Integrand size = 31, antiderivative size = 115 \[ \int \frac {1}{\sqrt {e+f x} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d) \sqrt {b e-a f}}+\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d) \sqrt {d e-c f}} \] Output:
-2*b^(1/2)*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/(-a*d+b*c)/(-a* f+b*e)^(1/2)+2*d^(1/2)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/(-a *d+b*c)/(-c*f+d*e)^(1/2)
Time = 0.22 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {e+f x} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{\sqrt {-b e+a f}}-\frac {2 \sqrt {d} \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{\sqrt {-d e+c f}}}{b c-a d} \] Input:
Integrate[1/(Sqrt[e + f*x]*(a*c + (b*c + a*d)*x + b*d*x^2)),x]
Output:
((2*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/Sqrt[-(b*e ) + a*f] - (2*Sqrt[d]*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/ Sqrt[-(d*e) + c*f])/(b*c - a*d)
Time = 0.24 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1149, 1406, 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.
\(\displaystyle \int \frac {1}{\sqrt {e+f x} \left (x (a d+b c)+a c+b d x^2\right )} \, dx\) |
\(\Big \downarrow \) 1149 |
\(\displaystyle 2 f \int \frac {1}{b d (e+f x)^2-(2 b d e-b c f-a d f) (e+f x)+(b e-a f) (d e-c f)}d\sqrt {e+f x}\) |
\(\Big \downarrow \) 1406 |
\(\displaystyle 2 f \left (\frac {b d \int \frac {1}{b d (e+f x)-d (b e-a f)}d\sqrt {e+f x}}{f (b c-a d)}-\frac {b d \int \frac {1}{b d (e+f x)-b (d e-c f)}d\sqrt {e+f x}}{f (b c-a d)}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle 2 f \left (\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{f (b c-a d) \sqrt {d e-c f}}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{f (b c-a d) \sqrt {b e-a f}}\right )\) |
Input:
Int[1/(Sqrt[e + f*x]*(a*c + (b*c + a*d)*x + b*d*x^2)),x]
Output:
2*f*(-((Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/((b*c - a*d)*f*Sqrt[b*e - a*f])) + (Sqrt[d]*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d *e - c*f]])/((b*c - a*d)*f*Sqrt[d*e - c*f]))
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[1/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Sym bol] :> Simp[2*e Subst[Int[1/(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[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c , 0] && PosQ[b^2 - 4*a*c]
Time = 1.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(\frac {-\frac {2 b \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{\sqrt {\left (a f -b e \right ) b}}+\frac {2 d \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\sqrt {\left (c f -d e \right ) d}}}{a d -b c}\) | \(87\) |
derivativedivides | \(2 f \left (\frac {d \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{f \left (a d -b c \right ) \sqrt {\left (c f -d e \right ) d}}-\frac {b \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f \left (a d -b c \right ) \sqrt {\left (a f -b e \right ) b}}\right )\) | \(104\) |
default | \(2 f \left (\frac {d \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{f \left (a d -b c \right ) \sqrt {\left (c f -d e \right ) d}}-\frac {b \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f \left (a d -b c \right ) \sqrt {\left (a f -b e \right ) b}}\right )\) | \(104\) |
Input:
int(1/(f*x+e)^(1/2)/(a*c+(a*d+b*c)*x+b*d*x^2),x,method=_RETURNVERBOSE)
Output:
2/(a*d-b*c)*(-b/((a*f-b*e)*b)^(1/2)*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^( 1/2))+d/((c*f-d*e)*d)^(1/2)*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2)))
Time = 0.15 (sec) , antiderivative size = 488, normalized size of antiderivative = 4.24 \[ \int \frac {1}{\sqrt {e+f x} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\left [-\frac {\sqrt {\frac {b}{b e - a f}} \log \left (\frac {b f x + 2 \, b e - a f + 2 \, {\left (b e - a f\right )} \sqrt {f x + e} \sqrt {\frac {b}{b e - a f}}}{b x + a}\right ) + \sqrt {\frac {d}{d e - c f}} \log \left (\frac {d f x + 2 \, d e - c f - 2 \, {\left (d e - c f\right )} \sqrt {f x + e} \sqrt {\frac {d}{d e - c f}}}{d x + c}\right )}{b c - a d}, -\frac {2 \, \sqrt {-\frac {d}{d e - c f}} \arctan \left (\sqrt {f x + e} \sqrt {-\frac {d}{d e - c f}}\right ) + \sqrt {\frac {b}{b e - a f}} \log \left (\frac {b f x + 2 \, b e - a f + 2 \, {\left (b e - a f\right )} \sqrt {f x + e} \sqrt {\frac {b}{b e - a f}}}{b x + a}\right )}{b c - a d}, \frac {2 \, \sqrt {-\frac {b}{b e - a f}} \arctan \left (\sqrt {f x + e} \sqrt {-\frac {b}{b e - a f}}\right ) - \sqrt {\frac {d}{d e - c f}} \log \left (\frac {d f x + 2 \, d e - c f - 2 \, {\left (d e - c f\right )} \sqrt {f x + e} \sqrt {\frac {d}{d e - c f}}}{d x + c}\right )}{b c - a d}, \frac {2 \, {\left (\sqrt {-\frac {b}{b e - a f}} \arctan \left (\sqrt {f x + e} \sqrt {-\frac {b}{b e - a f}}\right ) - \sqrt {-\frac {d}{d e - c f}} \arctan \left (\sqrt {f x + e} \sqrt {-\frac {d}{d e - c f}}\right )\right )}}{b c - a d}\right ] \] Input:
integrate(1/(f*x+e)^(1/2)/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="fricas")
Output:
[-(sqrt(b/(b*e - a*f))*log((b*f*x + 2*b*e - a*f + 2*(b*e - a*f)*sqrt(f*x + e)*sqrt(b/(b*e - a*f)))/(b*x + a)) + sqrt(d/(d*e - c*f))*log((d*f*x + 2*d *e - c*f - 2*(d*e - c*f)*sqrt(f*x + e)*sqrt(d/(d*e - c*f)))/(d*x + c)))/(b *c - a*d), -(2*sqrt(-d/(d*e - c*f))*arctan(sqrt(f*x + e)*sqrt(-d/(d*e - c* f))) + sqrt(b/(b*e - a*f))*log((b*f*x + 2*b*e - a*f + 2*(b*e - a*f)*sqrt(f *x + e)*sqrt(b/(b*e - a*f)))/(b*x + a)))/(b*c - a*d), (2*sqrt(-b/(b*e - a* f))*arctan(sqrt(f*x + e)*sqrt(-b/(b*e - a*f))) - sqrt(d/(d*e - c*f))*log(( d*f*x + 2*d*e - c*f - 2*(d*e - c*f)*sqrt(f*x + e)*sqrt(d/(d*e - c*f)))/(d* x + c)))/(b*c - a*d), 2*(sqrt(-b/(b*e - a*f))*arctan(sqrt(f*x + e)*sqrt(-b /(b*e - a*f))) - sqrt(-d/(d*e - c*f))*arctan(sqrt(f*x + e)*sqrt(-d/(d*e - c*f))))/(b*c - a*d)]
Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (97) = 194\).
Time = 5.86 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.20 \[ \int \frac {1}{\sqrt {e+f x} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\begin {cases} \frac {2 \left (\frac {f \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{\sqrt {\frac {c f - d e}{d}} \left (a d - b c\right )} - \frac {f \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {a f - b e}{b}}} \right )}}{\sqrt {\frac {a f - b e}{b}} \left (a d - b c\right )}\right )}{f} & \text {for}\: f \neq 0 \\\frac {- \frac {2 b d \left (\begin {cases} \frac {x + \frac {a d + b c}{2 b d}}{a d} & \text {for}\: b = 0 \\- \frac {x + \frac {a d + b c}{2 b d}}{b c} & \text {for}\: d = 0 \\- \frac {\log {\left (a d - b c - 2 b d \left (x + \frac {a d + b c}{2 b d}\right ) \right )}}{2 b d} & \text {otherwise} \end {cases}\right )}{a d - b c} - \frac {2 b d \left (\begin {cases} \frac {x + \frac {a d + b c}{2 b d}}{a d} & \text {for}\: b = 0 \\- \frac {x + \frac {a d + b c}{2 b d}}{b c} & \text {for}\: d = 0 \\\frac {\log {\left (a d - b c + 2 b d \left (x + \frac {a d + b c}{2 b d}\right ) \right )}}{2 b d} & \text {otherwise} \end {cases}\right )}{a d - b c}}{\sqrt {e}} & \text {otherwise} \end {cases} \] Input:
integrate(1/(f*x+e)**(1/2)/(a*c+(a*d+b*c)*x+b*d*x**2),x)
Output:
Piecewise((2*(f*atan(sqrt(e + f*x)/sqrt((c*f - d*e)/d))/(sqrt((c*f - d*e)/ d)*(a*d - b*c)) - f*atan(sqrt(e + f*x)/sqrt((a*f - b*e)/b))/(sqrt((a*f - b *e)/b)*(a*d - b*c)))/f, Ne(f, 0)), ((-2*b*d*Piecewise(((x + (a*d + b*c)/(2 *b*d))/(a*d), Eq(b, 0)), (-(x + (a*d + b*c)/(2*b*d))/(b*c), Eq(d, 0)), (-l og(a*d - b*c - 2*b*d*(x + (a*d + b*c)/(2*b*d)))/(2*b*d), True))/(a*d - b*c ) - 2*b*d*Piecewise(((x + (a*d + b*c)/(2*b*d))/(a*d), Eq(b, 0)), (-(x + (a *d + b*c)/(2*b*d))/(b*c), Eq(d, 0)), (log(a*d - b*c + 2*b*d*(x + (a*d + b* c)/(2*b*d)))/(2*b*d), True))/(a*d - b*c))/sqrt(e), True))
Exception generated. \[ \int \frac {1}{\sqrt {e+f x} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(f*x+e)^(1/2)/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m ore detail
Time = 0.41 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {e+f x} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {2 \, b \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{\sqrt {-b^{2} e + a b f} {\left (b c - a d\right )}} - \frac {2 \, d \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{\sqrt {-d^{2} e + c d f} {\left (b c - a d\right )}} \] Input:
integrate(1/(f*x+e)^(1/2)/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="giac")
Output:
2*b*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/(sqrt(-b^2*e + a*b*f)*(b* c - a*d)) - 2*d*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/(sqrt(-d^2*e + c*d*f)*(b*c - a*d))
Time = 6.52 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.93 \[ \int \frac {1}{\sqrt {e+f x} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {2\,c\,f\,\mathrm {atanh}\left (\frac {b\,\sqrt {e+f\,x}}{\sqrt {b^2\,e-a\,b\,f}}\right )\,\sqrt {b^2\,e-a\,b\,f}-2\,d\,e\,\mathrm {atanh}\left (\frac {b\,\sqrt {e+f\,x}}{\sqrt {b^2\,e-a\,b\,f}}\right )\,\sqrt {b^2\,e-a\,b\,f}-2\,a\,f\,\mathrm {atanh}\left (\frac {d\,\sqrt {e+f\,x}}{\sqrt {d^2\,e-c\,d\,f}}\right )\,\sqrt {d^2\,e-c\,d\,f}+2\,b\,e\,\mathrm {atanh}\left (\frac {d\,\sqrt {e+f\,x}}{\sqrt {d^2\,e-c\,d\,f}}\right )\,\sqrt {d^2\,e-c\,d\,f}}{-a^2\,c\,d\,f^2+a^2\,d^2\,e\,f+a\,b\,c^2\,f^2-a\,b\,d^2\,e^2-b^2\,c^2\,e\,f+b^2\,c\,d\,e^2} \] Input:
int(1/((e + f*x)^(1/2)*(a*c + x*(a*d + b*c) + b*d*x^2)),x)
Output:
(2*c*f*atanh((b*(e + f*x)^(1/2))/(b^2*e - a*b*f)^(1/2))*(b^2*e - a*b*f)^(1 /2) - 2*d*e*atanh((b*(e + f*x)^(1/2))/(b^2*e - a*b*f)^(1/2))*(b^2*e - a*b* f)^(1/2) - 2*a*f*atanh((d*(e + f*x)^(1/2))/(d^2*e - c*d*f)^(1/2))*(d^2*e - c*d*f)^(1/2) + 2*b*e*atanh((d*(e + f*x)^(1/2))/(d^2*e - c*d*f)^(1/2))*(d^ 2*e - c*d*f)^(1/2))/(a*b*c^2*f^2 - a*b*d^2*e^2 - a^2*c*d*f^2 + b^2*c*d*e^2 + a^2*d^2*e*f - b^2*c^2*e*f)
Time = 0.26 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.89 \[ \int \frac {1}{\sqrt {e+f x} \left (a c+(b c+a d) x+b d x^2\right )} \, dx=\frac {-2 \sqrt {b}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, b}{\sqrt {b}\, \sqrt {a f -b e}}\right ) c f +2 \sqrt {b}\, \sqrt {a f -b e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, b}{\sqrt {b}\, \sqrt {a f -b e}}\right ) d e +2 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) a f -2 \sqrt {d}\, \sqrt {c f -d e}\, \mathit {atan} \left (\frac {\sqrt {f x +e}\, d}{\sqrt {d}\, \sqrt {c f -d e}}\right ) b e}{a^{2} c d \,f^{2}-a^{2} d^{2} e f -a b \,c^{2} f^{2}+a b \,d^{2} e^{2}+b^{2} c^{2} e f -b^{2} c d \,e^{2}} \] Input:
int(1/(f*x+e)^(1/2)/(a*c+(a*d+b*c)*x+b*d*x^2),x)
Output:
(2*( - sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*c*f + sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt( a*f - b*e)))*d*e + sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d) *sqrt(c*f - d*e)))*a*f - sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(s qrt(d)*sqrt(c*f - d*e)))*b*e))/(a**2*c*d*f**2 - a**2*d**2*e*f - a*b*c**2*f **2 + a*b*d**2*e**2 + b**2*c**2*e*f - b**2*c*d*e**2)