\(\int \frac {1}{x \sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx\) [99]

Optimal result

Integrand size = 27, antiderivative size = 222 \[ \int \frac {1}{x \sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.00 (sec) , antiderivative size = 482, normalized size of antiderivative = 2.17 \[ \int \frac {1}{x \sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=-\frac {i (d+e x) \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {2+\frac {4 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )-\operatorname {EllipticPi}\left (\frac {d \left (2 c d-b e-\sqrt {\left (b^2-4 a c\right ) e^2}\right )}{2 \left (c d^2+e (-b d+a e)\right )},i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{d \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {a+x (b+c x)}} \] Input:

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

Output:

((-I)*(d + e*x)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sq 
rt[(b^2 - 4*a*c)*e^2])*(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))]*(EllipticF[I*ArcSin 
h[(Sqrt[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]))] - EllipticPi[(d*(2*c*d - b*e - Sqrt[(b 
^2 - 4*a*c)*e^2]))/(2*(c*d^2 + e*(-(b*d) + a*e))), I*ArcSinh[(Sqrt[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]))]))/(d*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 = 0.74 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.74, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1279, 187, 413, 413, 412}

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.

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

\(\Big \downarrow \) 1279

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

\(\Big \downarrow \) 187

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 413

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

\(\Big \downarrow \) 412

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

Input:

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

Output:

-((Sqrt[2]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[b - Sqrt[b^2 - 4*a 
*c] + 2*c*x]*Sqrt[b + Sqrt[b^2 - 4*a*c] + 2*c*x]*Sqrt[1 - (2*c*(d + e*x))/ 
(2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1 - (2*c*(d + e*x))/(2*c*d - (b 
+ Sqrt[b^2 - 4*a*c])*e)]*EllipticPi[(2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2 
*c*d), ArcSin[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 
 4*a*c])*e]], (2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)/(2*c*d - (b + Sqrt[b^2 - 
 4*a*c])*e)])/(Sqrt[c]*d*Sqrt[a + b*x + c*x^2]*Sqrt[b - Sqrt[b^2 - 4*a*c] 
- (2*c*d)/e + (2*c*(d + e*x))/e]*Sqrt[b + Sqrt[b^2 - 4*a*c] - (2*c*d)/e + 
(2*c*(d + e*x))/e]))
 

Defintions of rubi rules used

rule 187

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] &&  !SimplerQ[e + f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 1279

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[b 
 - q + 2*c*x]*(Sqrt[b + q + 2*c*x]/Sqrt[a + b*x + c*x^2])   Int[1/((d + e*x 
)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[ 
{a, b, c, d, e, f, g}, x]
 

Maple [A] (verified)

Time = 4.22 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.41

method	result	size
default	\(\frac {\left (\sqrt {-4 a c +b^{2}}\, e +b e -2 c d \right ) \operatorname {EllipticPi}\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}, -\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 d c}, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\right ) \sqrt {\frac {\left (2 c x +\sqrt {-4 a c +b^{2}}+b \right ) e}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}\, \sqrt {\frac {\left (-2 c x +\sqrt {-4 a c +b^{2}}-b \right ) e}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\, \sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {e x +d}}{c d \left (c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d \right )}\)	\(313\)
elliptic	\(-\frac {2 \sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, e \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, -\frac {\left (-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) e}{d}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}\, d}\)	\(355\)

Input:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/x/(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), x)
 

Giac [F]

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

integrate(1/x/(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), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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