\(\int \frac {\sqrt {a+b x+c x^2}}{b d+2 c d x} \, dx\) [155]

Optimal result

Integrand size = 26, antiderivative size = 83 \[ \int \frac {\sqrt {a+b x+c x^2}}{b d+2 c d x} \, dx=\frac {\sqrt {a+b x+c x^2}}{2 c d}-\frac {\sqrt {b^2-4 a c} \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{4 c^{3/2} d} \] Output:

1/2*(c*x^2+b*x+a)^(1/2)/c/d-1/4*(-4*a*c+b^2)^(1/2)*arctan(2*c^(1/2)*(c*x^2 
+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))/c^(3/2)/d
 

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {a+b x+c x^2}}{b d+2 c d x} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)}+\sqrt {b^2-4 a c} \arctan \left (\frac {\sqrt {c} \sqrt {b^2-4 a c} x}{\sqrt {a} (b+2 c x)-b \sqrt {a+x (b+c x)}}\right )}{2 c^{3/2} d} \] Input:

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

Output:

(Sqrt[c]*Sqrt[a + x*(b + c*x)] + Sqrt[b^2 - 4*a*c]*ArcTan[(Sqrt[c]*Sqrt[b^ 
2 - 4*a*c]*x)/(Sqrt[a]*(b + 2*c*x) - b*Sqrt[a + x*(b + c*x)])])/(2*c^(3/2) 
*d)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1109, 27, 1112, 218}

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.

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

\(\Big \downarrow \) 1109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1112

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {\sqrt {a+b x+c x^2}}{2 c d}-\frac {\sqrt {b^2-4 a c} \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{4 c^{3/2} d}\)

Input:

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

Output:

Sqrt[a + b*x + c*x^2]/(2*c*d) - (Sqrt[b^2 - 4*a*c]*ArcTan[(2*Sqrt[c]*Sqrt[ 
a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(4*c^(3/2)*d)
 

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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 1109

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[d*p*((b^2 - 4*a*c)/(b*e*(m + 2*p + 1)))   Int[(d + e*x)^m*(a + b*x 
 + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b* 
e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[p, 0] &&  !LtQ[m, -1] &&  !(IGtQ[(m - 1 
)/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p])) && RationalQ[m] && IntegerQ[2*p]
 

rule 1112

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symb 
ol] :> Simp[4*c   Subst[Int[1/(b^2*e - 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a 
+ b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
 

Maple [A] (verified)

Time = 1.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(\frac {\sqrt {c \,x^{2}+b x +a}-\frac {\left (4 a c -b^{2}\right ) \operatorname {arctanh}\left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, c}{\sqrt {\left (4 a c -b^{2}\right ) c}}\right )}{2 \sqrt {\left (4 a c -b^{2}\right ) c}}}{2 c d}\)	\(78\)
risch	\(\frac {\sqrt {c \,x^{2}+b x +a}}{2 c d}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{4 c^{2} \sqrt {\frac {4 a c -b^{2}}{c}}\, d}\)	\(132\)
default	\(\frac {\frac {\sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}}{2 d c}\)	\(149\)

Input:

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

Output:

1/2/c*((c*x^2+b*x+a)^(1/2)-1/2*(4*a*c-b^2)/((4*a*c-b^2)*c)^(1/2)*arctanh(2 
*(c*x^2+b*x+a)^(1/2)*c/((4*a*c-b^2)*c)^(1/2)))/d
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.33 \[ \int \frac {\sqrt {a+b x+c x^2}}{b d+2 c d x} \, dx=\left [\frac {\sqrt {-\frac {b^{2} - 4 \, a c}{c}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt {c x^{2} + b x + a} c \sqrt {-\frac {b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \, \sqrt {c x^{2} + b x + a}}{8 \, c d}, \frac {\sqrt {\frac {b^{2} - 4 \, a c}{c}} \arctan \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} c \sqrt {\frac {b^{2} - 4 \, a c}{c}}}{b^{2} - 4 \, a c}\right ) + 2 \, \sqrt {c x^{2} + b x + a}}{4 \, c d}\right ] \] Input:

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

Output:

[1/8*(sqrt(-(b^2 - 4*a*c)/c)*log(-(4*c^2*x^2 + 4*b*c*x - b^2 + 8*a*c - 4*s 
qrt(c*x^2 + b*x + a)*c*sqrt(-(b^2 - 4*a*c)/c))/(4*c^2*x^2 + 4*b*c*x + b^2) 
) + 4*sqrt(c*x^2 + b*x + a))/(c*d), 1/4*(sqrt((b^2 - 4*a*c)/c)*arctan(-2*s 
qrt(c*x^2 + b*x + a)*c*sqrt((b^2 - 4*a*c)/c)/(b^2 - 4*a*c)) + 2*sqrt(c*x^2 
 + b*x + a))/(c*d)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((c*x^2+b*x+a)^(1/2)/(2*c*d*x+b*d),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 = 0.13 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {a+b x+c x^2}}{b d+2 c d x} \, dx=-\frac {{\left (b^{2} - 4 \, a c\right )} \arctan \left (-\frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c + b \sqrt {c}}{\sqrt {b^{2} c - 4 \, a c^{2}}}\right )}{2 \, \sqrt {b^{2} c - 4 \, a c^{2}} c d} + \frac {\sqrt {c x^{2} + b x + a}}{2 \, c d} \] Input:

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

Output:

-1/2*(b^2 - 4*a*c)*arctan(-(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c + b*sq 
rt(c))/sqrt(b^2*c - 4*a*c^2))/(sqrt(b^2*c - 4*a*c^2)*c*d) + 1/2*sqrt(c*x^2 
 + b*x + a)/(c*d)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.78 \[ \int \frac {\sqrt {a+b x+c x^2}}{b d+2 c d x} \, dx=\frac {\sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {-\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right )-\sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right )+2 \sqrt {c \,x^{2}+b x +a}\, c}{4 c^{2} d} \] Input:

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

Output:

(sqrt(c)*sqrt(4*a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a 
+ b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2)) - sqrt(c)*sqrt(4*a*c - b* 
*2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x 
)/sqrt(4*a*c - b**2)) + 2*sqrt(a + b*x + c*x**2)*c)/(4*c**2*d)