\(\int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx\) [573]

Optimal result

Integrand size = 22, antiderivative size = 248 \[ \int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx=\frac {(2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{128 c^4}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}+\frac {e \left (192 c^2 d^2+35 b^2 e^2-2 c e (75 b d+16 a e)+42 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{240 c^3}-\frac {\left (b^2-4 a c\right ) (2 c d-b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{9/2}} \] Output:

1/128*(-b*e+2*c*d)*(16*c^2*d^2+7*b^2*e^2-4*c*e*(3*a*e+4*b*d))*(2*c*x+b)*(c 
*x^2+b*x+a)^(1/2)/c^4+1/5*e*(e*x+d)^2*(c*x^2+b*x+a)^(3/2)/c+1/240*e*(192*c 
^2*d^2+35*b^2*e^2-2*c*e*(16*a*e+75*b*d)+42*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x+ 
a)^(3/2)/c^3-1/256*(-4*a*c+b^2)*(-b*e+2*c*d)*(16*c^2*d^2+7*b^2*e^2-4*c*e*( 
3*a*e+4*b*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 = 2.31 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.19 \[ \int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (-105 b^4 e^3+10 b^3 c e^2 (45 d+7 e x)-4 b^2 c e \left (-115 a e^2+c \left (180 d^2+75 d e x+14 e^2 x^2\right )\right )+8 b c^2 \left (-a e^2 (195 d+29 e x)+6 c \left (10 d^3+10 d^2 e x+5 d e^2 x^2+e^3 x^3\right )\right )+16 c^2 \left (-16 a^2 e^3+a c e \left (120 d^2+45 d e x+8 e^2 x^2\right )+6 c^2 x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )\right )+15 \left (b^2-4 a c\right ) (-2 c d+b e) \left (16 c^2 d^2+7 b^2 e^2-4 c e (4 b d+3 a e)\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{1920 c^{9/2}} \] Input:

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

Output:

(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^4*e^3 + 10*b^3*c*e^2*(45*d + 7*e*x) 
 - 4*b^2*c*e*(-115*a*e^2 + c*(180*d^2 + 75*d*e*x + 14*e^2*x^2)) + 8*b*c^2* 
(-(a*e^2*(195*d + 29*e*x)) + 6*c*(10*d^3 + 10*d^2*e*x + 5*d*e^2*x^2 + e^3* 
x^3)) + 16*c^2*(-16*a^2*e^3 + a*c*e*(120*d^2 + 45*d*e*x + 8*e^2*x^2) + 6*c 
^2*x*(10*d^3 + 20*d^2*e*x + 15*d*e^2*x^2 + 4*e^3*x^3))) + 15*(b^2 - 4*a*c) 
*(-2*c*d + b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*ArcTanh[( 
Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/(1920*c^(9/2))
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1166, 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.

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

\(\Big \downarrow \) 1166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {5 (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \int \sqrt {c x^2+b x+a}dx}{16 c^2}+\frac {e \left (a+b x+c x^2\right )^{3/2} \left (-2 c e (16 a e+75 b d)+35 b^2 e^2+42 c e x (2 c d-b e)+192 c^2 d^2\right )}{24 c^2}}{10 c}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {5 (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c^2}+\frac {e \left (a+b x+c x^2\right )^{3/2} \left (-2 c e (16 a e+75 b d)+35 b^2 e^2+42 c e x (2 c d-b e)+192 c^2 d^2\right )}{24 c^2}}{10 c}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {5 (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c^2}+\frac {e \left (a+b x+c x^2\right )^{3/2} \left (-2 c e (16 a e+75 b d)+35 b^2 e^2+42 c e x (2 c d-b e)+192 c^2 d^2\right )}{24 c^2}}{10 c}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {5 (2 c d-b e) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right ) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right )}{16 c^2}+\frac {e \left (a+b x+c x^2\right )^{3/2} \left (-2 c e (16 a e+75 b d)+35 b^2 e^2+42 c e x (2 c d-b e)+192 c^2 d^2\right )}{24 c^2}}{10 c}+\frac {e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c}\)

Input:

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

Output:

(e*(d + e*x)^2*(a + b*x + c*x^2)^(3/2))/(5*c) + ((e*(192*c^2*d^2 + 35*b^2* 
e^2 - 2*c*e*(75*b*d + 16*a*e) + 42*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^ 
(3/2))/(24*c^2) + (5*(2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d 
+ 3*a*e))*(((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*c) - ((b^2 - 4*a*c)*ArcT 
anh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(3/2))))/(16*c^2) 
)/(10*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 1166

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[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, 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 = 1.06 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.58

method	result	size
risch	\(-\frac {\left (-384 e^{3} c^{4} x^{4}-48 b \,c^{3} e^{3} x^{3}-1440 c^{4} d \,e^{2} x^{3}-128 a \,c^{3} e^{3} x^{2}+56 b^{2} c^{2} e^{3} x^{2}-240 b \,c^{3} d \,e^{2} x^{2}-1920 c^{4} d^{2} e \,x^{2}+232 a b \,c^{2} e^{3} x -720 a \,c^{3} d \,e^{2} x -70 b^{3} c \,e^{3} x +300 b^{2} c^{2} d \,e^{2} x -480 b \,c^{3} d^{2} e x -960 c^{4} d^{3} x +256 a^{2} c^{2} e^{3}-460 a \,b^{2} c \,e^{3}+1560 a b \,c^{2} d \,e^{2}-1920 a \,c^{3} d^{2} e +105 e^{3} b^{4}-450 b^{3} c d \,e^{2}+720 b^{2} c^{2} d^{2} e -480 b \,c^{3} d^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{1920 c^{4}}+\frac {\left (48 a^{2} b \,c^{2} e^{3}-96 a^{2} c^{3} d \,e^{2}-40 a \,b^{3} c \,e^{3}+144 a \,b^{2} c^{2} d \,e^{2}-192 a b \,d^{2} e \,c^{3}+128 a \,c^{4} d^{3}+7 b^{5} e^{3}-30 b^{4} c d \,e^{2}+48 b^{3} c^{2} d^{2} e -32 b^{2} c^{3} d^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{256 c^{\frac {9}{2}}}\)	\(391\)
default	\(d^{3} \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 )+e^{3} \left (\frac {x^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{5 c}-\frac {7 b \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \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 )}{2 c}\right )}{8 c}-\frac {a \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 )}{4 c}\right )}{10 c}-\frac {2 a \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \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 )}{2 c}\right )}{5 c}\right )+3 d \,e^{2} \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \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 )}{2 c}\right )}{8 c}-\frac {a \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 )}{4 c}\right )+3 d^{2} e \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \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 )}{2 c}\right )\)	\(661\)

Input:

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

Output:

-1/1920*(-384*c^4*e^3*x^4-48*b*c^3*e^3*x^3-1440*c^4*d*e^2*x^3-128*a*c^3*e^ 
3*x^2+56*b^2*c^2*e^3*x^2-240*b*c^3*d*e^2*x^2-1920*c^4*d^2*e*x^2+232*a*b*c^ 
2*e^3*x-720*a*c^3*d*e^2*x-70*b^3*c*e^3*x+300*b^2*c^2*d*e^2*x-480*b*c^3*d^2 
*e*x-960*c^4*d^3*x+256*a^2*c^2*e^3-460*a*b^2*c*e^3+1560*a*b*c^2*d*e^2-1920 
*a*c^3*d^2*e+105*b^4*e^3-450*b^3*c*d*e^2+720*b^2*c^2*d^2*e-480*b*c^3*d^3)/ 
c^4*(c*x^2+b*x+a)^(1/2)+1/256*(48*a^2*b*c^2*e^3-96*a^2*c^3*d*e^2-40*a*b^3* 
c*e^3+144*a*b^2*c^2*d*e^2-192*a*b*c^3*d^2*e+128*a*c^4*d^3+7*b^5*e^3-30*b^4 
*c*d*e^2+48*b^3*c^2*d^2*e-32*b^2*c^3*d^3)/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 = 787, normalized size of antiderivative = 3.17 \[ \int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/7680*(15*(32*(b^2*c^3 - 4*a*c^4)*d^3 - 48*(b^3*c^2 - 4*a*b*c^3)*d^2*e 
+ 6*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*d*e^2 - (7*b^5 - 40*a*b^3*c + 48 
*a^2*b*c^2)*e^3)*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*(384*c^5*e^3*x^4 + 480*b*c^4*d^3 
- 240*(3*b^2*c^3 - 8*a*c^4)*d^2*e + 30*(15*b^3*c^2 - 52*a*b*c^3)*d*e^2 - ( 
105*b^4*c - 460*a*b^2*c^2 + 256*a^2*c^3)*e^3 + 48*(30*c^5*d*e^2 + b*c^4*e^ 
3)*x^3 + 8*(240*c^5*d^2*e + 30*b*c^4*d*e^2 - (7*b^2*c^3 - 16*a*c^4)*e^3)*x 
^2 + 2*(480*c^5*d^3 + 240*b*c^4*d^2*e - 30*(5*b^2*c^3 - 12*a*c^4)*d*e^2 + 
(35*b^3*c^2 - 116*a*b*c^3)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^5, 1/3840*(15* 
(32*(b^2*c^3 - 4*a*c^4)*d^3 - 48*(b^3*c^2 - 4*a*b*c^3)*d^2*e + 6*(5*b^4*c 
- 24*a*b^2*c^2 + 16*a^2*c^3)*d*e^2 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*e 
^3)*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*(384*c^5*e^3*x^4 + 480*b*c^4*d^3 - 240*(3*b^2*c^3 - 
8*a*c^4)*d^2*e + 30*(15*b^3*c^2 - 52*a*b*c^3)*d*e^2 - (105*b^4*c - 460*a*b 
^2*c^2 + 256*a^2*c^3)*e^3 + 48*(30*c^5*d*e^2 + b*c^4*e^3)*x^3 + 8*(240*c^5 
*d^2*e + 30*b*c^4*d*e^2 - (7*b^2*c^3 - 16*a*c^4)*e^3)*x^2 + 2*(480*c^5*d^3 
 + 240*b*c^4*d^2*e - 30*(5*b^2*c^3 - 12*a*c^4)*d*e^2 + (35*b^3*c^2 - 116*a 
*b*c^3)*e^3)*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. 887 vs. \(2 (241) = 482\).

Time = 1.10 (sec) , antiderivative size = 887, normalized size of antiderivative = 3.58 \[ \int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)^3*(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 = 0.36 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.58 \[ \int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx=\frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, e^{3} x + \frac {30 \, c^{4} d e^{2} + b c^{3} e^{3}}{c^{4}}\right )} x + \frac {240 \, c^{4} d^{2} e + 30 \, b c^{3} d e^{2} - 7 \, b^{2} c^{2} e^{3} + 16 \, a c^{3} e^{3}}{c^{4}}\right )} x + \frac {480 \, c^{4} d^{3} + 240 \, b c^{3} d^{2} e - 150 \, b^{2} c^{2} d e^{2} + 360 \, a c^{3} d e^{2} + 35 \, b^{3} c e^{3} - 116 \, a b c^{2} e^{3}}{c^{4}}\right )} x + \frac {480 \, b c^{3} d^{3} - 720 \, b^{2} c^{2} d^{2} e + 1920 \, a c^{3} d^{2} e + 450 \, b^{3} c d e^{2} - 1560 \, a b c^{2} d e^{2} - 105 \, b^{4} e^{3} + 460 \, a b^{2} c e^{3} - 256 \, a^{2} c^{2} e^{3}}{c^{4}}\right )} + \frac {{\left (32 \, b^{2} c^{3} d^{3} - 128 \, a c^{4} d^{3} - 48 \, b^{3} c^{2} d^{2} e + 192 \, a b c^{3} d^{2} e + 30 \, b^{4} c d e^{2} - 144 \, a b^{2} c^{2} d e^{2} + 96 \, a^{2} c^{3} d e^{2} - 7 \, b^{5} e^{3} + 40 \, a b^{3} c e^{3} - 48 \, a^{2} b c^{2} e^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {9}{2}}} \] Input:

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

Output:

1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*(8*e^3*x + (30*c^4*d*e^2 + b*c^3*e^3 
)/c^4)*x + (240*c^4*d^2*e + 30*b*c^3*d*e^2 - 7*b^2*c^2*e^3 + 16*a*c^3*e^3) 
/c^4)*x + (480*c^4*d^3 + 240*b*c^3*d^2*e - 150*b^2*c^2*d*e^2 + 360*a*c^3*d 
*e^2 + 35*b^3*c*e^3 - 116*a*b*c^2*e^3)/c^4)*x + (480*b*c^3*d^3 - 720*b^2*c 
^2*d^2*e + 1920*a*c^3*d^2*e + 450*b^3*c*d*e^2 - 1560*a*b*c^2*d*e^2 - 105*b 
^4*e^3 + 460*a*b^2*c*e^3 - 256*a^2*c^2*e^3)/c^4) + 1/256*(32*b^2*c^3*d^3 - 
 128*a*c^4*d^3 - 48*b^3*c^2*d^2*e + 192*a*b*c^3*d^2*e + 30*b^4*c*d*e^2 - 1 
44*a*b^2*c^2*d*e^2 + 96*a^2*c^3*d*e^2 - 7*b^5*e^3 + 40*a*b^3*c*e^3 - 48*a^ 
2*b*c^2*e^3)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c 
^(9/2)
 

Mupad [B] (verification not implemented)

Time = 6.28 (sec) , antiderivative size = 632, normalized size of antiderivative = 2.55 \[ \int (d+e x)^3 \sqrt {a+b x+c x^2} \, dx=d^3\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {7\,b\,e^3\,\left (\frac {5\,b\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}-\frac {x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c}+\frac {a\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}\right )}{10\,c}+\frac {e^3\,x^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{5\,c}+\frac {d^3\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}-\frac {2\,a\,e^3\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{5\,c}-\frac {3\,a\,d\,e^2\,\left (\left (\frac {x}{2}+\frac {b}{4\,c}\right )\,\sqrt {c\,x^2+b\,x+a}+\frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x+a}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{2\,c^{3/2}}\right )}{4\,c}+\frac {3\,d^2\,e\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}-\frac {15\,b\,d\,e^2\,\left (\frac {\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x+a}\right )\,\left (b^3-4\,a\,b\,c\right )}{16\,c^{5/2}}+\frac {\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{24\,c^2}\right )}{8\,c}+\frac {d^2\,e\,\left (-3\,b^2+2\,c\,x\,b+8\,c\,\left (c\,x^2+a\right )\right )\,\sqrt {c\,x^2+b\,x+a}}{8\,c^2}+\frac {3\,d\,e^2\,x\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{4\,c} \] Input:

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

Output:

d^3*(x/2 + b/(4*c))*(a + b*x + c*x^2)^(1/2) + (7*b*e^3*((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) + (e^3*x^2*(a + b*x + c*x^2)^(3/2))/( 
5*c) + (d^3*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)) - (2*a*e^3*((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) - (3*a*d*e^2*((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) + (3*d^2*e*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)) - (15*b*d*e^2* 
((log((b + 2*c*x)/c^(1/2) + 2*(a + b*x + c*x^2)^(1/2))*(b^3 - 4*a*b*c))/(1 
6*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) + (d^2*e*(8*c*(a + c*x^2) - 3*b^2 + 2*b*c*x)*(a + b*x + 
c*x^2)^(1/2))/(8*c^2) + (3*d*e^2*x*(a + b*x + c*x^2)^(3/2))/(4*c)
 

Reduce [F]

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

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

Output:

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