\(\int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx\) [98]

Optimal result

Integrand size = 24, antiderivative size = 179 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

2*2^(1/2)*(-4*a*c+b^2)^(1/2)*(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^ 
(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticF(1/2*(1+(2*c*x+b)/(-4 
*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+ 
b^2)^(1/2))*e))^(1/2))/c/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.69 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.72 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\frac {i (d+e x) \sqrt {2-\frac {4 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {1+\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{e \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {a+x (b+c x)}} \] Input:

Integrate[1/(Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]),x]
 

Output:

(I*(d + e*x)*Sqrt[2 - (4*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[ 
(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/(( 
-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticF[I*ArcSinh[(S 
qrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2 
])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b 
*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/(e*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c* 
d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[a + x*(b + c*x)])
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1172, 321}

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.

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

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}\)

Input:

Int[1/(Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]),x]
 

Output:

(2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4 
*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[S 
qrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[ 
b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(c*Sqrt[d + e*x]*Sqr 
t[a + b*x + c*x^2])
 

Defintions of rubi rules used

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 1172

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

Maple [A] (verified)

Time = 2.84 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.60

method	result	size
default	\(\frac {\left (-\sqrt {-4 a c +b^{2}}\, e -b e +2 c d \right ) \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\right ) \sqrt {\frac {\left (2 c x +\sqrt {-4 a c +b^{2}}+b \right ) e}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}\, \sqrt {\frac {\left (-2 c x +\sqrt {-4 a c +b^{2}}-b \right ) e}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\, \sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {e x +d}}{c e \left (c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d \right )}\)	\(287\)
elliptic	\(\frac {2 \sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}\)	\(321\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\frac {2 \, \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )}{c e} \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\sqrt {d + e x} \sqrt {a + b x + c x^{2}}}\, dx \] Input:

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

Output:

Integral(1/(sqrt(d + e*x)*sqrt(a + b*x + c*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {e x + d}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} \sqrt {e x + d}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\sqrt {d+e\,x}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}d x \] Input:

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

Output:

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