\(\int \frac {(b+2 c x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx\) [628]

Optimal result

Integrand size = 28, antiderivative size = 245 \[ \int \frac {(b+2 c x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx=\frac {(2 c d-b e) (d+e x)^2 \sqrt {a+b x+c x^2}}{4 c}+\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}+\frac {\left (32 c^3 d^3-15 b^3 e^3+4 b c e^2 (12 b d+13 a e)-8 c^2 d e (5 b d+16 a e)+2 c e \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{32 c^3}+\frac {3 \left (b^2-4 a c\right ) e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{64 c^{7/2}} \] Output:

1/4*(-b*e+2*c*d)*(e*x+d)^2*(c*x^2+b*x+a)^(1/2)/c+1/2*(e*x+d)^3*(c*x^2+b*x+ 
a)^(1/2)+1/32*(32*c^3*d^3-15*b^3*e^3+4*b*c*e^2*(13*a*e+12*b*d)-8*c^2*d*e*( 
16*a*e+5*b*d)+2*c*e*(8*c^2*d^2+5*b^2*e^2-4*c*e*(3*a*e+2*b*d))*x)*(c*x^2+b* 
x+a)^(1/2)/c^3+3/64*(-4*a*c+b^2)*e*(16*c^2*d^2+5*b^2*e^2-4*c*e*(a*e+4*b*d) 
)*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(7/2)
 

Mathematica [A] (verified)

Time = 1.83 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.82 \[ \int \frac {(b+2 c x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (-15 b^3 e^3+2 b c e^2 (24 b d+26 a e+5 b e x)+16 c^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-8 c^2 e \left (a e (16 d+3 e x)+b \left (6 d^2+4 d e x+e^2 x^2\right )\right )\right )+3 \left (b^2-4 a c\right ) e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{32 c^{7/2}} \] Input:

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

Output:

(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-15*b^3*e^3 + 2*b*c*e^2*(24*b*d + 26*a*e + 
 5*b*e*x) + 16*c^3*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - 8*c^2*e*( 
a*e*(16*d + 3*e*x) + b*(6*d^2 + 4*d*e*x + e^2*x^2))) + 3*(b^2 - 4*a*c)*e*( 
16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a 
] + Sqrt[a + x*(b + c*x)])])/(32*c^(7/2))
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1236, 27, 1236, 27, 1225, 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.

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

\(\Big \downarrow \) 1236

\(\displaystyle \frac {\int \frac {3 c (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\sqrt {c x^2+b x+a}}dx}{4 c}+\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{4} \int \frac {(d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\sqrt {c x^2+b x+a}}dx+\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}\)

\(\Big \downarrow \) 1236

\(\displaystyle \frac {3}{4} \left (\frac {\int \frac {(d+e x) \left (d e b^2+4 \left (c d^2+a e^2\right ) b-20 a c d e+\left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right )}{2 \sqrt {c x^2+b x+a}}dx}{3 c}+\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}\right )+\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{4} \left (\frac {\int \frac {(d+e x) \left (d e b^2+4 \left (c d^2+a e^2\right ) b-20 a c d e+\left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x\right )}{\sqrt {c x^2+b x+a}}dx}{6 c}+\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}\right )+\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {3}{4} \left (\frac {\frac {3 e \left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c^2}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (16 a e+5 b d)+4 b c e^2 (13 a e+12 b d)-15 b^3 e^3+32 c^3 d^3\right )}{4 c^2}}{6 c}+\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}\right )+\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {3}{4} \left (\frac {\frac {3 e \left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c^2}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (16 a e+5 b d)+4 b c e^2 (13 a e+12 b d)-15 b^3 e^3+32 c^3 d^3\right )}{4 c^2}}{6 c}+\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}\right )+\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {3}{4} \left (\frac {\frac {3 e \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )}{8 c^{5/2}}+\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )-8 c^2 d e (16 a e+5 b d)+4 b c e^2 (13 a e+12 b d)-15 b^3 e^3+32 c^3 d^3\right )}{4 c^2}}{6 c}+\frac {(d+e x)^2 \sqrt {a+b x+c x^2} (2 c d-b e)}{3 c}\right )+\frac {1}{2} (d+e x)^3 \sqrt {a+b x+c x^2}\)

Input:

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

Output:

((d + e*x)^3*Sqrt[a + b*x + c*x^2])/2 + (3*(((2*c*d - b*e)*(d + e*x)^2*Sqr 
t[a + b*x + c*x^2])/(3*c) + (((32*c^3*d^3 - 15*b^3*e^3 + 4*b*c*e^2*(12*b*d 
 + 13*a*e) - 8*c^2*d*e*(5*b*d + 16*a*e) + 2*c*e*(8*c^2*d^2 + 5*b^2*e^2 - 4 
*c*e*(2*b*d + 3*a*e))*x)*Sqrt[a + b*x + c*x^2])/(4*c^2) + (3*(b^2 - 4*a*c) 
*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*S 
qrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(5/2)))/(6*c)))/4
 

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 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 [A] (verified)

Time = 1.92 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.01

method	result	size
risch	\(\frac {\left (16 c^{3} x^{3} e^{3}-8 b \,c^{2} e^{3} x^{2}+64 c^{3} d \,e^{2} x^{2}-24 a \,c^{2} e^{3} x +10 x \,b^{2} c \,e^{3}-32 b \,c^{2} d \,e^{2} x +96 d^{2} e \,c^{3} x +52 a b c \,e^{3}-128 d \,e^{2} a \,c^{2}-15 b^{3} e^{3}+48 d \,e^{2} b^{2} c -48 d^{2} e b \,c^{2}+64 d^{3} c^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{32 c^{3}}+\frac {3 e \left (16 a^{2} c^{2} e^{2}-24 a \,b^{2} c \,e^{2}+64 a b \,c^{2} d e -64 a \,c^{3} d^{2}+5 b^{4} e^{2}-16 b^{3} c d e +16 b^{2} c^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{64 c^{\frac {7}{2}}}\)	\(248\)
default	\(\frac {b \,d^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{\sqrt {c}}+e^{2} \left (b e +6 c d \right ) \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )+3 d e \left (b e +2 c d \right ) \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+d^{2} \left (3 b e +2 c d \right ) \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )+2 e^{3} c \left (\frac {x^{3} \sqrt {c \,x^{2}+b x +a}}{4 c}-\frac {7 b \left (\frac {x^{2} \sqrt {c \,x^{2}+b x +a}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{6 c}-\frac {2 a \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{3 c}\right )}{8 c}-\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+b x +a}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x +a}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}-\frac {a \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )\)	\(737\)

Input:

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

Output:

1/32*(16*c^3*e^3*x^3-8*b*c^2*e^3*x^2+64*c^3*d*e^2*x^2-24*a*c^2*e^3*x+10*b^ 
2*c*e^3*x-32*b*c^2*d*e^2*x+96*c^3*d^2*e*x+52*a*b*c*e^3-128*a*c^2*d*e^2-15* 
b^3*e^3+48*b^2*c*d*e^2-48*b*c^2*d^2*e+64*c^3*d^3)/c^3*(c*x^2+b*x+a)^(1/2)+ 
3/64*e*(16*a^2*c^2*e^2-24*a*b^2*c*e^2+64*a*b*c^2*d*e-64*a*c^3*d^2+5*b^4*e^ 
2-16*b^3*c*d*e+16*b^2*c^2*d^2)/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a 
)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.22 \[ \int \frac {(b+2 c x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx=\left [\frac {3 \, {\left (16 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 16 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} e^{3}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (16 \, c^{4} e^{3} x^{3} + 64 \, c^{4} d^{3} - 48 \, b c^{3} d^{2} e + 16 \, {\left (3 \, b^{2} c^{2} - 8 \, a c^{3}\right )} d e^{2} - {\left (15 \, b^{3} c - 52 \, a b c^{2}\right )} e^{3} + 8 \, {\left (8 \, c^{4} d e^{2} - b c^{3} e^{3}\right )} x^{2} + 2 \, {\left (48 \, c^{4} d^{2} e - 16 \, b c^{3} d e^{2} + {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{128 \, c^{4}}, -\frac {3 \, {\left (16 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} e - 16 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} d e^{2} + {\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) - 2 \, {\left (16 \, c^{4} e^{3} x^{3} + 64 \, c^{4} d^{3} - 48 \, b c^{3} d^{2} e + 16 \, {\left (3 \, b^{2} c^{2} - 8 \, a c^{3}\right )} d e^{2} - {\left (15 \, b^{3} c - 52 \, a b c^{2}\right )} e^{3} + 8 \, {\left (8 \, c^{4} d e^{2} - b c^{3} e^{3}\right )} x^{2} + 2 \, {\left (48 \, c^{4} d^{2} e - 16 \, b c^{3} d e^{2} + {\left (5 \, b^{2} c^{2} - 12 \, a c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{64 \, c^{4}}\right ] \] Input:

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

Output:

[1/128*(3*(16*(b^2*c^2 - 4*a*c^3)*d^2*e - 16*(b^3*c - 4*a*b*c^2)*d*e^2 + ( 
5*b^4 - 24*a*b^2*c + 16*a^2*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*(16*c^4*e^3* 
x^3 + 64*c^4*d^3 - 48*b*c^3*d^2*e + 16*(3*b^2*c^2 - 8*a*c^3)*d*e^2 - (15*b 
^3*c - 52*a*b*c^2)*e^3 + 8*(8*c^4*d*e^2 - b*c^3*e^3)*x^2 + 2*(48*c^4*d^2*e 
 - 16*b*c^3*d*e^2 + (5*b^2*c^2 - 12*a*c^3)*e^3)*x)*sqrt(c*x^2 + b*x + a))/ 
c^4, -1/64*(3*(16*(b^2*c^2 - 4*a*c^3)*d^2*e - 16*(b^3*c - 4*a*b*c^2)*d*e^2 
 + (5*b^4 - 24*a*b^2*c + 16*a^2*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*(16*c^4*e^3*x^ 
3 + 64*c^4*d^3 - 48*b*c^3*d^2*e + 16*(3*b^2*c^2 - 8*a*c^3)*d*e^2 - (15*b^3 
*c - 52*a*b*c^2)*e^3 + 8*(8*c^4*d*e^2 - b*c^3*e^3)*x^2 + 2*(48*c^4*d^2*e - 
 16*b*c^3*d*e^2 + (5*b^2*c^2 - 12*a*c^3)*e^3)*x)*sqrt(c*x^2 + b*x + a))/c^ 
4]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 806 vs. \(2 (241) = 482\).

Time = 1.12 (sec) , antiderivative size = 806, normalized size of antiderivative = 3.29 \[ \int \frac {(b+2 c x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:

integrate((2*c*x+b)*(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**3/2 + x**2*(-3*b*e**3/4 + 6*c*d 
*e**2)/(3*c) + x*(-3*a*e**3/2 + 3*b*d*e**2 - 5*b*(-3*b*e**3/4 + 6*c*d*e**2 
)/(6*c) + 6*c*d**2*e)/(2*c) + (-2*a*(-3*b*e**3/4 + 6*c*d*e**2)/(3*c) + 3*b 
*d**2*e - 3*b*(-3*a*e**3/2 + 3*b*d*e**2 - 5*b*(-3*b*e**3/4 + 6*c*d*e**2)/( 
6*c) + 6*c*d**2*e)/(4*c) + 2*c*d**3)/c) + (-a*(-3*a*e**3/2 + 3*b*d*e**2 - 
5*b*(-3*b*e**3/4 + 6*c*d*e**2)/(6*c) + 6*c*d**2*e)/(2*c) + b*d**3 - b*(-2* 
a*(-3*b*e**3/4 + 6*c*d*e**2)/(3*c) + 3*b*d**2*e - 3*b*(-3*a*e**3/2 + 3*b*d 
*e**2 - 5*b*(-3*b*e**3/4 + 6*c*d*e**2)/(6*c) + 6*c*d**2*e)/(4*c) + 2*c*d** 
3)/(2*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqr 
t(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*(2*c*e**3*(a + b*x)**(9/2)/(9*b**4) + 
(a + b*x)**(7/2)*(-8*a*c*e**3 + b**2*e**3 + 6*b*c*d*e**2)/(7*b**4) + (a + 
b*x)**(5/2)*(12*a**2*c*e**3 - 3*a*b**2*e**3 - 18*a*b*c*d*e**2 + 3*b**3*d*e 
**2 + 6*b**2*c*d**2*e)/(5*b**4) + (a + b*x)**(3/2)*(-8*a**3*c*e**3 + 3*a** 
2*b**2*e**3 + 18*a**2*b*c*d*e**2 - 6*a*b**3*d*e**2 - 12*a*b**2*c*d**2*e + 
3*b**4*d**2*e + 2*b**3*c*d**3)/(3*b**4) + sqrt(a + b*x)*(2*a**4*c*e**3 - a 
**3*b**2*e**3 - 6*a**3*b*c*d*e**2 + 3*a**2*b**3*d*e**2 + 6*a**2*b**2*c*d** 
2*e - 3*a*b**4*d**2*e - 2*a*b**3*c*d**3 + b**5*d**3)/b**4)/b, Ne(b, 0)), ( 
2*c*(d**3*x**2/2 + d**2*e*x**3 + 3*d*e**2*x**4/4 + e**3*x**5/5)/sqrt(a), T 
rue))
 

Maxima [F(-2)]

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

integrate((2*c*x+b)*(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 = 0.17 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.07 \[ \int \frac {(b+2 c x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx=\frac {1}{32} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, e^{3} x + \frac {8 \, c^{3} d e^{2} - b c^{2} e^{3}}{c^{3}}\right )} x + \frac {48 \, c^{3} d^{2} e - 16 \, b c^{2} d e^{2} + 5 \, b^{2} c e^{3} - 12 \, a c^{2} e^{3}}{c^{3}}\right )} x + \frac {64 \, c^{3} d^{3} - 48 \, b c^{2} d^{2} e + 48 \, b^{2} c d e^{2} - 128 \, a c^{2} d e^{2} - 15 \, b^{3} e^{3} + 52 \, a b c e^{3}}{c^{3}}\right )} - \frac {3 \, {\left (16 \, b^{2} c^{2} d^{2} e - 64 \, a c^{3} d^{2} e - 16 \, b^{3} c d e^{2} + 64 \, a b c^{2} d e^{2} + 5 \, b^{4} e^{3} - 24 \, a b^{2} c e^{3} + 16 \, a^{2} c^{2} e^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{64 \, c^{\frac {7}{2}}} \] Input:

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

Output:

1/32*sqrt(c*x^2 + b*x + a)*(2*(4*(2*e^3*x + (8*c^3*d*e^2 - b*c^2*e^3)/c^3) 
*x + (48*c^3*d^2*e - 16*b*c^2*d*e^2 + 5*b^2*c*e^3 - 12*a*c^2*e^3)/c^3)*x + 
 (64*c^3*d^3 - 48*b*c^2*d^2*e + 48*b^2*c*d*e^2 - 128*a*c^2*d*e^2 - 15*b^3* 
e^3 + 52*a*b*c*e^3)/c^3) - 3/64*(16*b^2*c^2*d^2*e - 64*a*c^3*d^2*e - 16*b^ 
3*c*d*e^2 + 64*a*b*c^2*d*e^2 + 5*b^4*e^3 - 24*a*b^2*c*e^3 + 16*a^2*c^2*e^3 
)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(7/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.55 \[ \int \frac {(b+2 c x) (d+e x)^3}{\sqrt {a+b x+c x^2}} \, dx=\frac {104 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{2} e^{3}-256 \sqrt {c \,x^{2}+b x +a}\, a \,c^{3} d \,e^{2}-48 \sqrt {c \,x^{2}+b x +a}\, a \,c^{3} e^{3} x -30 \sqrt {c \,x^{2}+b x +a}\, b^{3} c \,e^{3}+96 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{2} d \,e^{2}+20 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{2} e^{3} x -96 \sqrt {c \,x^{2}+b x +a}\, b \,c^{3} d^{2} e -64 \sqrt {c \,x^{2}+b x +a}\, b \,c^{3} d \,e^{2} x -16 \sqrt {c \,x^{2}+b x +a}\, b \,c^{3} e^{3} x^{2}+128 \sqrt {c \,x^{2}+b x +a}\, c^{4} d^{3}+192 \sqrt {c \,x^{2}+b x +a}\, c^{4} d^{2} e x +128 \sqrt {c \,x^{2}+b x +a}\, c^{4} d \,e^{2} x^{2}+32 \sqrt {c \,x^{2}+b x +a}\, c^{4} e^{3} x^{3}+48 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{2} c^{2} e^{3}-72 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} c \,e^{3}+192 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a b \,c^{2} d \,e^{2}-192 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,c^{3} d^{2} e +15 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{4} e^{3}-48 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{3} c d \,e^{2}+48 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2} c^{2} d^{2} e}{64 c^{4}} \] Input:

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

Output:

(104*sqrt(a + b*x + c*x**2)*a*b*c**2*e**3 - 256*sqrt(a + b*x + c*x**2)*a*c 
**3*d*e**2 - 48*sqrt(a + b*x + c*x**2)*a*c**3*e**3*x - 30*sqrt(a + b*x + c 
*x**2)*b**3*c*e**3 + 96*sqrt(a + b*x + c*x**2)*b**2*c**2*d*e**2 + 20*sqrt( 
a + b*x + c*x**2)*b**2*c**2*e**3*x - 96*sqrt(a + b*x + c*x**2)*b*c**3*d**2 
*e - 64*sqrt(a + b*x + c*x**2)*b*c**3*d*e**2*x - 16*sqrt(a + b*x + c*x**2) 
*b*c**3*e**3*x**2 + 128*sqrt(a + b*x + c*x**2)*c**4*d**3 + 192*sqrt(a + b* 
x + c*x**2)*c**4*d**2*e*x + 128*sqrt(a + b*x + c*x**2)*c**4*d*e**2*x**2 + 
32*sqrt(a + b*x + c*x**2)*c**4*e**3*x**3 + 48*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*c**2*e**3 - 72*sqr 
t(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2) 
)*a*b**2*c*e**3 + 192*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*c**2*d*e**2 - 192*sqrt(c)*log((2*sqrt(c)*sq 
rt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a*c**3*d**2*e + 15*s 
qrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b** 
2))*b**4*e**3 - 48*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c 
*x)/sqrt(4*a*c - b**2))*b**3*c*d*e**2 + 48*sqrt(c)*log((2*sqrt(c)*sqrt(a + 
 b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*c**2*d**2*e)/(64*c**4 
)