\(\int x (d+e x) (a+b x+c x^2)^{3/2} \, dx\) [76]

Optimal result

Integrand size = 21, antiderivative size = 198 \[ \int x (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {\left (b^2-4 a c\right ) \left (12 b c d-7 b^2 e+4 a c e\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{512 c^4}-\frac {\left (12 b c d-7 b^2 e+4 a c e\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{192 c^3}+\frac {(12 c d-7 b e+10 c e x) \left (a+b x+c x^2\right )^{5/2}}{60 c^2}-\frac {\left (b^2-4 a c\right )^2 \left (12 b c d-7 b^2 e+4 a c e\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{1024 c^{9/2}} \] Output:

1/512*(-4*a*c+b^2)*(4*a*c*e-7*b^2*e+12*b*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(1/2 
)/c^4-1/192*(4*a*c*e-7*b^2*e+12*b*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^3+1 
/60*(10*c*e*x-7*b*e+12*c*d)*(c*x^2+b*x+a)^(5/2)/c^2-1/1024*(-4*a*c+b^2)^2* 
(4*a*c*e-7*b^2*e+12*b*c*d)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/ 
2))/c^(9/2)
 

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.28 \[ \int x (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {2 \sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^5 e+10 b^4 c (18 d+7 e x)+8 b^3 c (95 a e-c x (15 d+7 e x))+48 b^2 c^2 \left (c x^2 (2 d+e x)-a (25 d+9 e x)\right )+16 b c^2 \left (-81 a^2 e+6 a c x (7 d+3 e x)+4 c^2 x^3 (33 d+26 e x)\right )+32 c^3 \left (8 c^2 x^4 (6 d+5 e x)+3 a^2 (16 d+5 e x)+2 a c x^2 (48 d+35 e x)\right )\right )-15 \left (b^2-4 a c\right )^2 \left (-12 b c d+7 b^2 e-4 a c e\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{15360 c^{9/2}} \] Input:

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

Output:

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^5*e + 10*b^4*c*(18*d + 7*e*x) + 8 
*b^3*c*(95*a*e - c*x*(15*d + 7*e*x)) + 48*b^2*c^2*(c*x^2*(2*d + e*x) - a*( 
25*d + 9*e*x)) + 16*b*c^2*(-81*a^2*e + 6*a*c*x*(7*d + 3*e*x) + 4*c^2*x^3*( 
33*d + 26*e*x)) + 32*c^3*(8*c^2*x^4*(6*d + 5*e*x) + 3*a^2*(16*d + 5*e*x) + 
 2*a*c*x^2*(48*d + 35*e*x))) - 15*(b^2 - 4*a*c)^2*(-12*b*c*d + 7*b^2*e - 4 
*a*c*e)*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(15360*c^(9/2))
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {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.

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

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-7 b e+12 c d+10 c e x)}{60 c^2}-\frac {\left (4 a c e-7 b^2 e+12 b c d\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{24 c^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-7 b e+12 c d+10 c e x)}{60 c^2}-\frac {\left (4 a c e-7 b^2 e+12 b c d\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{24 c^2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-7 b e+12 c d+10 c e x)}{60 c^2}-\frac {\left (4 a c e-7 b^2 e+12 b c d\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{24 c^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-7 b e+12 c d+10 c e x)}{60 c^2}-\frac {\left (4 a c e-7 b^2 e+12 b c d\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{24 c^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (a+b x+c x^2\right )^{5/2} (-7 b e+12 c d+10 c e x)}{60 c^2}-\frac {\left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right ) \left (4 a c e-7 b^2 e+12 b c d\right )}{24 c^2}\)

Input:

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

Output:

((12*c*d - 7*b*e + 10*c*e*x)*(a + b*x + c*x^2)^(5/2))/(60*c^2) - ((12*b*c* 
d - 7*b^2*e + 4*a*c*e)*(((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)*A 
rcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(3/2))))/(16*c 
)))/(24*c^2)
 

Defintions of rubi rules used

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]
 

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.60

method	result	size
risch	\(-\frac {\left (-1280 c^{5} e \,x^{5}-1664 b \,c^{4} e \,x^{4}-1536 c^{5} d \,x^{4}-2240 a \,c^{4} e \,x^{3}-48 b^{2} c^{3} e \,x^{3}-2112 b \,c^{4} d \,x^{3}-288 b \,c^{3} a e \,x^{2}-3072 a \,c^{4} d \,x^{2}+56 b^{3} c^{2} e \,x^{2}-96 b^{2} c^{3} d \,x^{2}-480 a^{2} c^{3} e x +432 a \,b^{2} c^{2} e x -672 a b \,c^{3} d x -70 b^{4} c e x +120 b^{3} c^{2} d x +1296 a^{2} b \,c^{2} e -1536 a^{2} c^{3} d -760 a \,b^{3} c e +1200 a \,b^{2} c^{2} d +105 b^{5} e -180 b^{4} c d \right ) \sqrt {c \,x^{2}+b x +a}}{7680 c^{4}}-\frac {\left (64 a^{3} c^{3} e -144 a^{2} b^{2} c^{2} e +192 a^{2} b \,c^{3} d +60 a \,b^{4} c e -96 a \,b^{3} c^{2} d -7 b^{6} e +12 b^{5} c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{1024 c^{\frac {9}{2}}}\)	\(316\)
default	\(d \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )+e \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{6 c}-\frac {7 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{5 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{2 c}\right )}{12 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{6 c}\right )\)	\(394\)

Input:

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

Output:

-1/7680/c^4*(-1280*c^5*e*x^5-1664*b*c^4*e*x^4-1536*c^5*d*x^4-2240*a*c^4*e* 
x^3-48*b^2*c^3*e*x^3-2112*b*c^4*d*x^3-288*a*b*c^3*e*x^2-3072*a*c^4*d*x^2+5 
6*b^3*c^2*e*x^2-96*b^2*c^3*d*x^2-480*a^2*c^3*e*x+432*a*b^2*c^2*e*x-672*a*b 
*c^3*d*x-70*b^4*c*e*x+120*b^3*c^2*d*x+1296*a^2*b*c^2*e-1536*a^2*c^3*d-760* 
a*b^3*c*e+1200*a*b^2*c^2*d+105*b^5*e-180*b^4*c*d)*(c*x^2+b*x+a)^(1/2)-1/10 
24*(64*a^3*c^3*e-144*a^2*b^2*c^2*e+192*a^2*b*c^3*d+60*a*b^4*c*e-96*a*b^3*c 
^2*d-7*b^6*e+12*b^5*c*d)/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2 
))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 689, normalized size of antiderivative = 3.48 \[ \int x (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/30720*(15*(12*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d - (7*b^6 - 60*a*b^ 
4*c + 144*a^2*b^2*c^2 - 64*a^3*c^3)*e)*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 
*x^5 + 128*(12*c^6*d + 13*b*c^5*e)*x^4 + 16*(132*b*c^5*d + (3*b^2*c^4 + 14 
0*a*c^5)*e)*x^3 + 8*(12*(b^2*c^4 + 32*a*c^5)*d - (7*b^3*c^3 - 36*a*b*c^4)* 
e)*x^2 + 12*(15*b^4*c^2 - 100*a*b^2*c^3 + 128*a^2*c^4)*d - (105*b^5*c - 76 
0*a*b^3*c^2 + 1296*a^2*b*c^3)*e - 2*(12*(5*b^3*c^3 - 28*a*b*c^4)*d - (35*b 
^4*c^2 - 216*a*b^2*c^3 + 240*a^2*c^4)*e)*x)*sqrt(c*x^2 + b*x + a))/c^5, 1/ 
15360*(15*(12*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d - (7*b^6 - 60*a*b^4*c 
 + 144*a^2*b^2*c^2 - 64*a^3*c^3)*e)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + 
 a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)) + 2*(1280*c^6*e*x^5 + 12 
8*(12*c^6*d + 13*b*c^5*e)*x^4 + 16*(132*b*c^5*d + (3*b^2*c^4 + 140*a*c^5)* 
e)*x^3 + 8*(12*(b^2*c^4 + 32*a*c^5)*d - (7*b^3*c^3 - 36*a*b*c^4)*e)*x^2 + 
12*(15*b^4*c^2 - 100*a*b^2*c^3 + 128*a^2*c^4)*d - (105*b^5*c - 760*a*b^3*c 
^2 + 1296*a^2*b*c^3)*e - 2*(12*(5*b^3*c^3 - 28*a*b*c^4)*d - (35*b^4*c^2 - 
216*a*b^2*c^3 + 240*a^2*c^4)*e)*x)*sqrt(c*x^2 + b*x + a))/c^5]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1175 vs. \(2 (199) = 398\).

Time = 0.81 (sec) , antiderivative size = 1175, normalized size of antiderivative = 5.93 \[ \int x (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise(((-a*(a**2*e + 2*a*b*d - 3*a*(7*a*c*e/6 + b**2*e + 2*b*c*d - 9*b 
*(13*b*c*e/12 + c**2*d)/(10*c))/(4*c) - 5*b*(2*a*b*e + 2*a*c*d - 4*a*(13*b 
*c*e/12 + c**2*d)/(5*c) + b**2*d - 7*b*(7*a*c*e/6 + b**2*e + 2*b*c*d - 9*b 
*(13*b*c*e/12 + c**2*d)/(10*c))/(8*c))/(6*c))/(2*c) - b*(a**2*d - 2*a*(2*a 
*b*e + 2*a*c*d - 4*a*(13*b*c*e/12 + c**2*d)/(5*c) + b**2*d - 7*b*(7*a*c*e/ 
6 + b**2*e + 2*b*c*d - 9*b*(13*b*c*e/12 + c**2*d)/(10*c))/(8*c))/(3*c) - 3 
*b*(a**2*e + 2*a*b*d - 3*a*(7*a*c*e/6 + b**2*e + 2*b*c*d - 9*b*(13*b*c*e/1 
2 + c**2*d)/(10*c))/(4*c) - 5*b*(2*a*b*e + 2*a*c*d - 4*a*(13*b*c*e/12 + c* 
*2*d)/(5*c) + b**2*d - 7*b*(7*a*c*e/6 + b**2*e + 2*b*c*d - 9*b*(13*b*c*e/1 
2 + c**2*d)/(10*c))/(8*c))/(6*c))/(4*c))/(2*c))*Piecewise((log(b + 2*sqrt( 
c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(a - b**2/(4*c), 0)), ((b/(2 
*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True)) + sqrt(a + b*x 
+ c*x**2)*(c*e*x**5/6 + x**4*(13*b*c*e/12 + c**2*d)/(5*c) + x**3*(7*a*c*e/ 
6 + b**2*e + 2*b*c*d - 9*b*(13*b*c*e/12 + c**2*d)/(10*c))/(4*c) + x**2*(2* 
a*b*e + 2*a*c*d - 4*a*(13*b*c*e/12 + c**2*d)/(5*c) + b**2*d - 7*b*(7*a*c*e 
/6 + b**2*e + 2*b*c*d - 9*b*(13*b*c*e/12 + c**2*d)/(10*c))/(8*c))/(3*c) + 
x*(a**2*e + 2*a*b*d - 3*a*(7*a*c*e/6 + b**2*e + 2*b*c*d - 9*b*(13*b*c*e/12 
 + c**2*d)/(10*c))/(4*c) - 5*b*(2*a*b*e + 2*a*c*d - 4*a*(13*b*c*e/12 + c** 
2*d)/(5*c) + b**2*d - 7*b*(7*a*c*e/6 + b**2*e + 2*b*c*d - 9*b*(13*b*c*e/12 
 + c**2*d)/(10*c))/(8*c))/(6*c))/(2*c) + (a**2*d - 2*a*(2*a*b*e + 2*a*c...
 

Maxima [F(-2)]

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

integrate(x*(e*x+d)*(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 [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.66 \[ \int x (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {1}{7680} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c e x + \frac {12 \, c^{6} d + 13 \, b c^{5} e}{c^{5}}\right )} x + \frac {132 \, b c^{5} d + 3 \, b^{2} c^{4} e + 140 \, a c^{5} e}{c^{5}}\right )} x + \frac {12 \, b^{2} c^{4} d + 384 \, a c^{5} d - 7 \, b^{3} c^{3} e + 36 \, a b c^{4} e}{c^{5}}\right )} x - \frac {60 \, b^{3} c^{3} d - 336 \, a b c^{4} d - 35 \, b^{4} c^{2} e + 216 \, a b^{2} c^{3} e - 240 \, a^{2} c^{4} e}{c^{5}}\right )} x + \frac {180 \, b^{4} c^{2} d - 1200 \, a b^{2} c^{3} d + 1536 \, a^{2} c^{4} d - 105 \, b^{5} c e + 760 \, a b^{3} c^{2} e - 1296 \, a^{2} b c^{3} e}{c^{5}}\right )} + \frac {{\left (12 \, b^{5} c d - 96 \, a b^{3} c^{2} d + 192 \, a^{2} b c^{3} d - 7 \, b^{6} e + 60 \, a b^{4} c e - 144 \, a^{2} b^{2} c^{2} e + 64 \, a^{3} c^{3} e\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \] Input:

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

Output:

1/7680*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(10*c*e*x + (12*c^6*d + 13*b*c^5* 
e)/c^5)*x + (132*b*c^5*d + 3*b^2*c^4*e + 140*a*c^5*e)/c^5)*x + (12*b^2*c^4 
*d + 384*a*c^5*d - 7*b^3*c^3*e + 36*a*b*c^4*e)/c^5)*x - (60*b^3*c^3*d - 33 
6*a*b*c^4*d - 35*b^4*c^2*e + 216*a*b^2*c^3*e - 240*a^2*c^4*e)/c^5)*x + (18 
0*b^4*c^2*d - 1200*a*b^2*c^3*d + 1536*a^2*c^4*d - 105*b^5*c*e + 760*a*b^3* 
c^2*e - 1296*a^2*b*c^3*e)/c^5) + 1/1024*(12*b^5*c*d - 96*a*b^3*c^2*d + 192 
*a^2*b*c^3*d - 7*b^6*e + 60*a*b^4*c*e - 144*a^2*b^2*c^2*e + 64*a^3*c^3*e)* 
log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(9/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 2.74 (sec) , antiderivative size = 775, normalized size of antiderivative = 3.91 \[ \int x (d+e x) \left (a+b x+c x^2\right )^{3/2} \, dx=\frac {3072 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{4} d -210 \sqrt {c \,x^{2}+b x +a}\, b^{5} c e +360 \sqrt {c \,x^{2}+b x +a}\, b^{4} c^{2} d +3072 \sqrt {c \,x^{2}+b x +a}\, c^{6} d \,x^{4}+2560 \sqrt {c \,x^{2}+b x +a}\, c^{6} e \,x^{5}+105 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{6} e -2400 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{3} d +6144 \sqrt {c \,x^{2}+b x +a}\, a \,c^{5} d \,x^{2}+4480 \sqrt {c \,x^{2}+b x +a}\, a \,c^{5} e \,x^{3}+140 \sqrt {c \,x^{2}+b x +a}\, b^{4} c^{2} e x -240 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{3} d x -112 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{3} e \,x^{2}+192 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{4} d \,x^{2}+96 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{4} e \,x^{3}+4224 \sqrt {c \,x^{2}+b x +a}\, b \,c^{5} d \,x^{3}+3328 \sqrt {c \,x^{2}+b x +a}\, b \,c^{5} e \,x^{4}-960 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{3} c^{3} e -180 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{5} c d -864 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{3} e x +1344 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{4} d x +576 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{4} e \,x^{2}+2160 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b^{2} c^{2} e -2880 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b \,c^{3} d -900 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{4} c e +1440 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{3} c^{2} d -2592 \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,c^{3} e +960 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{4} e x +1520 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} c^{2} e}{15360 c^{5}} \] Input:

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

Output:

( - 2592*sqrt(a + b*x + c*x**2)*a**2*b*c**3*e + 3072*sqrt(a + b*x + c*x**2 
)*a**2*c**4*d + 960*sqrt(a + b*x + c*x**2)*a**2*c**4*e*x + 1520*sqrt(a + b 
*x + c*x**2)*a*b**3*c**2*e - 2400*sqrt(a + b*x + c*x**2)*a*b**2*c**3*d - 8 
64*sqrt(a + b*x + c*x**2)*a*b**2*c**3*e*x + 1344*sqrt(a + b*x + c*x**2)*a* 
b*c**4*d*x + 576*sqrt(a + b*x + c*x**2)*a*b*c**4*e*x**2 + 6144*sqrt(a + b* 
x + c*x**2)*a*c**5*d*x**2 + 4480*sqrt(a + b*x + c*x**2)*a*c**5*e*x**3 - 21 
0*sqrt(a + b*x + c*x**2)*b**5*c*e + 360*sqrt(a + b*x + c*x**2)*b**4*c**2*d 
 + 140*sqrt(a + b*x + c*x**2)*b**4*c**2*e*x - 240*sqrt(a + b*x + c*x**2)*b 
**3*c**3*d*x - 112*sqrt(a + b*x + c*x**2)*b**3*c**3*e*x**2 + 192*sqrt(a + 
b*x + c*x**2)*b**2*c**4*d*x**2 + 96*sqrt(a + b*x + c*x**2)*b**2*c**4*e*x** 
3 + 4224*sqrt(a + b*x + c*x**2)*b*c**5*d*x**3 + 3328*sqrt(a + b*x + c*x**2 
)*b*c**5*e*x**4 + 3072*sqrt(a + b*x + c*x**2)*c**6*d*x**4 + 2560*sqrt(a + 
b*x + c*x**2)*c**6*e*x**5 - 960*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x* 
*2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*c**3*e + 2160*sqrt(c)*log((2*sqr 
t(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**2*c** 
2*e - 2880*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt 
(4*a*c - b**2))*a**2*b*c**3*d - 900*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + 
c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**4*c*e + 1440*sqrt(c)*log((2* 
sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*b**3*c** 
2*d + 105*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sq...