\(\int (d+e x)^2 (f+g x) (a+b x+c x^2)^{3/2} \, dx\) [881]

Optimal result

Integrand size = 27, antiderivative size = 438 \[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{3/2} \, dx=-\frac {\left (b^2-4 a c\right ) \left (48 c^3 d^2 f-9 b^3 e^2 g+2 b c e (6 a e g+7 b (e f+2 d g))-8 c^2 (3 b d (2 e f+d g)+a e (e f+2 d g))\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{1024 c^5}+\frac {\left (48 c^3 d^2 f-9 b^3 e^2 g+2 b c e (6 a e g+7 b (e f+2 d g))-8 c^2 (3 b d (2 e f+d g)+a e (e f+2 d g))\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{384 c^4}+\frac {g (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}+\frac {\left (63 b^2 e^2 g+48 c^2 d (7 e f+d g)-2 c e (24 a e g+49 b (e f+2 d g))+10 c e (14 c e f+4 c d g-9 b e g) x\right ) \left (a+b x+c x^2\right )^{5/2}}{840 c^3}+\frac {\left (b^2-4 a c\right )^2 \left (48 c^3 d^2 f-9 b^3 e^2 g+2 b c e (6 a e g+7 b (e f+2 d g))-8 c^2 (3 b d (2 e f+d g)+a e (e f+2 d g))\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2048 c^{11/2}} \] Output:

-1/1024*(-4*a*c+b^2)*(48*c^3*d^2*f-9*b^3*e^2*g+2*b*c*e*(6*a*e*g+7*b*(2*d*g 
+e*f))-8*c^2*(3*b*d*(d*g+2*e*f)+a*e*(2*d*g+e*f)))*(2*c*x+b)*(c*x^2+b*x+a)^ 
(1/2)/c^5+1/384*(48*c^3*d^2*f-9*b^3*e^2*g+2*b*c*e*(6*a*e*g+7*b*(2*d*g+e*f) 
)-8*c^2*(3*b*d*(d*g+2*e*f)+a*e*(2*d*g+e*f)))*(2*c*x+b)*(c*x^2+b*x+a)^(3/2) 
/c^4+1/7*g*(e*x+d)^2*(c*x^2+b*x+a)^(5/2)/c+1/840*(63*b^2*e^2*g+48*c^2*d*(d 
*g+7*e*f)-2*c*e*(24*a*e*g+49*b*(2*d*g+e*f))+10*c*e*(-9*b*e*g+4*c*d*g+14*c* 
e*f)*x)*(c*x^2+b*x+a)^(5/2)/c^3+1/2048*(-4*a*c+b^2)^2*(48*c^3*d^2*f-9*b^3* 
e^2*g+2*b*c*e*(6*a*e*g+7*b*(2*d*g+e*f))-8*c^2*(3*b*d*(d*g+2*e*f)+a*e*(2*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 = 7.10 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.52 \[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (945 b^6 e^2 g-210 b^5 c e (7 e f+14 d g+3 e g x)+28 b^4 c \left (-270 a e^2 g+c \left (90 d^2 g+10 d e (18 f+7 g x)+e^2 x (35 f+18 g x)\right )\right )-16 b^3 c^2 \left (-7 a e (95 e f+190 d g+39 e g x)+c \left (105 d^2 (3 f+g x)+14 d e x (15 f+7 g x)+e^2 x^2 (49 f+27 g x)\right )\right )+48 b^2 c^2 \left (343 a^2 e^2 g+2 c^2 x \left (14 d e x (2 f+g x)+7 d^2 (5 f+2 g x)+e^2 x^2 (7 f+4 g x)\right )-2 a c \left (175 d^2 g+14 d e (25 f+9 g x)+e^2 x (63 f+31 g x)\right )\right )+32 b c^3 \left (-3 a^2 e (189 e f+378 d g+73 e g x)+6 a c \left (14 d e x (7 f+3 g x)+7 d^2 (25 f+7 g x)+e^2 x^2 (21 f+11 g x)\right )+4 c^2 x^2 \left (21 d^2 (15 f+11 g x)+14 d e x (33 f+26 g x)+2 e^2 x^2 (91 f+75 g x)\right )\right )+64 c^3 \left (-96 a^3 e^2 g+4 c^3 x^3 \left (21 d^2 (5 f+4 g x)+28 d e x (6 f+5 g x)+10 e^2 x^2 (7 f+6 g x)\right )+3 a^2 c \left (112 d^2 g+14 d e (16 f+5 g x)+e^2 x (35 f+16 g x)\right )+2 a c^2 x \left (21 d^2 (25 f+16 g x)+14 d e x (48 f+35 g x)+e^2 x^2 (245 f+192 g x)\right )\right )\right )+105 \left (b^2-4 a c\right )^2 \left (-48 c^3 d^2 f+9 b^3 e^2 g-2 b c e (6 a e g+7 b (e f+2 d g))+8 c^2 (3 b d (2 e f+d g)+a e (e f+2 d g))\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{215040 c^{11/2}} \] Input:

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

Output:

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^6*e^2*g - 210*b^5*c*e*(7*e*f + 14* 
d*g + 3*e*g*x) + 28*b^4*c*(-270*a*e^2*g + c*(90*d^2*g + 10*d*e*(18*f + 7*g 
*x) + e^2*x*(35*f + 18*g*x))) - 16*b^3*c^2*(-7*a*e*(95*e*f + 190*d*g + 39* 
e*g*x) + c*(105*d^2*(3*f + g*x) + 14*d*e*x*(15*f + 7*g*x) + e^2*x^2*(49*f 
+ 27*g*x))) + 48*b^2*c^2*(343*a^2*e^2*g + 2*c^2*x*(14*d*e*x*(2*f + g*x) + 
7*d^2*(5*f + 2*g*x) + e^2*x^2*(7*f + 4*g*x)) - 2*a*c*(175*d^2*g + 14*d*e*( 
25*f + 9*g*x) + e^2*x*(63*f + 31*g*x))) + 32*b*c^3*(-3*a^2*e*(189*e*f + 37 
8*d*g + 73*e*g*x) + 6*a*c*(14*d*e*x*(7*f + 3*g*x) + 7*d^2*(25*f + 7*g*x) + 
 e^2*x^2*(21*f + 11*g*x)) + 4*c^2*x^2*(21*d^2*(15*f + 11*g*x) + 14*d*e*x*( 
33*f + 26*g*x) + 2*e^2*x^2*(91*f + 75*g*x))) + 64*c^3*(-96*a^3*e^2*g + 4*c 
^3*x^3*(21*d^2*(5*f + 4*g*x) + 28*d*e*x*(6*f + 5*g*x) + 10*e^2*x^2*(7*f + 
6*g*x)) + 3*a^2*c*(112*d^2*g + 14*d*e*(16*f + 5*g*x) + e^2*x*(35*f + 16*g* 
x)) + 2*a*c^2*x*(21*d^2*(25*f + 16*g*x) + 14*d*e*x*(48*f + 35*g*x) + e^2*x 
^2*(245*f + 192*g*x)))) + 105*(b^2 - 4*a*c)^2*(-48*c^3*d^2*f + 9*b^3*e^2*g 
 - 2*b*c*e*(6*a*e*g + 7*b*(e*f + 2*d*g)) + 8*c^2*(3*b*d*(2*e*f + d*g) + a* 
e*(e*f + 2*d*g)))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(21504 
0*c^(11/2))
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.73, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1236, 27, 1225, 1087, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\int \frac {1}{2} (d+e x) (14 c d f-5 b d g-4 a e g+(14 c e f+4 c d g-9 b e g) x) \left (c x^2+b x+a\right )^{3/2}dx}{7 c}+\frac {g (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (d+e x) (14 c d f-5 b d g-4 a e g+(14 c e f+4 c d g-9 b e g) x) \left (c x^2+b x+a\right )^{3/2}dx}{14 c}+\frac {g (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {7 \left (-8 c^2 (a e (2 d g+e f)+3 b d (d g+2 e f))+2 b c e (6 a e g+7 b (2 d g+e f))-9 b^3 e^2 g+48 c^3 d^2 f\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{24 c^2}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (-2 c e (24 a e g+49 b (2 d g+e f))+63 b^2 e^2 g+10 c e x (-9 b e g+4 c d g+14 c e f)+48 c^2 d (d g+7 e f)\right )}{60 c^2}}{14 c}+\frac {g (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {7 \left (-8 c^2 (a e (2 d g+e f)+3 b d (d g+2 e f))+2 b c e (6 a e g+7 b (2 d g+e f))-9 b^3 e^2 g+48 c^3 d^2 f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{24 c^2}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (-2 c e (24 a e g+49 b (2 d g+e f))+63 b^2 e^2 g+10 c e x (-9 b e g+4 c d g+14 c e f)+48 c^2 d (d g+7 e f)\right )}{60 c^2}}{14 c}+\frac {g (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {7 \left (-8 c^2 (a e (2 d g+e f)+3 b d (d g+2 e f))+2 b c e (6 a e g+7 b (2 d g+e f))-9 b^3 e^2 g+48 c^3 d^2 f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{24 c^2}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (-2 c e (24 a e g+49 b (2 d g+e f))+63 b^2 e^2 g+10 c e x (-9 b e g+4 c d g+14 c e f)+48 c^2 d (d g+7 e f)\right )}{60 c^2}}{14 c}+\frac {g (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {7 \left (-8 c^2 (a e (2 d g+e f)+3 b d (d g+2 e f))+2 b c e (6 a e g+7 b (2 d g+e f))-9 b^3 e^2 g+48 c^3 d^2 f\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{24 c^2}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (-2 c e (24 a e g+49 b (2 d g+e f))+63 b^2 e^2 g+10 c e x (-9 b e g+4 c d g+14 c e f)+48 c^2 d (d g+7 e f)\right )}{60 c^2}}{14 c}+\frac {g (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {7 \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right ) \left (-8 c^2 (a e (2 d g+e f)+3 b d (d g+2 e f))+2 b c e (6 a e g+7 b (2 d g+e f))-9 b^3 e^2 g+48 c^3 d^2 f\right )}{24 c^2}+\frac {\left (a+b x+c x^2\right )^{5/2} \left (-2 c e (24 a e g+49 b (2 d g+e f))+63 b^2 e^2 g+10 c e x (-9 b e g+4 c d g+14 c e f)+48 c^2 d (d g+7 e f)\right )}{60 c^2}}{14 c}+\frac {g (d+e x)^2 \left (a+b x+c x^2\right )^{5/2}}{7 c}\)

Input:

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

Output:

(g*(d + e*x)^2*(a + b*x + c*x^2)^(5/2))/(7*c) + (((63*b^2*e^2*g + 48*c^2*d 
*(7*e*f + d*g) - 2*c*e*(24*a*e*g + 49*b*(e*f + 2*d*g)) + 10*c*e*(14*c*e*f 
+ 4*c*d*g - 9*b*e*g)*x)*(a + b*x + c*x^2)^(5/2))/(60*c^2) + (7*(48*c^3*d^2 
*f - 9*b^3*e^2*g + 2*b*c*e*(6*a*e*g + 7*b*(e*f + 2*d*g)) - 8*c^2*(3*b*d*(2 
*e*f + d*g) + a*e*(e*f + 2*d*g)))*(((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/( 
8*c) - (3*(b^2 - 4*a*c)*(((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)))/(24*c^2))/(14*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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(943\) vs. \(2(412)=824\).

Time = 2.27 (sec) , antiderivative size = 944, normalized size of antiderivative = 2.16

method	result	size
default	\(\text {Expression too large to display}\)	\(944\)
risch	\(\text {Expression too large to display}\)	\(1116\)

Input:

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

Output:

d^2*f*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+ 
b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c 
*x^2+b*x+a)^(1/2))))+e*(2*d*g+e*f)*(1/6*x*(c*x^2+b*x+a)^(5/2)/c-7/12*b/c*( 
1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/1 
6*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/ 
2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/6*a/c*(1/8*(2*c*x+b)*( 
c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2) 
/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))+ 
d*(d*g+2*e*f)*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)*(c*x^2+b*x 
+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4 
*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))+e^2*g*(1/ 
7*x^2*(c*x^2+b*x+a)^(5/2)/c-9/14*b/c*(1/6*x*(c*x^2+b*x+a)^(5/2)/c-7/12*b/c 
*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3 
/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^( 
3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))))-1/6*a/c*(1/8*(2*c*x+b) 
*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/ 
2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))) 
)-2/7*a/c*(1/5*(c*x^2+b*x+a)^(5/2)/c-1/2*b/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^ 
(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c 
-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 980 vs. \(2 (412) = 824\).

Time = 0.39 (sec) , antiderivative size = 1963, normalized size of antiderivative = 4.48 \[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/430080*(105*(2*(24*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^2 - 24*(b^5*c 
^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e + (7*b^6*c - 60*a*b^4*c^2 + 144*a^2*b 
^2*c^3 - 64*a^3*c^4)*e^2)*f - (24*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d 
^2 - 4*(7*b^6*c - 60*a*b^4*c^2 + 144*a^2*b^2*c^3 - 64*a^3*c^4)*d*e + 3*(3* 
b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*c^3)*e^2)*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*(15360*c^7*e^2*g*x^6 + 1280*(14*c^7*e^2*f + (28*c^7*d*e + 15*b*c^6 
*e^2)*g)*x^5 + 128*(14*(24*c^7*d*e + 13*b*c^6*e^2)*f + (168*c^7*d^2 + 364* 
b*c^6*d*e + 3*(b^2*c^5 + 64*a*c^6)*e^2)*g)*x^4 + 16*(14*(120*c^7*d^2 + 264 
*b*c^6*d*e + (3*b^2*c^5 + 140*a*c^6)*e^2)*f + (1848*b*c^6*d^2 + 28*(3*b^2* 
c^5 + 140*a*c^6)*d*e - 3*(9*b^3*c^4 - 44*a*b*c^5)*e^2)*g)*x^3 + 8*(14*(360 
*b*c^6*d^2 + 24*(b^2*c^5 + 32*a*c^6)*d*e - (7*b^3*c^4 - 36*a*b*c^5)*e^2)*f 
 + (168*(b^2*c^5 + 32*a*c^6)*d^2 - 28*(7*b^3*c^4 - 36*a*b*c^5)*d*e + 3*(21 
*b^4*c^3 - 124*a*b^2*c^4 + 128*a^2*c^5)*e^2)*g)*x^2 - 14*(120*(3*b^3*c^4 - 
 20*a*b*c^5)*d^2 - 24*(15*b^4*c^3 - 100*a*b^2*c^4 + 128*a^2*c^5)*d*e + (10 
5*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^2*b*c^4)*e^2)*f + (168*(15*b^4*c^3 - 10 
0*a*b^2*c^4 + 128*a^2*c^5)*d^2 - 28*(105*b^5*c^2 - 760*a*b^3*c^3 + 1296*a^ 
2*b*c^4)*d*e + 3*(315*b^6*c - 2520*a*b^4*c^2 + 5488*a^2*b^2*c^3 - 2048*a^3 
*c^4)*e^2)*g + 2*(14*(120*(b^2*c^5 + 20*a*c^6)*d^2 - 24*(5*b^3*c^4 - 28*a* 
b*c^5)*d*e + (35*b^4*c^3 - 216*a*b^2*c^4 + 240*a^2*c^5)*e^2)*f - (168*(...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4854 vs. \(2 (454) = 908\).

Time = 1.07 (sec) , antiderivative size = 4854, normalized size of antiderivative = 11.08 \[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(c*e**2*g*x**6/7 + x**5*(15*b*c*e**2*g/1 
4 + 2*c**2*d*e*g + c**2*e**2*f)/(6*c) + x**4*(8*a*c*e**2*g/7 + b**2*e**2*g 
 + 4*b*c*d*e*g + 2*b*c*e**2*f - 11*b*(15*b*c*e**2*g/14 + 2*c**2*d*e*g + c* 
*2*e**2*f)/(12*c) + c**2*d**2*g + 2*c**2*d*e*f)/(5*c) + x**3*(2*a*b*e**2*g 
 + 4*a*c*d*e*g + 2*a*c*e**2*f - 5*a*(15*b*c*e**2*g/14 + 2*c**2*d*e*g + c** 
2*e**2*f)/(6*c) + 2*b**2*d*e*g + b**2*e**2*f + 2*b*c*d**2*g + 4*b*c*d*e*f 
- 9*b*(8*a*c*e**2*g/7 + b**2*e**2*g + 4*b*c*d*e*g + 2*b*c*e**2*f - 11*b*(1 
5*b*c*e**2*g/14 + 2*c**2*d*e*g + c**2*e**2*f)/(12*c) + c**2*d**2*g + 2*c** 
2*d*e*f)/(10*c) + c**2*d**2*f)/(4*c) + x**2*(a**2*e**2*g + 4*a*b*d*e*g + 2 
*a*b*e**2*f + 2*a*c*d**2*g + 4*a*c*d*e*f - 4*a*(8*a*c*e**2*g/7 + b**2*e**2 
*g + 4*b*c*d*e*g + 2*b*c*e**2*f - 11*b*(15*b*c*e**2*g/14 + 2*c**2*d*e*g + 
c**2*e**2*f)/(12*c) + c**2*d**2*g + 2*c**2*d*e*f)/(5*c) + b**2*d**2*g + 2* 
b**2*d*e*f + 2*b*c*d**2*f - 7*b*(2*a*b*e**2*g + 4*a*c*d*e*g + 2*a*c*e**2*f 
 - 5*a*(15*b*c*e**2*g/14 + 2*c**2*d*e*g + c**2*e**2*f)/(6*c) + 2*b**2*d*e* 
g + b**2*e**2*f + 2*b*c*d**2*g + 4*b*c*d*e*f - 9*b*(8*a*c*e**2*g/7 + b**2* 
e**2*g + 4*b*c*d*e*g + 2*b*c*e**2*f - 11*b*(15*b*c*e**2*g/14 + 2*c**2*d*e* 
g + c**2*e**2*f)/(12*c) + c**2*d**2*g + 2*c**2*d*e*f)/(10*c) + c**2*d**2*f 
)/(8*c))/(3*c) + x*(2*a**2*d*e*g + a**2*e**2*f + 2*a*b*d**2*g + 4*a*b*d*e* 
f + 2*a*c*d**2*f - 3*a*(2*a*b*e**2*g + 4*a*c*d*e*g + 2*a*c*e**2*f - 5*a*(1 
5*b*c*e**2*g/14 + 2*c**2*d*e*g + c**2*e**2*f)/(6*c) + 2*b**2*d*e*g + b*...
 

Maxima [F(-2)]

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

integrate((e*x+d)^2*(g*x+f)*(c*x^2+b*x+a)^(3/2),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. 1067 vs. \(2 (412) = 824\).

Time = 0.21 (sec) , antiderivative size = 1067, normalized size of antiderivative = 2.44 \[ \int (d+e x)^2 (f+g x) \left (a+b x+c x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/107520*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*(12*c*e^2*g*x + (14*c^7*e^2 
*f + 28*c^7*d*e*g + 15*b*c^6*e^2*g)/c^6)*x + (336*c^7*d*e*f + 182*b*c^6*e^ 
2*f + 168*c^7*d^2*g + 364*b*c^6*d*e*g + 3*b^2*c^5*e^2*g + 192*a*c^6*e^2*g) 
/c^6)*x + (1680*c^7*d^2*f + 3696*b*c^6*d*e*f + 42*b^2*c^5*e^2*f + 1960*a*c 
^6*e^2*f + 1848*b*c^6*d^2*g + 84*b^2*c^5*d*e*g + 3920*a*c^6*d*e*g - 27*b^3 
*c^4*e^2*g + 132*a*b*c^5*e^2*g)/c^6)*x + (5040*b*c^6*d^2*f + 336*b^2*c^5*d 
*e*f + 10752*a*c^6*d*e*f - 98*b^3*c^4*e^2*f + 504*a*b*c^5*e^2*f + 168*b^2* 
c^5*d^2*g + 5376*a*c^6*d^2*g - 196*b^3*c^4*d*e*g + 1008*a*b*c^5*d*e*g + 63 
*b^4*c^3*e^2*g - 372*a*b^2*c^4*e^2*g + 384*a^2*c^5*e^2*g)/c^6)*x + (1680*b 
^2*c^5*d^2*f + 33600*a*c^6*d^2*f - 1680*b^3*c^4*d*e*f + 9408*a*b*c^5*d*e*f 
 + 490*b^4*c^3*e^2*f - 3024*a*b^2*c^4*e^2*f + 3360*a^2*c^5*e^2*f - 840*b^3 
*c^4*d^2*g + 4704*a*b*c^5*d^2*g + 980*b^4*c^3*d*e*g - 6048*a*b^2*c^4*d*e*g 
 + 6720*a^2*c^5*d*e*g - 315*b^5*c^2*e^2*g + 2184*a*b^3*c^3*e^2*g - 3504*a^ 
2*b*c^4*e^2*g)/c^6)*x - (5040*b^3*c^4*d^2*f - 33600*a*b*c^5*d^2*f - 5040*b 
^4*c^3*d*e*f + 33600*a*b^2*c^4*d*e*f - 43008*a^2*c^5*d*e*f + 1470*b^5*c^2* 
e^2*f - 10640*a*b^3*c^3*e^2*f + 18144*a^2*b*c^4*e^2*f - 2520*b^4*c^3*d^2*g 
 + 16800*a*b^2*c^4*d^2*g - 21504*a^2*c^5*d^2*g + 2940*b^5*c^2*d*e*g - 2128 
0*a*b^3*c^3*d*e*g + 36288*a^2*b*c^4*d*e*g - 945*b^6*c*e^2*g + 7560*a*b^4*c 
^2*e^2*g - 16464*a^2*b^2*c^3*e^2*g + 6144*a^3*c^4*e^2*g)/c^6) - 1/2048*(48 
*b^4*c^3*d^2*f - 384*a*b^2*c^4*d^2*f + 768*a^2*c^5*d^2*f - 48*b^5*c^2*d...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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