\(\int \sqrt {d x} \sqrt {a+b x+c x^2} \, dx\) [311]

Optimal result

Integrand size = 22, antiderivative size = 478 \[ \int \sqrt {d x} \sqrt {a+b x+c x^2} \, dx=\frac {2 b \sqrt {d x} \sqrt {a+b x+c x^2}}{15 c}+\frac {2 (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 d}-\frac {\sqrt {2} \left (b^2-3 a c\right ) \sqrt {-b+\sqrt {b^2-4 a c}} \left (b+\sqrt {b^2-4 a c}\right ) \sqrt {d} \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} E\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right )|\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{15 c^{5/2} \sqrt {a+x (b+c x)}}+\frac {\sqrt {2} \sqrt {-b+\sqrt {b^2-4 a c}} \left (b^3-4 a b c+\sqrt {b^2-4 a c} \left (b^2-3 a c\right )\right ) \sqrt {d} \sqrt {1+\frac {2 c x}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x}{b+\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d x}}{\sqrt {-b+\sqrt {b^2-4 a c}} \sqrt {d}}\right ),\frac {b-\sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}}\right )}{15 c^{5/2} \sqrt {a+x (b+c x)}} \] Output:

2/15*b*(d*x)^(1/2)*(c*x^2+b*x+a)^(1/2)/c+2/5*(d*x)^(3/2)*(c*x^2+b*x+a)^(1/ 
2)/d-1/15*2^(1/2)*(-3*a*c+b^2)*(-b+(-4*a*c+b^2)^(1/2))^(1/2)*(b+(-4*a*c+b^ 
2)^(1/2))*d^(1/2)*(1+2*c*x/(b-(-4*a*c+b^2)^(1/2)))^(1/2)*(1+2*c*x/(b+(-4*a 
*c+b^2)^(1/2)))^(1/2)*EllipticE(2^(1/2)*c^(1/2)*(d*x)^(1/2)/(-b+(-4*a*c+b^ 
2)^(1/2))^(1/2)/d^(1/2),((b-(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1 
/2))/c^(5/2)/(a+x*(c*x+b))^(1/2)+1/15*2^(1/2)*(-b+(-4*a*c+b^2)^(1/2))^(1/2 
)*(b^3-4*a*b*c+(-4*a*c+b^2)^(1/2)*(-3*a*c+b^2))*d^(1/2)*(1+2*c*x/(b-(-4*a* 
c+b^2)^(1/2)))^(1/2)*(1+2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(2^(1 
/2)*c^(1/2)*(d*x)^(1/2)/(-b+(-4*a*c+b^2)^(1/2))^(1/2)/d^(1/2),((b-(-4*a*c+ 
b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2))/c^(5/2)/(a+x*(c*x+b))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.89 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.03 \[ \int \sqrt {d x} \sqrt {a+b x+c x^2} \, dx=\frac {d \left (-4 \left (b^2-3 a c\right ) (a+x (b+c x))+2 c x (b+3 c x) (a+x (b+c x))+\frac {i \left (b^2-3 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}+\frac {i \left (b^3-4 a b c-b^2 \sqrt {b^2-4 a c}+3 a c \sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}\right )}{15 c^2 \sqrt {d x} \sqrt {a+x (b+c x)}} \] Input:

Integrate[Sqrt[d*x]*Sqrt[a + b*x + c*x^2],x]
 

Output:

(d*(-4*(b^2 - 3*a*c)*(a + x*(b + c*x)) + 2*c*x*(b + 3*c*x)*(a + x*(b + c*x 
)) + (I*(b^2 - 3*a*c)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b 
^2 - 4*a*c])*x)]*x^(3/2)*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqr 
t[b^2 - 4*a*c]*x)]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a 
*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/ 
(b + Sqrt[b^2 - 4*a*c])] + (I*(b^3 - 4*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 3*a 
*c*Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(3/2)* 
Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*Ellipt 
icF[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqr 
t[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]) 
)/(15*c^2*Sqrt[d*x]*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.79, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1162, 25, 27, 1236, 27, 1241, 1240, 1511, 27, 1416, 1509}

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.

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

\(\Big \downarrow \) 1162

\(\displaystyle \frac {2 (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 d}-\frac {\int -\frac {d \sqrt {d x} (2 a+b x)}{\sqrt {c x^2+b x+a}}dx}{5 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {d \sqrt {d x} (2 a+b x)}{\sqrt {c x^2+b x+a}}dx}{5 d}+\frac {2 (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \int \frac {\sqrt {d x} (2 a+b x)}{\sqrt {c x^2+b x+a}}dx+\frac {2 (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 d}\)

\(\Big \downarrow \) 1236

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1241

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

\(\Big \downarrow \) 1240

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

\(\displaystyle \frac {1}{5} \left (\frac {2 b \sqrt {d x} \sqrt {a+b x+c x^2}}{3 c}-\frac {2 d \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\frac {2 \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} b\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {2 \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{\sqrt {c x^2+b x+a}}d\sqrt {x}}{\sqrt {c}}\right )}{3 c \sqrt {d x}}\right )+\frac {2 (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 d}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {1}{5} \left (\frac {2 b \sqrt {d x} \sqrt {a+b x+c x^2}}{3 c}-\frac {2 d \sqrt {x} \left (\frac {\sqrt [4]{a} \left (\frac {2 \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} b\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {2 \left (b^2-3 a c\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x+c x^2}}-\frac {\sqrt {x} \sqrt {a+b x+c x^2}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {c}}\right )}{3 c \sqrt {d x}}\right )+\frac {2 (d x)^{3/2} \sqrt {a+b x+c x^2}}{5 d}\)

Input:

Int[Sqrt[d*x]*Sqrt[a + b*x + c*x^2],x]
 

Output:

(2*(d*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(5*d) + ((2*b*Sqrt[d*x]*Sqrt[a + b*x 
 + c*x^2])/(3*c) - (2*d*Sqrt[x]*((-2*(b^2 - 3*a*c)*(-((Sqrt[x]*Sqrt[a + b* 
x + c*x^2])/(Sqrt[a] + Sqrt[c]*x)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x)*Sqrt[( 
a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt 
[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x + c*x^2 
])))/Sqrt[c] + (a^(1/4)*(Sqrt[a]*b + (2*(b^2 - 3*a*c))/Sqrt[c])*(Sqrt[a] + 
 Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*Ar 
cTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(1/4)* 
Sqrt[a + b*x + c*x^2])))/(3*c*Sqrt[d*x]))/5
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 1162

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x 
] - Simp[p/(e*(m + 2*p + 1))   Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - 
b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x 
] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && 
!ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

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])
 

rule 1240

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), 
x_Symbol] :> Simp[2   Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, 
 Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
 

rule 1241

Int[((f_) + (g_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_ 
)^2]), x_Symbol] :> Simp[Sqrt[x]/Sqrt[e*x]   Int[(f + g*x)/(Sqrt[x]*Sqrt[a 
+ b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, e, f, g}, x]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

Maple [A] (verified)

Time = 1.90 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.60

method	result	size
risch	\(\frac {2 \left (3 c x +b \right ) x \sqrt {c \,x^{2}+b x +a}\, d}{15 c \sqrt {d x}}-\frac {\left (-\frac {\left (6 a c -2 b^{2}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{c \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}+\frac {a b \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{c \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}\right ) d \sqrt {d x \left (c \,x^{2}+b x +a \right )}}{15 c \sqrt {d x}\, \sqrt {c \,x^{2}+b x +a}}\)	\(764\)
elliptic	\(\frac {\sqrt {d x}\, \sqrt {d x \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 x \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{5}+\frac {2 b \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{15 c}-\frac {a d b \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{15 c^{2} \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}+\frac {\left (\frac {2 a d}{5}-\frac {2 b^{2} d}{15 c}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{c \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}\right )}{d x \sqrt {c \,x^{2}+b x +a}}\)	\(786\)
default	\(\text {Expression too large to display}\)	\(1335\)

Input:

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

Output:

2/15*(3*c*x+b)*x*(c*x^2+b*x+a)^(1/2)/c*d/(d*x)^(1/2)-1/15/c*(-(6*a*c-2*b^2 
)*(b+(-4*a*c+b^2)^(1/2))/c*2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(- 
4*a*c+b^2)^(1/2))*c)^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4 
*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*(-2*c*x/(b+(-4*a* 
c+b^2)^(1/2)))^(1/2)/(c*d*x^3+b*d*x^2+a*d*x)^(1/2)*((-1/2*(b+(-4*a*c+b^2)^ 
(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(2^(1/2)*((x+1/2*(b+(-4*a 
*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2),1/2*(-2*(b+(-4*a*c+b^2)^ 
(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1 
/2))+1/2/c*(-b+(-4*a*c+b^2)^(1/2))*EllipticF(2^(1/2)*((x+1/2*(b+(-4*a*c+b^ 
2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2),1/2*(-2*(b+(-4*a*c+b^2)^(1/2) 
)/c/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))) 
+a*b*(b+(-4*a*c+b^2)^(1/2))/c*2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b 
+(-4*a*c+b^2)^(1/2))*c)^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+ 
(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*(-2*c*x/(b+(-4 
*a*c+b^2)^(1/2)))^(1/2)/(c*d*x^3+b*d*x^2+a*d*x)^(1/2)*EllipticF(2^(1/2)*(( 
x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2),1/2*(-2*(b 
+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^ 
2)^(1/2))))^(1/2)))*d*(d*x*(c*x^2+b*x+a))^(1/2)/(d*x)^(1/2)/(c*x^2+b*x+a)^ 
(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.39 \[ \int \sqrt {d x} \sqrt {a+b x+c x^2} \, dx=\frac {2 \, {\left ({\left (2 \, b^{3} - 9 \, a b c\right )} \sqrt {c d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 6 \, {\left (b^{2} c - 3 \, a c^{2}\right )} \sqrt {c d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) + 3 \, {\left (3 \, c^{3} x + b c^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {d x}\right )}}{45 \, c^{3}} \] Input:

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

Output:

2/45*((2*b^3 - 9*a*b*c)*sqrt(c*d)*weierstrassPInverse(4/3*(b^2 - 3*a*c)/c^ 
2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1/3*(3*c*x + b)/c) + 6*(b^2*c - 3*a*c^2)*s 
qrt(c*d)*weierstrassZeta(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^ 
3, weierstrassPInverse(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 
 1/3*(3*c*x + b)/c)) + 3*(3*c^3*x + b*c^2)*sqrt(c*x^2 + b*x + a)*sqrt(d*x) 
)/c^3
 

Sympy [F]

\[ \int \sqrt {d x} \sqrt {a+b x+c x^2} \, dx=\int \sqrt {d x} \sqrt {a + b x + c x^{2}}\, dx \] Input:

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

Output:

Integral(sqrt(d*x)*sqrt(a + b*x + c*x**2), x)
 

Maxima [F]

\[ \int \sqrt {d x} \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} \sqrt {d x} \,d x } \] Input:

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

Output:

integrate(sqrt(c*x^2 + b*x + a)*sqrt(d*x), x)
 

Giac [F]

\[ \int \sqrt {d x} \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} \sqrt {d x} \,d x } \] Input:

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

Output:

integrate(sqrt(c*x^2 + b*x + a)*sqrt(d*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {d x} \sqrt {a+b x+c x^2} \, dx=\int \sqrt {d\,x}\,\sqrt {c\,x^2+b\,x+a} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \sqrt {d x} \sqrt {a+b x+c x^2} \, dx=\frac {\sqrt {d}\, \left (2 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, a +2 \sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, b x -3 \left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x}{c \,x^{2}+b x +a}d x \right ) a c +\left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, x}{c \,x^{2}+b x +a}d x \right ) b^{2}-\left (\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}}{c \,x^{3}+b \,x^{2}+a x}d x \right ) a^{2}\right )}{5 b} \] Input:

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

Output:

(sqrt(d)*(2*sqrt(x)*sqrt(a + b*x + c*x**2)*a + 2*sqrt(x)*sqrt(a + b*x + c* 
x**2)*b*x - 3*int((sqrt(x)*sqrt(a + b*x + c*x**2)*x)/(a + b*x + c*x**2),x) 
*a*c + int((sqrt(x)*sqrt(a + b*x + c*x**2)*x)/(a + b*x + c*x**2),x)*b**2 - 
 int((sqrt(x)*sqrt(a + b*x + c*x**2))/(a*x + b*x**2 + c*x**3),x)*a**2))/(5 
*b)