+b e f)-32 c^3 d (3 a e (d g+e f)+2 b d (d g+3 e f))+21 b^4 e^3 g+128 c^4 d^3 f\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c e x \left (-4 c e (5 a e g+9 b d g+7 b e f)+21 b^2 e^2 g+8 c^2 d (d g+7 e f)\right )+8 c^2 e (16 a e (3 d g+e f)+3 b d (19 d g+25 e f))-28 b c e^2 (7 a e g+5 b (3 d g+e f))+105 b^3 e^3 g-64 c^3 d^2 (d g+12 e f)\right )}{24 c^2}}{10 c}+\frac {(d+e x)^2 \left (a+b x+c x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{5 c}}{4 c}+\frac {g (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\frac {5 \left (8 c^2 e \left (2 a^2 e^2 g+6 a b e (3 d g+e f)+15 b^2 d (d g+e f)\right )-28 b^2 c e^2 (2 a e g+3 b d g+b e f)-32 c^3 d (3 a e (d g+e f)+2 b d (d g+3 e f))+21 b^4 e^3 g+128 c^4 d^3 f\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c e x \left (-4 c e (5 a e g+9 b d g+7 b e f)+21 b^2 e^2 g+8 c^2 d (d g+7 e f)\right )+8 c^2 e (16 a e (3 d g+e f)+3 b d (19 d g+25 e f))-28 b c e^2 (7 a e g+5 b (3 d g+e f))+105 b^3 e^3 g-64 c^3 d^2 (d g+12 e f)\right )}{24 c^2}}{10 c}+\frac {(d+e x)^2 \left (a+b x+c x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{5 c}}{4 c}+\frac {g (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {5 \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right ) \left (8 c^2 e \left (2 a^2 e^2 g+6 a b e (3 d g+e f)+15 b^2 d (d g+e f)\right )-28 b^2 c e^2 (2 a e g+3 b d g+b e f)-32 c^3 d (3 a e (d g+e f)+2 b d (d g+3 e f))+21 b^4 e^3 g+128 c^4 d^3 f\right )}{16 c^2}-\frac {\left (a+b x+c x^2\right )^{3/2} \left (-6 c e x \left (-4 c e (5 a e g+9 b d g+7 b e f)+21 b^2 e^2 g+8 c^2 d (d g+7 e f)\right )+8 c^2 e (16 a e (3 d g+e f)+3 b d (19 d g+25 e f))-28 b c e^2 (7 a e g+5 b (3 d g+e f))+105 b^3 e^3 g-64 c^3 d^2 (d g+12 e f)\right )}{24 c^2}}{10 c}+\frac {(d+e x)^2 \left (a+b x+c x^2\right )^{3/2} (-3 b e g+2 c d g+4 c e f)}{5 c}}{4 c}+\frac {g (d+e x)^3 \left (a+b x+c x^2\right )^{3/2}}{6 c}\)

Input:

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

Output:

(g*(d + e*x)^3*(a + b*x + c*x^2)^(3/2))/(6*c) + (((4*c*e*f + 2*c*d*g - 3*b 
*e*g)*(d + e*x)^2*(a + b*x + c*x^2)^(3/2))/(5*c) + (-1/24*((105*b^3*e^3*g 
- 64*c^3*d^2*(12*e*f + d*g) - 28*b*c*e^2*(7*a*e*g + 5*b*(e*f + 3*d*g)) + 8 
*c^2*e*(16*a*e*(e*f + 3*d*g) + 3*b*d*(25*e*f + 19*d*g)) - 6*c*e*(21*b^2*e^ 
2*g + 8*c^2*d*(7*e*f + d*g) - 4*c*e*(7*b*e*f + 9*b*d*g + 5*a*e*g))*x)*(a + 
 b*x + c*x^2)^(3/2))/c^2 + (5*(128*c^4*d^3*f + 21*b^4*e^3*g - 28*b^2*c*e^2 
*(b*e*f + 3*b*d*g + 2*a*e*g) - 32*c^3*d*(3*a*e*(e*f + d*g) + 2*b*d*(3*e*f 
+ d*g)) + 8*c^2*e*(2*a^2*e^2*g + 15*b^2*d*(e*f + d*g) + 6*a*b*e*(e*f + 3*d 
*g)))*(((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[ 
(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(3/2))))/(16*c^2))/(1 
0*c))/(4*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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

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 1225

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

rule 1236

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

Maple [A] (verified)

Time = 2.45 (sec) , antiderivative size = 981, normalized size of antiderivative = 1.85

method	result	size
risch	\(\frac {\left (1280 g \,e^{3} c^{5} x^{5}+128 b \,c^{4} e^{3} g \,x^{4}+4608 c^{5} d \,e^{2} g \,x^{4}+1536 c^{5} e^{3} f \,x^{4}+320 a \,c^{4} e^{3} g \,x^{3}-144 b^{2} c^{3} e^{3} g \,x^{3}+576 b \,c^{4} d \,e^{2} g \,x^{3}+192 b \,c^{4} e^{3} f \,x^{3}+5760 c^{5} d^{2} e g \,x^{3}+5760 c^{5} d \,e^{2} f \,x^{3}-544 a b \,c^{3} e^{3} g \,x^{2}+1536 a \,c^{4} d \,e^{2} g \,x^{2}+512 a \,c^{4} e^{3} f \,x^{2}+168 b^{3} c^{2} e^{3} g \,x^{2}-672 b^{2} c^{3} d \,e^{2} g \,x^{2}-224 b^{2} c^{3} e^{3} f \,x^{2}+960 b \,c^{4} d^{2} e g \,x^{2}+960 b \,c^{4} d \,e^{2} f \,x^{2}+2560 c^{5} d^{3} g \,x^{2}+7680 c^{5} d^{2} e f \,x^{2}-480 a^{2} c^{3} e^{3} g x +896 a \,b^{2} c^{2} e^{3} g x -2784 a b \,c^{3} d \,e^{2} g x -928 a b \,c^{3} e^{3} f x +2880 a \,c^{4} d^{2} e g x +2880 a \,c^{4} d \,e^{2} f x -210 b^{4} c \,e^{3} g x +840 b^{3} c^{2} d \,e^{2} g x +280 b^{3} c^{2} e^{3} f x -1200 b^{2} c^{3} d^{2} e g x -1200 b^{2} c^{3} d \,e^{2} f x +640 b \,c^{4} d^{3} g x +1920 b \,c^{4} d^{2} e f x +3840 c^{5} d^{3} f x +1808 a^{2} b \,c^{2} e^{3} g -3072 a^{2} c^{3} d \,e^{2} g -1024 a^{2} c^{3} e^{3} f -1680 a \,b^{3} c \,e^{3} g +5520 a \,b^{2} c^{2} d \,e^{2} g +1840 a \,b^{2} c^{2} e^{3} f -6240 a b \,c^{3} d^{2} e g -6240 a b \,c^{3} d \,e^{2} f +2560 a \,c^{4} d^{3} g +7680 a \,c^{4} d^{2} e f +315 b^{5} e^{3} g -1260 b^{4} c d \,e^{2} g -420 b^{4} c \,e^{3} f +1800 b^{3} c^{2} d^{2} e g +1800 b^{3} c^{2} d \,e^{2} f -960 b^{2} c^{3} d^{3} g -2880 b^{2} c^{3} d^{2} e f +1920 b \,c^{4} d^{3} f \right ) \sqrt {c \,x^{2}+b x +a}}{7680 c^{5}}+\frac {\left (64 a^{3} c^{3} e^{3} g -240 a^{2} b^{2} c^{2} e^{3} g +576 a^{2} b \,c^{3} d \,e^{2} g +192 a^{2} b \,c^{3} e^{3} f -384 a^{2} c^{4} d^{2} e g -384 a^{2} c^{4} d \,e^{2} f +140 a \,b^{4} c \,e^{3} g -480 a \,b^{3} c^{2} d \,e^{2} g -160 a \,b^{3} c^{2} e^{3} f +576 a \,b^{2} c^{3} d^{2} e g +576 a \,b^{2} c^{3} d \,e^{2} f -256 a b \,c^{4} d^{3} g -768 a b \,c^{4} d^{2} e f +512 a \,c^{5} d^{3} f -21 b^{6} e^{3} g +84 b^{5} c d \,e^{2} g +28 b^{5} c \,e^{3} f -120 b^{4} c^{2} d^{2} e g -120 b^{4} c^{2} d \,e^{2} f +64 b^{3} c^{3} d^{3} g +192 b^{3} c^{3} d^{2} e f -128 b^{2} c^{4} d^{3} f \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {11}{2}}}\)	\(981\)
default	\(\text {Expression too large to display}\)	\(1206\)

Input:

int((e*x+d)^3*(g*x+f)*(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/7680*(1280*c^5*e^3*g*x^5+128*b*c^4*e^3*g*x^4+4608*c^5*d*e^2*g*x^4+1536*c 
^5*e^3*f*x^4+320*a*c^4*e^3*g*x^3-144*b^2*c^3*e^3*g*x^3+576*b*c^4*d*e^2*g*x 
^3+192*b*c^4*e^3*f*x^3+5760*c^5*d^2*e*g*x^3+5760*c^5*d*e^2*f*x^3-544*a*b*c 
^3*e^3*g*x^2+1536*a*c^4*d*e^2*g*x^2+512*a*c^4*e^3*f*x^2+168*b^3*c^2*e^3*g* 
x^2-672*b^2*c^3*d*e^2*g*x^2-224*b^2*c^3*e^3*f*x^2+960*b*c^4*d^2*e*g*x^2+96 
0*b*c^4*d*e^2*f*x^2+2560*c^5*d^3*g*x^2+7680*c^5*d^2*e*f*x^2-480*a^2*c^3*e^ 
3*g*x+896*a*b^2*c^2*e^3*g*x-2784*a*b*c^3*d*e^2*g*x-928*a*b*c^3*e^3*f*x+288 
0*a*c^4*d^2*e*g*x+2880*a*c^4*d*e^2*f*x-210*b^4*c*e^3*g*x+840*b^3*c^2*d*e^2 
*g*x+280*b^3*c^2*e^3*f*x-1200*b^2*c^3*d^2*e*g*x-1200*b^2*c^3*d*e^2*f*x+640 
*b*c^4*d^3*g*x+1920*b*c^4*d^2*e*f*x+3840*c^5*d^3*f*x+1808*a^2*b*c^2*e^3*g- 
3072*a^2*c^3*d*e^2*g-1024*a^2*c^3*e^3*f-1680*a*b^3*c*e^3*g+5520*a*b^2*c^2* 
d*e^2*g+1840*a*b^2*c^2*e^3*f-6240*a*b*c^3*d^2*e*g-6240*a*b*c^3*d*e^2*f+256 
0*a*c^4*d^3*g+7680*a*c^4*d^2*e*f+315*b^5*e^3*g-1260*b^4*c*d*e^2*g-420*b^4* 
c*e^3*f+1800*b^3*c^2*d^2*e*g+1800*b^3*c^2*d*e^2*f-960*b^2*c^3*d^3*g-2880*b 
^2*c^3*d^2*e*f+1920*b*c^4*d^3*f)/c^5*(c*x^2+b*x+a)^(1/2)+1/1024*(64*a^3*c^ 
3*e^3*g-240*a^2*b^2*c^2*e^3*g+576*a^2*b*c^3*d*e^2*g+192*a^2*b*c^3*e^3*f-38 
4*a^2*c^4*d^2*e*g-384*a^2*c^4*d*e^2*f+140*a*b^4*c*e^3*g-480*a*b^3*c^2*d*e^ 
2*g-160*a*b^3*c^2*e^3*f+576*a*b^2*c^3*d^2*e*g+576*a*b^2*c^3*d*e^2*f-256*a* 
b*c^4*d^3*g-768*a*b*c^4*d^2*e*f+512*a*c^5*d^3*f-21*b^6*e^3*g+84*b^5*c*d*e^ 
2*g+28*b^5*c*e^3*f-120*b^4*c^2*d^2*e*g-120*b^4*c^2*d*e^2*f+64*b^3*c^3*d...
 

Fricas [A] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 1757, normalized size of antiderivative = 3.32 \[ \int (d+e x)^3 (f+g x) \sqrt {a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^3*(g*x+f)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

[-1/30720*(15*(4*(32*(b^2*c^4 - 4*a*c^5)*d^3 - 48*(b^3*c^3 - 4*a*b*c^4)*d^ 
2*e + 6*(5*b^4*c^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*d*e^2 - (7*b^5*c - 40*a*b^ 
3*c^2 + 48*a^2*b*c^3)*e^3)*f - (64*(b^3*c^3 - 4*a*b*c^4)*d^3 - 24*(5*b^4*c 
^2 - 24*a*b^2*c^3 + 16*a^2*c^4)*d^2*e + 12*(7*b^5*c - 40*a*b^3*c^2 + 48*a^ 
2*b*c^3)*d*e^2 - (21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2 - 64*a^3*c^3)*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*(1280*c^6*e^3*g*x^5 + 128*(12*c^6*e^3*f + (36 
*c^6*d*e^2 + b*c^5*e^3)*g)*x^4 + 16*(12*(30*c^6*d*e^2 + b*c^5*e^3)*f + (36 
0*c^6*d^2*e + 36*b*c^5*d*e^2 - (9*b^2*c^4 - 20*a*c^5)*e^3)*g)*x^3 + 8*(4*( 
240*c^6*d^2*e + 30*b*c^5*d*e^2 - (7*b^2*c^4 - 16*a*c^5)*e^3)*f + (320*c^6* 
d^3 + 120*b*c^5*d^2*e - 12*(7*b^2*c^4 - 16*a*c^5)*d*e^2 + (21*b^3*c^3 - 68 
*a*b*c^4)*e^3)*g)*x^2 + 4*(480*b*c^5*d^3 - 240*(3*b^2*c^4 - 8*a*c^5)*d^2*e 
 + 30*(15*b^3*c^3 - 52*a*b*c^4)*d*e^2 - (105*b^4*c^2 - 460*a*b^2*c^3 + 256 
*a^2*c^4)*e^3)*f - (320*(3*b^2*c^4 - 8*a*c^5)*d^3 - 120*(15*b^3*c^3 - 52*a 
*b*c^4)*d^2*e + 12*(105*b^4*c^2 - 460*a*b^2*c^3 + 256*a^2*c^4)*d*e^2 - (31 
5*b^5*c - 1680*a*b^3*c^2 + 1808*a^2*b*c^3)*e^3)*g + 2*(4*(480*c^6*d^3 + 24 
0*b*c^5*d^2*e - 30*(5*b^2*c^4 - 12*a*c^5)*d*e^2 + (35*b^3*c^3 - 116*a*b*c^ 
4)*e^3)*f + (320*b*c^5*d^3 - 120*(5*b^2*c^4 - 12*a*c^5)*d^2*e + 12*(35*b^3 
*c^3 - 116*a*b*c^4)*d*e^2 - (105*b^4*c^2 - 448*a*b^2*c^3 + 240*a^2*c^4)*e^ 
3)*g)*x)*sqrt(c*x^2 + b*x + a))/c^6, 1/15360*(15*(4*(32*(b^2*c^4 - 4*a*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2484 vs. \(2 (563) = 1126\).

Time = 1.17 (sec) , antiderivative size = 2484, normalized size of antiderivative = 4.69 \[ \int (d+e x)^3 (f+g x) \sqrt {a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)**3*(g*x+f)*(c*x**2+b*x+a)**(1/2),x)
 

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(e**3*g*x**5/6 + x**4*(b*e**3*g/12 + 3*c 
*d*e**2*g + c*e**3*f)/(5*c) + x**3*(a*e**3*g/6 + 3*b*d*e**2*g + b*e**3*f - 
 9*b*(b*e**3*g/12 + 3*c*d*e**2*g + c*e**3*f)/(10*c) + 3*c*d**2*e*g + 3*c*d 
*e**2*f)/(4*c) + x**2*(3*a*d*e**2*g + a*e**3*f - 4*a*(b*e**3*g/12 + 3*c*d* 
e**2*g + c*e**3*f)/(5*c) + 3*b*d**2*e*g + 3*b*d*e**2*f - 7*b*(a*e**3*g/6 + 
 3*b*d*e**2*g + b*e**3*f - 9*b*(b*e**3*g/12 + 3*c*d*e**2*g + c*e**3*f)/(10 
*c) + 3*c*d**2*e*g + 3*c*d*e**2*f)/(8*c) + c*d**3*g + 3*c*d**2*e*f)/(3*c) 
+ x*(3*a*d**2*e*g + 3*a*d*e**2*f - 3*a*(a*e**3*g/6 + 3*b*d*e**2*g + b*e**3 
*f - 9*b*(b*e**3*g/12 + 3*c*d*e**2*g + c*e**3*f)/(10*c) + 3*c*d**2*e*g + 3 
*c*d*e**2*f)/(4*c) + b*d**3*g + 3*b*d**2*e*f - 5*b*(3*a*d*e**2*g + a*e**3* 
f - 4*a*(b*e**3*g/12 + 3*c*d*e**2*g + c*e**3*f)/(5*c) + 3*b*d**2*e*g + 3*b 
*d*e**2*f - 7*b*(a*e**3*g/6 + 3*b*d*e**2*g + b*e**3*f - 9*b*(b*e**3*g/12 + 
 3*c*d*e**2*g + c*e**3*f)/(10*c) + 3*c*d**2*e*g + 3*c*d*e**2*f)/(8*c) + c* 
d**3*g + 3*c*d**2*e*f)/(6*c) + c*d**3*f)/(2*c) + (a*d**3*g + 3*a*d**2*e*f 
- 2*a*(3*a*d*e**2*g + a*e**3*f - 4*a*(b*e**3*g/12 + 3*c*d*e**2*g + c*e**3* 
f)/(5*c) + 3*b*d**2*e*g + 3*b*d*e**2*f - 7*b*(a*e**3*g/6 + 3*b*d*e**2*g + 
b*e**3*f - 9*b*(b*e**3*g/12 + 3*c*d*e**2*g + c*e**3*f)/(10*c) + 3*c*d**2*e 
*g + 3*c*d*e**2*f)/(8*c) + c*d**3*g + 3*c*d**2*e*f)/(3*c) + b*d**3*f - 3*b 
*(3*a*d**2*e*g + 3*a*d*e**2*f - 3*a*(a*e**3*g/6 + 3*b*d*e**2*g + b*e**3*f 
- 9*b*(b*e**3*g/12 + 3*c*d*e**2*g + c*e**3*f)/(10*c) + 3*c*d**2*e*g + 3...
 

Maxima [F(-2)]

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

integrate((e*x+d)^3*(g*x+f)*(c*x^2+b*x+a)^(1/2),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 [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 945, normalized size of antiderivative = 1.78 \[ \int (d+e x)^3 (f+g x) \sqrt {a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

integrate((e*x+d)^3*(g*x+f)*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

Output:

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*e^3*g*x + (12*c^5*e^3*f + 36* 
c^5*d*e^2*g + b*c^4*e^3*g)/c^5)*x + (360*c^5*d*e^2*f + 12*b*c^4*e^3*f + 36 
0*c^5*d^2*e*g + 36*b*c^4*d*e^2*g - 9*b^2*c^3*e^3*g + 20*a*c^4*e^3*g)/c^5)* 
x + (960*c^5*d^2*e*f + 120*b*c^4*d*e^2*f - 28*b^2*c^3*e^3*f + 64*a*c^4*e^3 
*f + 320*c^5*d^3*g + 120*b*c^4*d^2*e*g - 84*b^2*c^3*d*e^2*g + 192*a*c^4*d* 
e^2*g + 21*b^3*c^2*e^3*g - 68*a*b*c^3*e^3*g)/c^5)*x + (1920*c^5*d^3*f + 96 
0*b*c^4*d^2*e*f - 600*b^2*c^3*d*e^2*f + 1440*a*c^4*d*e^2*f + 140*b^3*c^2*e 
^3*f - 464*a*b*c^3*e^3*f + 320*b*c^4*d^3*g - 600*b^2*c^3*d^2*e*g + 1440*a* 
c^4*d^2*e*g + 420*b^3*c^2*d*e^2*g - 1392*a*b*c^3*d*e^2*g - 105*b^4*c*e^3*g 
 + 448*a*b^2*c^2*e^3*g - 240*a^2*c^3*e^3*g)/c^5)*x + (1920*b*c^4*d^3*f - 2 
880*b^2*c^3*d^2*e*f + 7680*a*c^4*d^2*e*f + 1800*b^3*c^2*d*e^2*f - 6240*a*b 
*c^3*d*e^2*f - 420*b^4*c*e^3*f + 1840*a*b^2*c^2*e^3*f - 1024*a^2*c^3*e^3*f 
 - 960*b^2*c^3*d^3*g + 2560*a*c^4*d^3*g + 1800*b^3*c^2*d^2*e*g - 6240*a*b* 
c^3*d^2*e*g - 1260*b^4*c*d*e^2*g + 5520*a*b^2*c^2*d*e^2*g - 3072*a^2*c^3*d 
*e^2*g + 315*b^5*e^3*g - 1680*a*b^3*c*e^3*g + 1808*a^2*b*c^2*e^3*g)/c^5) + 
 1/1024*(128*b^2*c^4*d^3*f - 512*a*c^5*d^3*f - 192*b^3*c^3*d^2*e*f + 768*a 
*b*c^4*d^2*e*f + 120*b^4*c^2*d*e^2*f - 576*a*b^2*c^3*d*e^2*f + 384*a^2*c^4 
*d*e^2*f - 28*b^5*c*e^3*f + 160*a*b^3*c^2*e^3*f - 192*a^2*b*c^3*e^3*f - 64 
*b^3*c^3*d^3*g + 256*a*b*c^4*d^3*g + 120*b^4*c^2*d^2*e*g - 576*a*b^2*c^3*d 
^2*e*g + 384*a^2*c^4*d^2*e*g - 84*b^5*c*d*e^2*g + 480*a*b^3*c^2*d*e^2*g...
 

Mupad [B] (verification not implemented)

Time = 15.31 (sec) , antiderivative size = 1716, normalized size of antiderivative = 3.24 \[ \int (d+e x)^3 (f+g x) \sqrt {a+b x+c x^2} \, dx=\text {Too large to display} \] Input:

int((f + g*x)*(d + e*x)^3*(a + b*x + c*x^2)^(1/2),x)
 

Output:

d^3*f*(x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (e^3*f*x^2*(a + b*x + c*x^ 
2)^(3/2))/(5*c) + (e^3*g*x^3*(a + b*x + c*x^2)^(3/2))/(6*c) + (d^3*f*log(( 
b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c - b^2/4))/(2*c^(3/2)) + 
 (d^3*g*log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b* 
c))/(16*c^(5/2)) - (2*a*e^3*f*((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x 
^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b 
*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)))/(5*c) + (d^3*g*(8*c*(a + c*x^2) 
- 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2) + (a*e^3*g*((5*b*((lo 
g((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^ 
(5/2)) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24 
*c^2)))/(8*c) - (x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*((x/2 + b/(4*c))*(a 
 + b*x + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2) 
)*(a*c - b^2/4))/(2*c^(3/2))))/(4*c)))/(2*c) + (7*b*e^3*f*((5*b*((log((b + 
 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2)) 
 + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c^2)) 
)/(8*c) - (x*(a + b*x + c*x^2)^(3/2))/(4*c) + (a*((x/2 + b/(4*c))*(a + b*x 
 + c*x^2)^(1/2) + (log((b/2 + c*x)/c^(1/2) + (a + b*x + c*x^2)^(1/2))*(a*c 
 - b^2/4))/(2*c^(3/2))))/(4*c)))/(10*c) - (3*b*e^3*g*((7*b*((5*b*((log((b 
+ 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(16*c^(5/2) 
) + ((8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + c*x^2)^(1/2))/(24*c...
 

Reduce [F]

\[ \int (d+e x)^3 (f+g x) \sqrt {a+b x+c x^2} \, dx=\int \left (e x +d \right )^{3} \left (g x +f \right ) \sqrt {c \,x^{2}+b x +a}d x \] Input:

int((e*x+d)^3*(g*x+f)*(c*x^2+b*x+a)^(1/2),x)
 

Output:

int((e*x+d)^3*(g*x+f)*(c*x^2+b*x+a)^(1/2),x)