Integrand size = 46, antiderivative size = 80 \[ \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {c d f-a e g} \sqrt {d+e x}}{\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{\sqrt {g} \sqrt {c d f-a e g}} \] Output:
-2*arctan(1/g^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*(-a*e*g+c*d*f) ^(1/2)*(e*x+d)^(1/2))/g^(1/2)/(-a*e*g+c*d*f)^(1/2)
Time = 0.10 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {a e+c d x} \sqrt {d+e x} \arctan \left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{\sqrt {g} \sqrt {c d f-a e g} \sqrt {(a e+c d x) (d+e x)}} \] Input:
Integrate[Sqrt[d + e*x]/((f + g*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e* x^2]),x]
Output:
(2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTan[(Sqrt[g]*Sqrt[a*e + c*d*x])/Sqrt [c*d*f - a*e*g]])/(Sqrt[g]*Sqrt[c*d*f - a*e*g]*Sqrt[(a*e + c*d*x)*(d + e*x )])
Time = 0.41 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1255, 218}
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.
\(\displaystyle \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \, dx\) |
\(\Big \downarrow \) 1255 |
\(\displaystyle 2 e^2 \int \frac {1}{(c d f-a e g) e^2+\frac {g \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right ) e^2}{d+e x}}d\frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 \arctan \left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{\sqrt {g} \sqrt {c d f-a e g}}\) |
Input:
Int[Sqrt[d + e*x]/((f + g*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]), x]
Output:
(2*ArcTan[(Sqrt[g]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(Sqrt[c*d* f - a*e*g]*Sqrt[d + e*x])])/(Sqrt[g]*Sqrt[c*d*f - a*e*g])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[Sqrt[(d_) + (e_.)*(x_)]/(((f_.) + (g_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2*e^2 Subst[Int[1/(c*(e*f + d*g) - b*e *g + e^2*g*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{ a, b, c, d, e, f, g}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Time = 2.76 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96
method | result | size |
default | \(-\frac {2 \sqrt {\left (e x +d \right ) \left (c d x +a e \right )}\, \operatorname {arctanh}\left (\frac {g \sqrt {c d x +a e}}{\sqrt {\left (a e g -d f c \right ) g}}\right )}{\sqrt {e x +d}\, \sqrt {c d x +a e}\, \sqrt {\left (a e g -d f c \right ) g}}\) | \(77\) |
Input:
int((e*x+d)^(1/2)/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2),x,method =_RETURNVERBOSE)
Output:
-2/(e*x+d)^(1/2)*((e*x+d)*(c*d*x+a*e))^(1/2)/(c*d*x+a*e)^(1/2)/((a*e*g-c*d *f)*g)^(1/2)*arctanh(g*(c*d*x+a*e)^(1/2)/((a*e*g-c*d*f)*g)^(1/2))
Time = 0.10 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.16 \[ \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [-\frac {\sqrt {-c d f g + a e g^{2}} \log \left (-\frac {c d e g x^{2} - c d^{2} f + 2 \, a d e g - {\left (c d e f - {\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d f g + a e g^{2}} \sqrt {e x + d}}{e g x^{2} + d f + {\left (e f + d g\right )} x}\right )}{c d f g - a e g^{2}}, -\frac {2 \, \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d f g - a e g^{2}} \sqrt {e x + d}}{c d^{2} f - a d e g + {\left (c d e f - a e^{2} g\right )} x}\right )}{\sqrt {c d f g - a e g^{2}}}\right ] \] Input:
integrate((e*x+d)^(1/2)/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")
Output:
[-sqrt(-c*d*f*g + a*e*g^2)*log(-(c*d*e*g*x^2 - c*d^2*f + 2*a*d*e*g - (c*d* e*f - (c*d^2 + 2*a*e^2)*g)*x - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)* x)*sqrt(-c*d*f*g + a*e*g^2)*sqrt(e*x + d))/(e*g*x^2 + d*f + (e*f + d*g)*x) )/(c*d*f*g - a*e*g^2), -2*arctan(-sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2) *x)*sqrt(c*d*f*g - a*e*g^2)*sqrt(e*x + d)/(c*d^2*f - a*d*e*g + (c*d*e*f - a*e^2*g)*x))/sqrt(c*d*f*g - a*e*g^2)]
\[ \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\sqrt {d + e x}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )}\, dx \] Input:
integrate((e*x+d)**(1/2)/(g*x+f)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/ 2),x)
Output:
Integral(sqrt(d + e*x)/(sqrt((d + e*x)*(a*e + c*d*x))*(f + g*x)), x)
\[ \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {\sqrt {e x + d}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}} \,d x } \] Input:
integrate((e*x+d)^(1/2)/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)), x)
Time = 0.13 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, e \arctan \left (\frac {\sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}} g}{\sqrt {c d f g - a e g^{2}} e}\right )}{\sqrt {c d f g - a e g^{2}} {\left | e \right |}} \] Input:
integrate((e*x+d)^(1/2)/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")
Output:
2*e*arctan(sqrt((e*x + d)*c*d*e - c*d^2*e + a*e^3)*g/(sqrt(c*d*f*g - a*e*g ^2)*e))/(sqrt(c*d*f*g - a*e*g^2)*abs(e))
Timed out. \[ \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\sqrt {d+e\,x}}{\left (f+g\,x\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \] Input:
int((d + e*x)^(1/2)/((f + g*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/ 2)),x)
Output:
int((d + e*x)^(1/2)/((f + g*x)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/ 2)), x)
Time = 0.47 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {2 \sqrt {g}\, \sqrt {-a e g +c d f}\, \mathit {atan} \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {g}\, \sqrt {-a e g +c d f}}\right )}{g \left (a e g -c d f \right )} \] Input:
int((e*x+d)^(1/2)/(g*x+f)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
Output:
( - 2*sqrt(g)*sqrt( - a*e*g + c*d*f)*atan((sqrt(a*e + c*d*x)*g)/(sqrt(g)*s qrt( - a*e*g + c*d*f))))/(g*(a*e*g - c*d*f))