\(\int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx\) [2291]

Optimal result

Integrand size = 22, antiderivative size = 198 \[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx=-\frac {\sqrt {2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {c} \sqrt {b^2-4 a c}}+\frac {\sqrt {2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {c} \sqrt {b^2-4 a c}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {713, 1144, 214} \[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx=\frac {\sqrt {2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {c} \sqrt {b^2-4 a c}}-\frac {\sqrt {2} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {c} \sqrt {b^2-4 a c}} \]

[In]

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

[Out]

-((Sqrt[2]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - S
qrt[b^2 - 4*a*c])*e]])/(Sqrt[c]*Sqrt[b^2 - 4*a*c])) + (Sqrt[2]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*ArcTanh
[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[c]*Sqrt[b^2 - 4*a*c])

Rule 214

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

Rule 713

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

Rule 1144

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2/2)*(b/q + 1), Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2/2)*(b/q - 1), Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

Rubi steps \begin{align*} \text {integral}& = (2 e) \text {Subst}\left (\int \frac {x^2}{c d^2-b d e+a e^2+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right ) \\ & = -\left (\left (-e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )\right )+\left (e-\frac {2 c d-b e}{\sqrt {b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {b^2-4 a c} e+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right ) \\ & = -\frac {\sqrt {2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \tanh ^{-1}\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 )}{\sqrt {c} \sqrt {b^2-4 a c}}+\frac {\sqrt {2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \tanh ^{-1}\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 )}{\sqrt {c} \sqrt {b^2-4 a c}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.62 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx=\frac {\sqrt {2} \left (\frac {\left (-2 i c d+\left (i b+\sqrt {-b^2+4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\left (2 i c d+\left (-i b+\sqrt {-b^2+4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}\right )}{\sqrt {c} \sqrt {-b^2+4 a c}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(\frac {e \sqrt {2}\, \left (\frac {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {-e^{2} \left (4 a c -b^{2}\right )}}\)	\(222\)
derivativedivides	\(8 e c \left (\frac {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\)	\(251\)
default	\(8 e c \left (\frac {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctan \left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\)	\(251\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (158) = 316\).

Time = 0.31 (sec) , antiderivative size = 715, normalized size of antiderivative = 3.61 \[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx=-\frac {1}{2} \, \sqrt {2} \sqrt {\frac {2 \, c d - b e + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\sqrt {2} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt {\frac {2 \, c d - b e + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt {e x + d} e\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {2 \, c d - b e + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (-\sqrt {2} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt {\frac {2 \, c d - b e + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt {e x + d} e\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {2 \, c d - b e - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\sqrt {2} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt {\frac {2 \, c d - b e - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt {e x + d} e\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {\frac {2 \, c d - b e - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (-\sqrt {2} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt {\frac {2 \, c d - b e - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt {e x + d} e\right ) \]

[In]

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

[Out]

-1/2*sqrt(2)*sqrt((2*c*d - b*e + (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2))*log(sqrt(
2)*(b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3))*sqrt((2*c*d - b*e + (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4
*a*c^3)))/(b^2*c - 4*a*c^2)) + 2*sqrt(e*x + d)*e) + 1/2*sqrt(2)*sqrt((2*c*d - b*e + (b^2*c - 4*a*c^2)*sqrt(e^2
/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2))*log(-sqrt(2)*(b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3))*sqrt((2
*c*d - b*e + (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2)) + 2*sqrt(e*x + d)*e) + 1/2*sq
rt(2)*sqrt((2*c*d - b*e - (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2))*log(sqrt(2)*(b^2
*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3))*sqrt((2*c*d - b*e - (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)
))/(b^2*c - 4*a*c^2)) + 2*sqrt(e*x + d)*e) - 1/2*sqrt(2)*sqrt((2*c*d - b*e - (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c
^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2))*log(-sqrt(2)*(b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3))*sqrt((2*c*d -
b*e - (b^2*c - 4*a*c^2)*sqrt(e^2/(b^2*c^2 - 4*a*c^3)))/(b^2*c - 4*a*c^2)) + 2*sqrt(e*x + d)*e)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (158) = 316\).

Time = 0.31 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.25 \[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx=\frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} - 4 \, a c\right )} e^{3} - {\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} + b^{2} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, c d - b e + \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{2} d^{2} - \sqrt {b^{2} - 4 \, a c} b c d e + \sqrt {b^{2} - 4 \, a c} a c e^{2}\right )} {\left | c \right |} {\left | e \right |}} - \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} - 4 \, a c\right )} e^{3} - {\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} + b^{2} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, c d - b e - \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{2} d^{2} - \sqrt {b^{2} - 4 \, a c} b c d e + \sqrt {b^{2} - 4 \, a c} a c e^{2}\right )} {\left | c \right |} {\left | e \right |}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.11 (sec) , antiderivative size = 709, normalized size of antiderivative = 3.58 \[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx=-2\,\mathrm {atanh}\left (\frac {2\,\left (\sqrt {d+e\,x}\,\left (-8\,b^2\,c\,e^4+16\,b\,c^2\,d\,e^3-16\,c^3\,d^2\,e^2+16\,a\,c^2\,e^4\right )+\frac {\sqrt {d+e\,x}\,\left (8\,b^3\,c^2\,e^3-16\,d\,b^2\,c^3\,e^2-32\,a\,b\,c^3\,e^3+64\,a\,d\,c^4\,e^2\right )\,\left (b^3\,e+e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+8\,a\,c^2\,d-2\,b^2\,c\,d-4\,a\,b\,c\,e\right )}{2\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}\right )\,\sqrt {-\frac {b^3\,e+e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+8\,a\,c^2\,d-2\,b^2\,c\,d-4\,a\,b\,c\,e}{2\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}}{16\,c^2\,d^2\,e^3-16\,b\,c\,d\,e^4+16\,a\,c\,e^5}\right )\,\sqrt {-\frac {b^3\,e+e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+8\,a\,c^2\,d-2\,b^2\,c\,d-4\,a\,b\,c\,e}{2\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}-2\,\mathrm {atanh}\left (\frac {2\,\left (\sqrt {d+e\,x}\,\left (-8\,b^2\,c\,e^4+16\,b\,c^2\,d\,e^3-16\,c^3\,d^2\,e^2+16\,a\,c^2\,e^4\right )-\frac {\sqrt {d+e\,x}\,\left (8\,b^3\,c^2\,e^3-16\,d\,b^2\,c^3\,e^2-32\,a\,b\,c^3\,e^3+64\,a\,d\,c^4\,e^2\right )\,\left (e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e-8\,a\,c^2\,d+2\,b^2\,c\,d+4\,a\,b\,c\,e\right )}{2\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}\right )\,\sqrt {\frac {e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e-8\,a\,c^2\,d+2\,b^2\,c\,d+4\,a\,b\,c\,e}{2\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}}{16\,c^2\,d^2\,e^3-16\,b\,c\,d\,e^4+16\,a\,c\,e^5}\right )\,\sqrt {\frac {e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e-8\,a\,c^2\,d+2\,b^2\,c\,d+4\,a\,b\,c\,e}{2\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}} \]

[In]

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

[Out]

- 2*atanh((2*((d + e*x)^(1/2)*(16*a*c^2*e^4 - 8*b^2*c*e^4 - 16*c^3*d^2*e^2 + 16*b*c^2*d*e^3) + ((d + e*x)^(1/2
)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2)*(b^3*e + e*(-(4*a*c - b^2)^3)^(1/2) + 8
*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e))/(2*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))*(-(b^3*e + e*(-(4*a*c - b^2)^3)^(1/
2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(2*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))^(1/2))/(16*c^2*d^2*e^3 + 16*a*
c*e^5 - 16*b*c*d*e^4))*(-(b^3*e + e*(-(4*a*c - b^2)^3)^(1/2) + 8*a*c^2*d - 2*b^2*c*d - 4*a*b*c*e)/(2*(b^4*c +
16*a^2*c^3 - 8*a*b^2*c^2)))^(1/2) - 2*atanh((2*((d + e*x)^(1/2)*(16*a*c^2*e^4 - 8*b^2*c*e^4 - 16*c^3*d^2*e^2 +
16*b*c^2*d*e^3) - ((d + e*x)^(1/2)*(8*b^3*c^2*e^3 - 16*b^2*c^3*d*e^2 - 32*a*b*c^3*e^3 + 64*a*c^4*d*e^2)*(e*(-
(4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c*d + 4*a*b*c*e))/(2*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))*(
(e*(-(4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b^2*c*d + 4*a*b*c*e)/(2*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)
))^(1/2))/(16*c^2*d^2*e^3 + 16*a*c*e^5 - 16*b*c*d*e^4))*((e*(-(4*a*c - b^2)^3)^(1/2) - b^3*e - 8*a*c^2*d + 2*b
^2*c*d + 4*a*b*c*e)/(2*(b^4*c + 16*a^2*c^3 - 8*a*b^2*c^2)))^(1/2)