\(\int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx\) [91]

Optimal result

Integrand size = 24, antiderivative size = 178 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.79 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=\frac {i \left (2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e\right ) \sqrt {\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {1-\frac {2 c (d+e x)}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}} \left (E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {d+e x}\right )|\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {d+e x}\right ),\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}\right )\right )}{\sqrt {2} c e \sqrt {\frac {c}{-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+x (b+c x)}} \] Input:

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

Output:

(I*(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)*Sqrt[(e*(b + Sqrt[b^2 - 4*a*c] + 2 
*c*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)]*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(-2* 
c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[d + e*x]], (2*c*d - (b + Sqrt[b^2 - 
 4*a*c])*e)/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)] - EllipticF[I*ArcSinh[Sq 
rt[2]*Sqrt[c/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[d + e*x]], (2*c*d 
- (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e)]))/(Sqrt 
[2]*c*e*Sqrt[c/(-2*c*d + (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + x*(b + c*x)] 
)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 188, 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, 327}

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.

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

\(\Big \downarrow \) 1172

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[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]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(745\) vs. \(2(156)=312\).

Time = 1.37 (sec) , antiderivative size = 746, normalized size of antiderivative = 4.19

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 d \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}+\frac {2 e \left (\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {d}{e}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}\)	\(746\)
default	\(\frac {\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}\, \left (\sqrt {-4 a c +b^{2}}\, e +b e -2 c d \right ) \sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}\, \sqrt {\frac {\left (-2 c x +\sqrt {-4 a c +b^{2}}-b \right ) e}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\, \sqrt {\frac {\left (2 c x +\sqrt {-4 a c +b^{2}}+b \right ) e}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}\, \left (\operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\right ) e b -2 d \operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\right ) c -\operatorname {EllipticF}\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\right ) e \sqrt {-4 a c +b^{2}}-\operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\right ) b e +2 \operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\right ) c d +\operatorname {EllipticE}\left (\sqrt {2}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}}, \sqrt {-\frac {\sqrt {-4 a c +b^{2}}\, e +b e -2 c d}{2 c d -b e +\sqrt {-4 a c +b^{2}}\, e}}\right ) \sqrt {-4 a c +b^{2}}\, e \right )}{2 e \left (c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+a e x +b d x +a d \right ) c^{2}}\)	\(747\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (160) = 320\).

Time = 0.09 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {c e} c e {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - {\left (2 \, c d - b e\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right )}}{3 \, c^{2} e} \] Input:

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

Output:

-2/3*(3*sqrt(c*e)*c*e*weierstrassZeta(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), 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))) - (2*c*d - b*e)*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^2*e)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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