\(\int (d+e x) (f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{5/2} \, dx\) [161]

Optimal result

Integrand size = 42, antiderivative size = 366 \[ \int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\frac {5 (2 c d-b e)^5 (16 c e f+2 c d g-9 b e g) (b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{16384 c^5 e}+\frac {5 (2 c d-b e)^3 (16 c e f+2 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{6144 c^4 e}+\frac {(2 c d-b e) (16 c e f+2 c d g-9 b e g) (b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{384 c^3 e}+\frac {(9 b e g-16 c (e f+d g)-14 c e g x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2}}{112 c^2 e^2}+\frac {5 (2 c d-b e)^7 (16 c e f+2 c d g-9 b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{16384 c^{11/2} e^2} \] Output:

5/16384*(-b*e+2*c*d)^5*(-9*b*e*g+2*c*d*g+16*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+5/6144*(-b*e+2*c*d)^3*(-9*b*e*g+2*c*d*g+16 
*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c^4/e+1/384*(-b*e 
+2*c*d)*(-9*b*e*g+2*c*d*g+16*c*e*f)*(2*c*x+b)*(d*(-b*e+c*d)-b*e^2*x-c*e^2* 
x^2)^(5/2)/c^3/e+1/112*(9*b*e*g-16*c*(d*g+e*f)-14*c*e*g*x)*(d*(-b*e+c*d)-b 
*e^2*x-c*e^2*x^2)^(7/2)/c^2/e^2+5/16384*(-b*e+2*c*d)^7*(-9*b*e*g+2*c*d*g+1 
6*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 [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(733\) vs. \(2(366)=732\).

Time = 4.70 (sec) , antiderivative size = 733, normalized size of antiderivative = 2.00 \[ \int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\frac {(2 c d-b e)^7 ((d+e x) (-b e+c (d-e x)))^{5/2} \left (\frac {\sqrt {c} \left (945 b^7 e^7 g-210 b^6 c e^6 (8 e f+58 d g+3 e g x)+28 b^5 c^2 e^5 \left (2363 d^2 g+38 d e (20 f+7 g x)+2 e^2 x (20 f+9 g x)\right )-128 c^7 \left (384 d^7 g-64 d e^6 x^5 (7 f+6 g x)-48 e^7 x^6 (8 f+7 g x)+3 d^6 e (128 f+35 g x)-24 d^5 e^2 x (77 f+48 g x)+16 d^3 e^4 x^3 (91 f+72 g x)+8 d^2 e^5 x^4 (144 f+119 g x)-2 d^4 e^3 x^2 (576 f+413 g x)\right )+64 b c^6 e \left (2967 d^6 g-8 d^2 e^4 x^3 (30 f+19 g x)+16 e^6 x^5 (116 f+99 g x)+6 d^5 e (692 f+181 g x)+16 d e^5 x^4 (284 f+235 g x)-24 d^3 e^3 x^2 (374 f+269 g x)-6 d^4 e^2 x (1156 f+739 g x)\right )+16 b^3 c^4 e^3 \left (20779 d^4 g+24 e^4 x^3 (2 f+g x)+8 d e^3 x^2 (74 f+33 g x)+20 d^2 e^2 x (192 f+73 g x)+4 d^3 e (5024 f+1431 g x)\right )+32 b^2 c^5 e^2 \left (-10434 d^5 g+1224 d^3 e^2 x (4 f+3 g x)+8 e^5 x^4 (296 f+243 g x)+16 d e^4 x^3 (583 f+455 g x)-3 d^4 e (4616 f+1227 g x)+4 d^2 e^3 x^2 (3276 f+2375 g x)\right )-8 b^4 c^3 e^4 \left (24372 d^3 g+2 e^3 x^2 (56 f+27 g x)+8 d e^2 x (203 f+85 g x)+d^2 e (14112 f+4523 g x)\right )\right )}{(2 c d-b e)^7 (d+e x)^2 (-c d+b e+c e x)^2}+\frac {105 (9 b e g-2 c (8 e f+d g)) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {c} \sqrt {d+e x}}\right )}{(d+e x)^{5/2} (-b e+c (d-e x))^{5/2}}\right )}{344064 c^{11/2} e^2} \] Input:

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

Output:

((2*c*d - b*e)^7*((d + e*x)*(-(b*e) + c*(d - e*x)))^(5/2)*((Sqrt[c]*(945*b 
^7*e^7*g - 210*b^6*c*e^6*(8*e*f + 58*d*g + 3*e*g*x) + 28*b^5*c^2*e^5*(2363 
*d^2*g + 38*d*e*(20*f + 7*g*x) + 2*e^2*x*(20*f + 9*g*x)) - 128*c^7*(384*d^ 
7*g - 64*d*e^6*x^5*(7*f + 6*g*x) - 48*e^7*x^6*(8*f + 7*g*x) + 3*d^6*e*(128 
*f + 35*g*x) - 24*d^5*e^2*x*(77*f + 48*g*x) + 16*d^3*e^4*x^3*(91*f + 72*g* 
x) + 8*d^2*e^5*x^4*(144*f + 119*g*x) - 2*d^4*e^3*x^2*(576*f + 413*g*x)) + 
64*b*c^6*e*(2967*d^6*g - 8*d^2*e^4*x^3*(30*f + 19*g*x) + 16*e^6*x^5*(116*f 
 + 99*g*x) + 6*d^5*e*(692*f + 181*g*x) + 16*d*e^5*x^4*(284*f + 235*g*x) - 
24*d^3*e^3*x^2*(374*f + 269*g*x) - 6*d^4*e^2*x*(1156*f + 739*g*x)) + 16*b^ 
3*c^4*e^3*(20779*d^4*g + 24*e^4*x^3*(2*f + g*x) + 8*d*e^3*x^2*(74*f + 33*g 
*x) + 20*d^2*e^2*x*(192*f + 73*g*x) + 4*d^3*e*(5024*f + 1431*g*x)) + 32*b^ 
2*c^5*e^2*(-10434*d^5*g + 1224*d^3*e^2*x*(4*f + 3*g*x) + 8*e^5*x^4*(296*f 
+ 243*g*x) + 16*d*e^4*x^3*(583*f + 455*g*x) - 3*d^4*e*(4616*f + 1227*g*x) 
+ 4*d^2*e^3*x^2*(3276*f + 2375*g*x)) - 8*b^4*c^3*e^4*(24372*d^3*g + 2*e^3* 
x^2*(56*f + 27*g*x) + 8*d*e^2*x*(203*f + 85*g*x) + d^2*e*(14112*f + 4523*g 
*x))))/((2*c*d - b*e)^7*(d + e*x)^2*(-(c*d) + b*e + c*e*x)^2) + (105*(9*b* 
e*g - 2*c*(8*e*f + d*g))*ArcTan[Sqrt[c*d - b*e - c*e*x]/(Sqrt[c]*Sqrt[d + 
e*x])])/((d + e*x)^(5/2)*(-(b*e) + c*(d - e*x))^(5/2))))/(344064*c^(11/2)* 
e^2)
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1225, 1087, 1087, 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.

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

\(\Big \downarrow \) 1225

\(\displaystyle \frac {(2 c d-b e) (-9 b e g+2 c d g+16 c e f) \int \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{5/2}dx}{32 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-16 c (d g+e f)-14 c e g x)}{112 c^2 e^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(2 c d-b e) (-9 b e g+2 c d g+16 c e f) \left (\frac {5 (2 c d-b e)^2 \int \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )^{3/2}dx}{24 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 c}\right )}{32 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-16 c (d g+e f)-14 c e g x)}{112 c^2 e^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(2 c d-b e) (-9 b e g+2 c d g+16 c e f) \left (\frac {5 (2 c d-b e)^2 \left (\frac {3 (2 c d-b e)^2 \int \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}dx}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{24 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 c}\right )}{32 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-16 c (d g+e f)-14 c e g x)}{112 c^2 e^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {(2 c d-b e) (-9 b e g+2 c d g+16 c e f) \left (\frac {5 (2 c d-b e)^2 \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{8 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{24 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 c}\right )}{32 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-16 c (d g+e f)-14 c e g x)}{112 c^2 e^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(2 c d-b e) (-9 b e g+2 c d g+16 c e f) \left (\frac {5 (2 c d-b e)^2 \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{4 c}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{24 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 c}\right )}{32 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-16 c (d g+e f)-14 c e g x)}{112 c^2 e^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(2 c d-b e) \left (\frac {5 (2 c d-b e)^2 \left (\frac {3 (2 c d-b e)^2 \left (\frac {(2 c d-b e)^2 \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{8 c^{3/2} e}+\frac {(b+2 c x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c}\right )}{16 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{8 c}\right )}{24 c}+\frac {(b+2 c x) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{12 c}\right ) (-9 b e g+2 c d g+16 c e f)}{32 c^2 e}+\frac {\left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{7/2} (9 b e g-16 c (d g+e f)-14 c e g x)}{112 c^2 e^2}\)

Input:

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

Output:

((9*b*e*g - 16*c*(e*f + d*g) - 14*c*e*g*x)*(d*(c*d - b*e) - b*e^2*x - c*e^ 
2*x^2)^(7/2))/(112*c^2*e^2) + ((2*c*d - b*e)*(16*c*e*f + 2*c*d*g - 9*b*e*g 
)*(((b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(12*c) + (5*( 
2*c*d - b*e)^2*(((b + 2*c*x)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/ 
(8*c) + (3*(2*c*d - b*e)^2*(((b + 2*c*x)*Sqrt[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)))/(16*c)))/(24*c)))/ 
(32*c^2*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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1411\) vs. \(2(342)=684\).

Time = 2.40 (sec) , antiderivative size = 1412, normalized size of antiderivative = 3.86

method	result	size
default	\(\text {Expression too large to display}\)	\(1412\)

Input:

int((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x,method=_RETUR 
NVERBOSE)
 

Output:

d*f*(-1/12*(-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)^(5/2) 
-5/24*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/8*(-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)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^ 
2*e^4)/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)))))+(d*g+ 
e*f)*(-1/7*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c/e^2-1/2*b/c*(-1/12*(-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)^(5/2)-5/24*(-4*c*e^ 
2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2*(-1/8*(-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)^(3/2)-3/16*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/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))))))+e*g*(-1/8*x*(-c*e^ 
2*x^2-b*e^2*x-b*d*e+c*d^2)^(7/2)/c/e^2-9/16*b/c*(-1/7*(-c*e^2*x^2-b*e^2*x- 
b*d*e+c*d^2)^(7/2)/c/e^2-1/2*b/c*(-1/12*(-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)^(5/2)-5/24*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/c/e^2 
*(-1/8*(-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)^(3/2)-3/1 
6*(-4*c*e^2*(-b*d*e+c*d^2)-b^2*e^4)/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^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1166 vs. \(2 (342) = 684\).

Time = 3.32 (sec) , antiderivative size = 2345, normalized size of antiderivative = 6.41 \[ \int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algori 
thm="fricas")
 

Output:

[-1/1376256*(105*(16*(128*c^8*d^7*e - 448*b*c^7*d^6*e^2 + 672*b^2*c^6*d^5* 
e^3 - 560*b^3*c^5*d^4*e^4 + 280*b^4*c^4*d^3*e^5 - 84*b^5*c^3*d^2*e^6 + 14* 
b^6*c^2*d*e^7 - b^7*c*e^8)*f + (256*c^8*d^8 - 2048*b*c^7*d^7*e + 5376*b^2* 
c^6*d^6*e^2 - 7168*b^3*c^5*d^5*e^3 + 5600*b^4*c^4*d^4*e^4 - 2688*b^5*c^3*d 
^3*e^5 + 784*b^6*c^2*d^2*e^6 - 128*b^7*c*d*e^7 + 9*b^8*e^8)*g)*sqrt(-c)*lo 
g(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*(43008* 
c^8*e^7*g*x^7 + 3072*(16*c^8*e^7*f + (16*c^8*d*e^6 + 33*b*c^7*e^7)*g)*x^6 
+ 256*(16*(14*c^8*d*e^6 + 29*b*c^7*e^7)*f - (476*c^8*d^2*e^5 - 940*b*c^7*d 
*e^6 - 243*b^2*c^6*e^7)*g)*x^5 - 128*(16*(72*c^8*d^2*e^5 - 142*b*c^7*d*e^6 
 - 37*b^2*c^6*e^7)*f + (1152*c^8*d^3*e^4 + 76*b*c^7*d^2*e^5 - 1820*b^2*c^6 
*d*e^6 - 3*b^3*c^5*e^7)*g)*x^4 - 16*(16*(728*c^8*d^3*e^4 + 60*b*c^7*d^2*e^ 
5 - 1166*b^2*c^6*d*e^6 - 3*b^3*c^5*e^7)*f - (6608*c^8*d^4*e^3 - 25824*b*c^ 
7*d^3*e^4 + 19000*b^2*c^6*d^2*e^5 + 264*b^3*c^5*d*e^6 - 27*b^4*c^4*e^7)*g) 
*x^3 + 8*(16*(1152*c^8*d^4*e^3 - 4488*b*c^7*d^3*e^4 + 3276*b^2*c^6*d^2*e^5 
 + 74*b^3*c^5*d*e^6 - 7*b^4*c^4*e^7)*f + (18432*c^8*d^5*e^2 - 35472*b*c^7* 
d^4*e^3 + 14688*b^2*c^6*d^3*e^4 + 2920*b^3*c^5*d^2*e^5 - 680*b^4*c^4*d*e^6 
 + 63*b^5*c^3*e^7)*g)*x^2 - 16*(3072*c^8*d^6*e - 16608*b*c^7*d^5*e^2 + 276 
96*b^2*c^6*d^4*e^3 - 20096*b^3*c^5*d^3*e^4 + 7056*b^4*c^4*d^2*e^5 - 1330*b 
^5*c^3*d*e^6 + 105*b^6*c^2*e^7)*f - (49152*c^8*d^7 - 189888*b*c^7*d^6*e...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11123 vs. \(2 (357) = 714\).

Time = 2.40 (sec) , antiderivative size = 11123, normalized size of antiderivative = 30.39 \[ \int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(-b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2)*(c**2*e**5*g*x** 
7/8 - x**6*(-33*b*c**2*e**7*g/16 - c**3*d*e**6*g - c**3*e**7*f)/(7*c*e**2) 
 - x**5*(-3*b**2*c*e**7*g - 6*b*c**2*d*e**6*g - 3*b*c**2*e**7*f - 13*b*(-3 
3*b*c**2*e**7*g/16 - c**3*d*e**6*g - c**3*e**7*f)/(14*c) + 3*c**3*d**2*e** 
5*g - c**3*d*e**6*f - c**2*e**5*g*(-7*b*d*e + 7*c*d**2)/8)/(6*c*e**2) - x* 
*4*(-b**3*e**7*g - 9*b**2*c*d*e**6*g - 3*b**2*c*e**7*f + 3*b*c**2*d**2*e** 
5*g - 6*b*c**2*d*e**6*f - 11*b*(-3*b**2*c*e**7*g - 6*b*c**2*d*e**6*g - 3*b 
*c**2*e**7*f - 13*b*(-33*b*c**2*e**7*g/16 - c**3*d*e**6*g - c**3*e**7*f)/( 
14*c) + 3*c**3*d**2*e**5*g - c**3*d*e**6*f - c**2*e**5*g*(-7*b*d*e + 7*c*d 
**2)/8)/(12*c) + 3*c**3*d**3*e**4*g + 3*c**3*d**2*e**5*f + (-6*b*d*e + 6*c 
*d**2)*(-33*b*c**2*e**7*g/16 - c**3*d*e**6*g - c**3*e**7*f)/(7*c*e**2))/(5 
*c*e**2) - x**3*(-4*b**3*d*e**6*g - b**3*e**7*f - 6*b**2*c*d**2*e**5*g - 9 
*b**2*c*d*e**6*f + 12*b*c**2*d**3*e**4*g + 3*b*c**2*d**2*e**5*f - 9*b*(-b* 
*3*e**7*g - 9*b**2*c*d*e**6*g - 3*b**2*c*e**7*f + 3*b*c**2*d**2*e**5*g - 6 
*b*c**2*d*e**6*f - 11*b*(-3*b**2*c*e**7*g - 6*b*c**2*d*e**6*g - 3*b*c**2*e 
**7*f - 13*b*(-33*b*c**2*e**7*g/16 - c**3*d*e**6*g - c**3*e**7*f)/(14*c) + 
 3*c**3*d**2*e**5*g - c**3*d*e**6*f - c**2*e**5*g*(-7*b*d*e + 7*c*d**2)/8) 
/(12*c) + 3*c**3*d**3*e**4*g + 3*c**3*d**2*e**5*f + (-6*b*d*e + 6*c*d**2)* 
(-33*b*c**2*e**7*g/16 - c**3*d*e**6*g - c**3*e**7*f)/(7*c*e**2))/(10*c) - 
3*c**3*d**4*e**3*g + 3*c**3*d**3*e**4*f + (-5*b*d*e + 5*c*d**2)*(-3*b**...
 

Maxima [F(-2)]

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

integrate((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algori 
thm="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. 1234 vs. \(2 (342) = 684\).

Time = 0.41 (sec) , antiderivative size = 1234, normalized size of antiderivative = 3.37 \[ \int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)*(g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(5/2),x, algori 
thm="giac")
 

Output:

1/344064*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*(4*(2*(8*(2*(12*(14 
*c^2*e^5*g*x + (16*c^9*e^17*f + 16*c^9*d*e^16*g + 33*b*c^8*e^17*g)/(c^7*e^ 
12))*x + (224*c^9*d*e^16*f + 464*b*c^8*e^17*f - 476*c^9*d^2*e^15*g + 940*b 
*c^8*d*e^16*g + 243*b^2*c^7*e^17*g)/(c^7*e^12))*x - (1152*c^9*d^2*e^15*f - 
 2272*b*c^8*d*e^16*f - 592*b^2*c^7*e^17*f + 1152*c^9*d^3*e^14*g + 76*b*c^8 
*d^2*e^15*g - 1820*b^2*c^7*d*e^16*g - 3*b^3*c^6*e^17*g)/(c^7*e^12))*x - (1 
1648*c^9*d^3*e^14*f + 960*b*c^8*d^2*e^15*f - 18656*b^2*c^7*d*e^16*f - 48*b 
^3*c^6*e^17*f - 6608*c^9*d^4*e^13*g + 25824*b*c^8*d^3*e^14*g - 19000*b^2*c 
^7*d^2*e^15*g - 264*b^3*c^6*d*e^16*g + 27*b^4*c^5*e^17*g)/(c^7*e^12))*x + 
(18432*c^9*d^4*e^13*f - 71808*b*c^8*d^3*e^14*f + 52416*b^2*c^7*d^2*e^15*f 
+ 1184*b^3*c^6*d*e^16*f - 112*b^4*c^5*e^17*f + 18432*c^9*d^5*e^12*g - 3547 
2*b*c^8*d^4*e^13*g + 14688*b^2*c^7*d^3*e^14*g + 2920*b^3*c^6*d^2*e^15*g - 
680*b^4*c^5*d*e^16*g + 63*b^5*c^4*e^17*g)/(c^7*e^12))*x + (118272*c^9*d^5* 
e^12*f - 221952*b*c^8*d^4*e^13*f + 78336*b^2*c^7*d^3*e^14*f + 30720*b^3*c^ 
6*d^2*e^15*f - 6496*b^4*c^5*d*e^16*f + 560*b^5*c^4*e^17*f - 6720*c^9*d^6*e 
^11*g + 34752*b*c^8*d^5*e^12*g - 58896*b^2*c^7*d^4*e^13*g + 45792*b^3*c^6* 
d^3*e^14*g - 18092*b^4*c^5*d^2*e^15*g + 3724*b^5*c^4*d*e^16*g - 315*b^6*c^ 
3*e^17*g)/(c^7*e^12))*x - (49152*c^9*d^6*e^11*f - 265728*b*c^8*d^5*e^12*f 
+ 443136*b^2*c^7*d^4*e^13*f - 321536*b^3*c^6*d^3*e^14*f + 112896*b^4*c^5*d 
^2*e^15*f - 21280*b^5*c^4*d*e^16*f + 1680*b^6*c^3*e^17*f + 49152*c^9*d^...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 8.50 (sec) , antiderivative size = 4284, normalized size of antiderivative = 11.70 \[ \int (d+e x) (f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*(945*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d)) 
*b**9*e**9*g - 15330*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - 
b*e + 2*c*d))*b**8*c*d*e**8*g - 1680*sqrt(c)*asinh((sqrt( - b*e + c*d - c* 
e*x)*i)/sqrt( - b*e + 2*c*d))*b**8*c*e**9*f + 109200*sqrt(c)*asinh((sqrt( 
- b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**7*c**2*d**2*e**7*g + 2688 
0*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**7* 
c**2*d*e**8*f - 446880*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( 
- b*e + 2*c*d))*b**6*c**3*d**3*e**6*g - 188160*sqrt(c)*asinh((sqrt( - b*e 
+ c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**6*c**3*d**2*e**7*f + 1152480*sq 
rt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**5*c**4 
*d**4*e**5*g + 752640*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - 
 b*e + 2*c*d))*b**5*c**4*d**3*e**6*f - 1928640*sqrt(c)*asinh((sqrt( - b*e 
+ c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**4*c**5*d**5*e**4*g - 1881600*sq 
rt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**4*c**5 
*d**4*e**5*f + 2069760*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( 
- b*e + 2*c*d))*b**3*c**6*d**6*e**3*g + 3010560*sqrt(c)*asinh((sqrt( - b*e 
 + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**3*c**6*d**5*e**4*f - 1344000*s 
qrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b**2*c** 
7*d**7*e**2*g - 3010560*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( 
 - b*e + 2*c*d))*b**2*c**7*d**6*e**3*f + 456960*sqrt(c)*asinh((sqrt( - ...