Integrand size = 22, antiderivative size = 198 \[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx=-\frac {\sqrt {2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {c} \sqrt {b^2-4 a c}}+\frac {\sqrt {2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {c} \sqrt {b^2-4 a c}} \] Output:
-2^(1/2)*(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)*arctanh(2^(1/2)*c^(1/2)*(e *x+d)^(1/2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2))/c^(1/2)/(-4*a*c+b^2)^( 1/2)+2^(1/2)*(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)^(1/2)*arctanh(2^(1/2)*c^(1/2 )*(e*x+d)^(1/2)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)^(1/2))/c^(1/2)/(-4*a*c+b^ 2)^(1/2)
Result contains complex when optimal does not.
Time = 0.99 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx=\frac {\sqrt {2} \left (\frac {\left (-2 i c d+\left (i b+\sqrt {-b^2+4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-2 c d+\left (b-i \sqrt {-b^2+4 a c}\right ) e}}+\frac {\left (2 i c d+\left (-i b+\sqrt {-b^2+4 a c}\right ) e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e+i \sqrt {-b^2+4 a c} e}}\right )}{\sqrt {-2 c d+\left (b+i \sqrt {-b^2+4 a c}\right ) e}}\right )}{\sqrt {c} \sqrt {-b^2+4 a c}} \] Input:
Integrate[Sqrt[d + e*x]/(a + b*x + c*x^2),x]
Output:
(Sqrt[2]*((((-2*I)*c*d + (I*b + Sqrt[-b^2 + 4*a*c])*e)*ArcTan[(Sqrt[2]*Sqr t[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*Sq rt[-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])
Time = 0.38 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.18, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1148, 1450, 221}
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.
| \(\displaystyle \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 1148 |
| \(\displaystyle 2 e \int \frac {d+e x}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}\) |
| \(\Big \downarrow \) 1450 |
| \(\displaystyle 2 e \left (\frac {1}{2} \left (\frac {2 c d-b e}{e \sqrt {b^2-4 a c}}+1\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}+\frac {1}{2} \left (1-\frac {2 c d-b e}{e \sqrt {b^2-4 a c}}\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle 2 e \left (-\frac {\left (\frac {2 c d-b e}{e \sqrt {b^2-4 a c}}+1\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {\left (1-\frac {2 c d-b e}{e \sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {c} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )\) |
Input:
Int[Sqrt[d + e*x]/(a + b*x + c*x^2),x]
Output:
2*e*(-(((1 + (2*c*d - b*e)/(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[2]*Sqrt[c] *Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e])) - ((1 - (2*c*d - b*e)/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b + Sq rt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[c]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c] )*e]))
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]
Int[Sqrt[(d_.) + (e_.)*(x_)]/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]
Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Wi th[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(d^2/2)*(b/q + 1) Int[(d*x)^(m - 2)/(b/ 2 + q/2 + c*x^2), x], x] - Simp[(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]
Time = 1.42 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.12
| method | result | size |
| pseudoelliptic | \(\frac {e \sqrt {2}\, \left (-\frac {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{\sqrt {-e^{2} \left (4 a c -b^{2}\right )}}\) | \(222\) |
| derivativedivides | \(8 e c \left (-\frac {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\) | \(251\) |
| default | \(8 e c \left (-\frac {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )\) | \(251\) |
Input:
int((e*x+d)^(1/2)/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
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)*arctanh((e*x+d)^(1/2)*c*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)*arctan(( e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (158) = 316\).
Time = 0.09 (sec) , antiderivative size = 715, normalized size of antiderivative = 3.61 \[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx=-\frac {1}{2} \, \sqrt {2} \sqrt {\frac {2 \, c d - b e + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\sqrt {2} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt {\frac {2 \, c d - b e + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt {e x + d} e\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {2 \, c d - b e + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (-\sqrt {2} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt {\frac {2 \, c d - b e + {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt {e x + d} e\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {2 \, c d - b e - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (\sqrt {2} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt {\frac {2 \, c d - b e - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt {e x + d} e\right ) - \frac {1}{2} \, \sqrt {2} \sqrt {\frac {2 \, c d - b e - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (-\sqrt {2} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}} \sqrt {\frac {2 \, c d - b e - {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {\frac {e^{2}}{b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} + 2 \, \sqrt {e x + d} e\right ) \] Input:
integrate((e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
-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*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)
\[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx=\int \frac {\sqrt {d + e x}}{a + b x + c x^{2}}\, dx \] Input:
integrate((e*x+d)**(1/2)/(c*x**2+b*x+a),x)
Output:
Integral(sqrt(d + e*x)/(a + b*x + c*x**2), x)
\[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx=\int { \frac {\sqrt {e x + d}}{c x^{2} + b x + a} \,d x } \] Input:
integrate((e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="maxima")
Output:
integrate(sqrt(e*x + d)/(c*x^2 + b*x + a), x)
Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (158) = 316\).
Time = 0.28 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.25 \[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx=\frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} - 4 \, a c\right )} e^{3} - {\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} + b^{2} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, c d - b e + \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{2} d^{2} - \sqrt {b^{2} - 4 \, a c} b c d e + \sqrt {b^{2} - 4 \, a c} a c e^{2}\right )} {\left | c \right |} {\left | e \right |}} - \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} - 4 \, a c\right )} e^{3} - {\left (4 \, c^{2} d^{2} e - 4 \, b c d e^{2} + b^{2} e^{3}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x + d}}{\sqrt {-\frac {2 \, c d - b e - \sqrt {{\left (2 \, c d - b e\right )}^{2} - 4 \, {\left (c d^{2} - b d e + a e^{2}\right )} c}}{c}}}\right )}{4 \, {\left (\sqrt {b^{2} - 4 \, a c} c^{2} d^{2} - \sqrt {b^{2} - 4 \, a c} b c d e + \sqrt {b^{2} - 4 \, a c} a c e^{2}\right )} {\left | c \right |} {\left | e \right |}} \] Input:
integrate((e*x+d)^(1/2)/(c*x^2+b*x+a),x, algorithm="giac")
Output:
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 - s qrt((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))
Time = 11.31 (sec) , antiderivative size = 709, normalized size of antiderivative = 3.58 \[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx=-2\,\mathrm {atanh}\left (\frac {2\,\left (\sqrt {d+e\,x}\,\left (-8\,b^2\,c\,e^4+16\,b\,c^2\,d\,e^3-16\,c^3\,d^2\,e^2+16\,a\,c^2\,e^4\right )+\frac {\sqrt {d+e\,x}\,\left (8\,b^3\,c^2\,e^3-16\,d\,b^2\,c^3\,e^2-32\,a\,b\,c^3\,e^3+64\,a\,d\,c^4\,e^2\right )\,\left (b^3\,e+e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+8\,a\,c^2\,d-2\,b^2\,c\,d-4\,a\,b\,c\,e\right )}{2\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}\right )\,\sqrt {-\frac {b^3\,e+e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+8\,a\,c^2\,d-2\,b^2\,c\,d-4\,a\,b\,c\,e}{2\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}}{16\,c^2\,d^2\,e^3-16\,b\,c\,d\,e^4+16\,a\,c\,e^5}\right )\,\sqrt {-\frac {b^3\,e+e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}+8\,a\,c^2\,d-2\,b^2\,c\,d-4\,a\,b\,c\,e}{2\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}-2\,\mathrm {atanh}\left (\frac {2\,\left (\sqrt {d+e\,x}\,\left (-8\,b^2\,c\,e^4+16\,b\,c^2\,d\,e^3-16\,c^3\,d^2\,e^2+16\,a\,c^2\,e^4\right )-\frac {\sqrt {d+e\,x}\,\left (8\,b^3\,c^2\,e^3-16\,d\,b^2\,c^3\,e^2-32\,a\,b\,c^3\,e^3+64\,a\,d\,c^4\,e^2\right )\,\left (e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e-8\,a\,c^2\,d+2\,b^2\,c\,d+4\,a\,b\,c\,e\right )}{2\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}\right )\,\sqrt {\frac {e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e-8\,a\,c^2\,d+2\,b^2\,c\,d+4\,a\,b\,c\,e}{2\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}}}{16\,c^2\,d^2\,e^3-16\,b\,c\,d\,e^4+16\,a\,c\,e^5}\right )\,\sqrt {\frac {e\,\sqrt {-{\left (4\,a\,c-b^2\right )}^3}-b^3\,e-8\,a\,c^2\,d+2\,b^2\,c\,d+4\,a\,b\,c\,e}{2\,\left (16\,a^2\,c^3-8\,a\,b^2\,c^2+b^4\,c\right )}} \] Input:
int((d + e*x)^(1/2)/(a + b*x + c*x^2),x)
Output:
- 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*atan h((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)
Time = 0.48 (sec) , antiderivative size = 1369, normalized size of antiderivative = 6.91 \[ \int \frac {\sqrt {d+e x}}{a+b x+c x^2} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^(1/2)/(c*x^2+b*x+a),x)
Output:
( - 4*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*sqrt(a*e **2 - b*d*e + c*d**2)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d* e + c*d**2) + b*e - 2*c*d))*c + 2*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d *e + c*d**2) + b*e - 2*c*d)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d **2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*b*e - 4*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a*e* *2 - b*d*e + c*d**2) + b*e - 2*c*d)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d *e + c*d**2) - b*e + 2*c*d) - 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt (a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*c*d + 4*sqrt(2*sqrt(c)*sqrt(a*e* *2 - b*d*e + c*d**2) + b*e - 2*c*d)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((sq rt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) + 2*sqrt(c)*sqrt (d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d))*c - 2*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c*d)*at an((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) + 2*sqrt(c )*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2*c* d))*b*e + 4*sqrt(c)*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b*e - 2 *c*d)*atan((sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e + 2*c*d) + 2*sqrt(c)*sqrt(d + e*x))/sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) + b* e - 2*c*d))*c*d + 2*sqrt(2*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2) - b*e ...