\(\int (d+e x)^3 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx\) [136]

Optimal result

Integrand size = 44, antiderivative size = 350 \[ \int (d+e x)^3 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {7 (2 c d-b e)^3 (4 c e f+2 c d g-3 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{512 c^5 e}-\frac {(4 c e f+2 c d g-3 b e g) (d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{20 c^2 e^2}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}-\frac {7 (2 c d-b e) (4 c e f+2 c d g-3 b e g) (16 c d-5 b e+6 c e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{960 c^4 e^2}+\frac {7 (2 c d-b e)^5 (4 c e f+2 c d g-3 b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{512 c^{11/2} e^2} \] Output:

7/512*(-b*e+2*c*d)^3*(-3*b*e*g+2*c*d*g+4*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b* 
e^2*x-c*e^2*x^2)^(1/2)/c^5/e-1/20*(-3*b*e*g+2*c*d*g+4*c*e*f)*(e*x+d)^2*(d* 
(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^2/e^2-1/6*g*(e*x+d)^3*(d*(-b*e+c*d)- 
b*e^2*x-c*e^2*x^2)^(3/2)/c/e^2-7/960*(-b*e+2*c*d)*(-3*b*e*g+2*c*d*g+4*c*e* 
f)*(6*c*e*x-5*b*e+16*c*d)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^4/e^2+7 
/512*(-b*e+2*c*d)^5*(-3*b*e*g+2*c*d*g+4*c*e*f)*arctan(c^(1/2)*(e*x+d)/(d*( 
-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/c^(11/2)/e^2
 

Mathematica [A] (verified)

Time = 1.61 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.28 \[ \int (d+e x)^3 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {(2 c d-b e)^5 \sqrt {(d+e x) (-b e+c (d-e x))} \left (\frac {\sqrt {c} \left (315 b^5 e^5 g-210 b^4 c e^4 (2 e f+14 d g+e g x)+56 b^3 c^2 e^3 \left (193 d^2 g+e^2 x (5 f+3 g x)+d e (65 f+31 g x)\right )-32 c^5 \left (176 d^5 g-36 d e^4 x^3 (5 f+4 g x)-8 e^5 x^4 (6 f+5 g x)-2 d^3 e^2 x (15 f+16 g x)-2 d^2 e^3 x^2 (112 f+85 g x)+d^4 e (272 f+105 g x)\right )+16 b c^4 e \left (1047 d^4 g+4 e^4 x^3 (3 f+2 g x)+4 d e^3 x^2 (23 f+14 g x)+2 d^2 e^2 x (179 f+95 g x)+2 d^3 e (559 f+227 g x)\right )-16 b^2 c^3 e^2 \left (1213 d^3 g+19 d e^2 x (7 f+4 g x)+e^3 x^2 (14 f+9 g x)+d^2 e (749 f+335 g x)\right )\right )}{(2 c d-b e)^5}-\frac {105 (4 c e f+2 c d g-3 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{\sqrt {d+e x} \sqrt {-b e+c (d-e x)}}\right )}{7680 c^{11/2} e^2} \] Input:

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

Output:

((2*c*d - b*e)^5*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((Sqrt[c]*(315*b^5 
*e^5*g - 210*b^4*c*e^4*(2*e*f + 14*d*g + e*g*x) + 56*b^3*c^2*e^3*(193*d^2* 
g + e^2*x*(5*f + 3*g*x) + d*e*(65*f + 31*g*x)) - 32*c^5*(176*d^5*g - 36*d* 
e^4*x^3*(5*f + 4*g*x) - 8*e^5*x^4*(6*f + 5*g*x) - 2*d^3*e^2*x*(15*f + 16*g 
*x) - 2*d^2*e^3*x^2*(112*f + 85*g*x) + d^4*e*(272*f + 105*g*x)) + 16*b*c^4 
*e*(1047*d^4*g + 4*e^4*x^3*(3*f + 2*g*x) + 4*d*e^3*x^2*(23*f + 14*g*x) + 2 
*d^2*e^2*x*(179*f + 95*g*x) + 2*d^3*e*(559*f + 227*g*x)) - 16*b^2*c^3*e^2* 
(1213*d^3*g + 19*d*e^2*x*(7*f + 4*g*x) + e^3*x^2*(14*f + 9*g*x) + d^2*e*(7 
49*f + 335*g*x))))/(2*c*d - b*e)^5 - (105*(4*c*e*f + 2*c*d*g - 3*b*e*g)*Ar 
cTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + e*x])])/(Sqrt[d + e*x]*Sqrt 
[-(b*e) + c*(d - e*x)])))/(7680*c^(11/2)*e^2)
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.159, Rules used = {1221, 1134, 1134, 1160, 1087, 1092, 217}

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.

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

\(\Big \downarrow \) 1221

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

\(\Big \downarrow \) 1134

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

\(\Big \downarrow \) 1134

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(-3 b e g+2 c d g+4 c e f) \left (\frac {7 (2 c d-b e) \left (\frac {5 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {(2 c d-b e)^2 \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{8 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e}\right )}{8 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 c e}\right )}{10 c}-\frac {(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e}\right )}{4 c e}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(-3 b e g+2 c d g+4 c e f) \left (\frac {7 (2 c d-b e) \left (\frac {5 (2 c d-b e) \left (\frac {(2 c d-b e) \left (\frac {(2 c d-b e)^2 \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{4 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{2 c}-\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e}\right )}{8 c}-\frac {(d+e x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{4 c e}\right )}{10 c}-\frac {(d+e x)^2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{5 c e}\right )}{4 c e}-\frac {g (d+e x)^3 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6 c e^2}\)

\(\Big \downarrow \) 217

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

Input:

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

Output:

-1/6*(g*(d + e*x)^3*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(c*e^2) + 
 ((4*c*e*f + 2*c*d*g - 3*b*e*g)*(-1/5*((d + e*x)^2*(d*(c*d - b*e) - b*e^2* 
x - c*e^2*x^2)^(3/2))/(c*e) + (7*(2*c*d - b*e)*(-1/4*((d + e*x)*(d*(c*d - 
b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(c*e) + (5*(2*c*d - b*e)*(-1/3*(d*(c*d 
- b*e) - b*e^2*x - c*e^2*x^2)^(3/2)/(c*e) + ((2*c*d - b*e)*(((b + 2*c*x)*S 
qrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])/(4*c) + ((2*c*d - b*e)^2*ArcTan[ 
(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/(8 
*c^(3/2)*e)))/(2*c)))/(8*c)))/(10*c)))/(4*c*e)
 

Defintions of rubi rules used

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[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 1134

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

rule 1160

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

rule 1221

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[(m*(g*(c*d - b*e) + c*e*f) + e*(p + 1)*(2*c 
*f - b*g))/(c*e*(m + 2*p + 2))   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] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
 && NeQ[m + 2*p + 2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2476\) vs. \(2(326)=652\).

Time = 3.42 (sec) , antiderivative size = 2477, normalized size of antiderivative = 7.08

method	result	size
default	\(\text {Expression too large to display}\)	\(2477\)

Input:

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

Output:

d^3*f*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2 
)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2) 
^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)))+e^2*(3*d*g+e*f 
)*(-1/5*x^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-7/10*b/c*(-1/4*x* 
(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-5/8*b/c*(-1/3*(-c*e^2*x^2-b*e 
^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-1/2*b/c*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e 
^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/ 
e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d 
*e+c*d^2)^(1/2))))+1/4*(-b*d*e+c*d^2)/c/e^2*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2 
*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e 
^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2 
*x-b*d*e+c*d^2)^(1/2))))+2/5*(-b*d*e+c*d^2)/c/e^2*(-1/3*(-c*e^2*x^2-b*e^2* 
x-b*d*e+c*d^2)^(3/2)/c/e^2-1/2*b/c*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2* 
x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2 
/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+ 
c*d^2)^(1/2)))))+3*d*e*(d*g+e*f)*(-1/4*x*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^ 
(3/2)/c/e^2-5/8*b/c*(-1/3*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/c/e^2-1/2 
*b/c*(-1/4*(-2*c*e^2*x-b*e^2)/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2) 
-1/8*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^ 
(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2))))+1/4*(-b*d*e...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 728 vs. \(2 (326) = 652\).

Time = 0.77 (sec) , antiderivative size = 1469, normalized size of antiderivative = 4.20 \[ \int (d+e x)^3 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/30720*(105*(4*(32*c^6*d^5*e - 80*b*c^5*d^4*e^2 + 80*b^2*c^4*d^3*e^3 - 
40*b^3*c^3*d^2*e^4 + 10*b^4*c^2*d*e^5 - b^5*c*e^6)*f + (64*c^6*d^6 - 256*b 
*c^5*d^5*e + 400*b^2*c^4*d^4*e^2 - 320*b^3*c^3*d^3*e^3 + 140*b^4*c^2*d^2*e 
^4 - 32*b^5*c*d*e^5 + 3*b^6*e^6)*g)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2 
*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e^2 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 
 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c)) - 4*(1280*c^6*e^5*g*x^5 + 128*(12*c^6* 
e^5*f + (36*c^6*d*e^4 + b*c^5*e^5)*g)*x^4 + 16*(12*(30*c^6*d*e^4 + b*c^5*e 
^5)*f + (340*c^6*d^2*e^3 + 56*b*c^5*d*e^4 - 9*b^2*c^4*e^5)*g)*x^3 + 8*(4*( 
224*c^6*d^2*e^3 + 46*b*c^5*d*e^4 - 7*b^2*c^4*e^5)*f + (128*c^6*d^3*e^2 + 3 
80*b*c^5*d^2*e^3 - 152*b^2*c^4*d*e^4 + 21*b^3*c^3*e^5)*g)*x^2 - 4*(2176*c^ 
6*d^4*e - 4472*b*c^5*d^3*e^2 + 2996*b^2*c^4*d^2*e^3 - 910*b^3*c^3*d*e^4 + 
105*b^4*c^2*e^5)*f - (5632*c^6*d^5 - 16752*b*c^5*d^4*e + 19408*b^2*c^4*d^3 
*e^2 - 10808*b^3*c^3*d^2*e^3 + 2940*b^4*c^2*d*e^4 - 315*b^5*c*e^5)*g + 2*( 
4*(120*c^6*d^3*e^2 + 716*b*c^5*d^2*e^3 - 266*b^2*c^4*d*e^4 + 35*b^3*c^3*e^ 
5)*f - (1680*c^6*d^4*e - 3632*b*c^5*d^3*e^2 + 2680*b^2*c^4*d^2*e^3 - 868*b 
^3*c^3*d*e^4 + 105*b^4*c^2*e^5)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - 
b*d*e))/(c^6*e^2), -1/15360*(105*(4*(32*c^6*d^5*e - 80*b*c^5*d^4*e^2 + 80* 
b^2*c^4*d^3*e^3 - 40*b^3*c^3*d^2*e^4 + 10*b^4*c^2*d*e^5 - b^5*c*e^6)*f + ( 
64*c^6*d^6 - 256*b*c^5*d^5*e + 400*b^2*c^4*d^4*e^2 - 320*b^3*c^3*d^3*e^3 + 
 140*b^4*c^2*d^2*e^4 - 32*b^5*c*d*e^5 + 3*b^6*e^6)*g)*sqrt(c)*arctan(1/...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2919 vs. \(2 (343) = 686\).

Time = 1.61 (sec) , antiderivative size = 2919, normalized size of antiderivative = 8.34 \[ \int (d+e x)^3 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)^3*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algo 
rithm="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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (326) = 652\).

Time = 0.16 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.16 \[ \int (d+e x)^3 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\frac {1}{7680} \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, e^{3} g x + \frac {12 \, c^{5} e^{11} f + 36 \, c^{5} d e^{10} g + b c^{4} e^{11} g}{c^{5} e^{8}}\right )} x + \frac {360 \, c^{5} d e^{10} f + 12 \, b c^{4} e^{11} f + 340 \, c^{5} d^{2} e^{9} g + 56 \, b c^{4} d e^{10} g - 9 \, b^{2} c^{3} e^{11} g}{c^{5} e^{8}}\right )} x + \frac {896 \, c^{5} d^{2} e^{9} f + 184 \, b c^{4} d e^{10} f - 28 \, b^{2} c^{3} e^{11} f + 128 \, c^{5} d^{3} e^{8} g + 380 \, b c^{4} d^{2} e^{9} g - 152 \, b^{2} c^{3} d e^{10} g + 21 \, b^{3} c^{2} e^{11} g}{c^{5} e^{8}}\right )} x + \frac {480 \, c^{5} d^{3} e^{8} f + 2864 \, b c^{4} d^{2} e^{9} f - 1064 \, b^{2} c^{3} d e^{10} f + 140 \, b^{3} c^{2} e^{11} f - 1680 \, c^{5} d^{4} e^{7} g + 3632 \, b c^{4} d^{3} e^{8} g - 2680 \, b^{2} c^{3} d^{2} e^{9} g + 868 \, b^{3} c^{2} d e^{10} g - 105 \, b^{4} c e^{11} g}{c^{5} e^{8}}\right )} x - \frac {8704 \, c^{5} d^{4} e^{7} f - 17888 \, b c^{4} d^{3} e^{8} f + 11984 \, b^{2} c^{3} d^{2} e^{9} f - 3640 \, b^{3} c^{2} d e^{10} f + 420 \, b^{4} c e^{11} f + 5632 \, c^{5} d^{5} e^{6} g - 16752 \, b c^{4} d^{4} e^{7} g + 19408 \, b^{2} c^{3} d^{3} e^{8} g - 10808 \, b^{3} c^{2} d^{2} e^{9} g + 2940 \, b^{4} c d e^{10} g - 315 \, b^{5} e^{11} g}{c^{5} e^{8}}\right )} - \frac {7 \, {\left (128 \, c^{6} d^{5} e f - 320 \, b c^{5} d^{4} e^{2} f + 320 \, b^{2} c^{4} d^{3} e^{3} f - 160 \, b^{3} c^{3} d^{2} e^{4} f + 40 \, b^{4} c^{2} d e^{5} f - 4 \, b^{5} c e^{6} f + 64 \, c^{6} d^{6} g - 256 \, b c^{5} d^{5} e g + 400 \, b^{2} c^{4} d^{4} e^{2} g - 320 \, b^{3} c^{3} d^{3} e^{3} g + 140 \, b^{4} c^{2} d^{2} e^{4} g - 32 \, b^{5} c d e^{5} g + 3 \, b^{6} e^{6} g\right )} \log \left ({\left | -b e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} {\left | e \right |} \right |}\right )}{1024 \, \sqrt {-c} c^{5} e {\left | e \right |}} \] Input:

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

Output:

1/7680*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*(2*(8*(10*e^3*g*x 
+ (12*c^5*e^11*f + 36*c^5*d*e^10*g + b*c^4*e^11*g)/(c^5*e^8))*x + (360*c^5 
*d*e^10*f + 12*b*c^4*e^11*f + 340*c^5*d^2*e^9*g + 56*b*c^4*d*e^10*g - 9*b^ 
2*c^3*e^11*g)/(c^5*e^8))*x + (896*c^5*d^2*e^9*f + 184*b*c^4*d*e^10*f - 28* 
b^2*c^3*e^11*f + 128*c^5*d^3*e^8*g + 380*b*c^4*d^2*e^9*g - 152*b^2*c^3*d*e 
^10*g + 21*b^3*c^2*e^11*g)/(c^5*e^8))*x + (480*c^5*d^3*e^8*f + 2864*b*c^4* 
d^2*e^9*f - 1064*b^2*c^3*d*e^10*f + 140*b^3*c^2*e^11*f - 1680*c^5*d^4*e^7* 
g + 3632*b*c^4*d^3*e^8*g - 2680*b^2*c^3*d^2*e^9*g + 868*b^3*c^2*d*e^10*g - 
 105*b^4*c*e^11*g)/(c^5*e^8))*x - (8704*c^5*d^4*e^7*f - 17888*b*c^4*d^3*e^ 
8*f + 11984*b^2*c^3*d^2*e^9*f - 3640*b^3*c^2*d*e^10*f + 420*b^4*c*e^11*f + 
 5632*c^5*d^5*e^6*g - 16752*b*c^4*d^4*e^7*g + 19408*b^2*c^3*d^3*e^8*g - 10 
808*b^3*c^2*d^2*e^9*g + 2940*b^4*c*d*e^10*g - 315*b^5*e^11*g)/(c^5*e^8)) - 
 7/1024*(128*c^6*d^5*e*f - 320*b*c^5*d^4*e^2*f + 320*b^2*c^4*d^3*e^3*f - 1 
60*b^3*c^3*d^2*e^4*f + 40*b^4*c^2*d*e^5*f - 4*b^5*c*e^6*f + 64*c^6*d^6*g - 
 256*b*c^5*d^5*e*g + 400*b^2*c^4*d^4*e^2*g - 320*b^3*c^3*d^3*e^3*g + 140*b 
^4*c^2*d^2*e^4*g - 32*b^5*c*d*e^5*g + 3*b^6*e^6*g)*log(abs(-b*e^2 + 2*(sqr 
t(-c*e^2)*x - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*sqrt(-c)*abs(e)) 
)/(sqrt(-c)*c^5*e*abs(e))
 

Mupad [B] (verification not implemented)

Time = 9.64 (sec) , antiderivative size = 3311, normalized size of antiderivative = 9.46 \[ \int (d+e x)^3 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 2.05 (sec) , antiderivative size = 2564, normalized size of antiderivative = 7.33 \[ \int (d+e x)^3 (f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*(315*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d)) 
*b**7*e**7*g - 3990*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b 
*e + 2*c*d))*b**6*c*d*e**6*g - 420*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e* 
x)*i)/sqrt( - b*e + 2*c*d))*b**6*c*e**7*f + 21420*sqrt(c)*asinh((sqrt( - b 
*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**5*c**2*d**2*e**5*g + 5040*sq 
rt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**5*c**2 
*d*e**6*f - 63000*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e 
 + 2*c*d))*b**4*c**3*d**3*e**4*g - 25200*sqrt(c)*asinh((sqrt( - b*e + c*d 
- c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**4*c**3*d**2*e**5*f + 109200*sqrt(c)*a 
sinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c**4*d**4*e 
**3*g + 67200*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2 
*c*d))*b**3*c**4*d**3*e**4*f - 110880*sqrt(c)*asinh((sqrt( - b*e + c*d - c 
*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**5*d**5*e**2*g - 100800*sqrt(c)*asin 
h((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c**5*d**4*e**3 
*f + 60480*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c* 
d))*b*c**6*d**6*e*g + 80640*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/s 
qrt( - b*e + 2*c*d))*b*c**6*d**5*e**2*f - 13440*sqrt(c)*asinh((sqrt( - b*e 
 + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**7*d**7*g - 26880*sqrt(c)*asinh 
((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**7*d**6*e*f + 315* 
sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d ...