Integrand size = 22, antiderivative size = 47 \[ \int \frac {1}{\sqrt {x} \sqrt {x \left (a+b x+c x^2\right )}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {x} (2 a+b x)}{2 \sqrt {a} \sqrt {a x+b x^2+c x^3}}\right )}{\sqrt {a}} \] Output:
-arctanh(1/2*x^(1/2)*(b*x+2*a)/a^(1/2)/(c*x^3+b*x^2+a*x)^(1/2))/a^(1/2)
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.49 \[ \int \frac {1}{\sqrt {x} \sqrt {x \left (a+b x+c x^2\right )}} \, dx=\frac {2 \sqrt {x} \sqrt {a+x (b+c x)} \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {x (a+x (b+c x))}} \] Input:
Integrate[1/(Sqrt[x]*Sqrt[x*(a + b*x + c*x^2)]),x]
Output:
(2*Sqrt[x]*Sqrt[a + x*(b + c*x)]*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x) ])/Sqrt[a]])/(Sqrt[a]*Sqrt[x*(a + x*(b + c*x))])
Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2035, 2093, 1951, 219}
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.
\(\displaystyle \int \frac {1}{\sqrt {x} \sqrt {x \left (a+b x+c x^2\right )}} \, dx\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle 2 \int \frac {1}{\sqrt {x \left (c x^2+b x+a\right )}}d\sqrt {x}\) |
\(\Big \downarrow \) 2093 |
\(\displaystyle 2 \int \frac {1}{\sqrt {c x^3+b x^2+a x}}d\sqrt {x}\) |
\(\Big \downarrow \) 1951 |
\(\displaystyle -2 \int \frac {1}{4 a-x}d\frac {\sqrt {x} (2 a+b x)}{\sqrt {c x^3+b x^2+a x}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {\sqrt {x} (2 a+b x)}{2 \sqrt {a} \sqrt {a x+b x^2+c x^3}}\right )}{\sqrt {a}}\) |
Input:
Int[1/(Sqrt[x]*Sqrt[x*(a + b*x + c*x^2)]),x]
Output:
-(ArcTanh[(Sqrt[x]*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x + b*x^2 + c*x^3])]/Sqr t[a])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(r_.)], x_Symbol] : > Simp[-2/(n - 2) Subst[Int[1/(4*a - x^2), x], x, x*((2*a + b*x^(n - 2))/ Sqrt[a*x^2 + b*x^n + c*x^r])], x] /; FreeQ[{a, b, c, n, r}, x] && EqQ[r, 2* n - 2] && PosQ[n - 2] && NeQ[b^2 - 4*a*c, 0]
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && G eneralizedTrinomialQ[u, x] && !GeneralizedTrinomialMatchQ[u, x]
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.36
method | result | size |
default | \(-\frac {\sqrt {x}\, \sqrt {c \,x^{2}+b x +a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {x \left (c \,x^{2}+b x +a \right )}\, \sqrt {a}}\) | \(64\) |
Input:
int(1/x^(1/2)/(x*(c*x^2+b*x+a))^(1/2),x,method=_RETURNVERBOSE)
Output:
-x^(1/2)/(x*(c*x^2+b*x+a))^(1/2)*(c*x^2+b*x+a)^(1/2)/a^(1/2)*ln((2*a+b*x+2 *a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)
Time = 0.14 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.79 \[ \int \frac {1}{\sqrt {x} \sqrt {x \left (a+b x+c x^2\right )}} \, dx=\left [\frac {\log \left (\frac {8 \, a b x^{2} + {\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{3} + b x^{2} + a x} {\left (b x + 2 \, a\right )} \sqrt {a} \sqrt {x}}{x^{3}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {c x^{3} + b x^{2} + a x} {\left (b x + 2 \, a\right )} \sqrt {-a} \sqrt {x}}{2 \, {\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right )}{a}\right ] \] Input:
integrate(1/x^(1/2)/(x*(c*x^2+b*x+a))^(1/2),x, algorithm="fricas")
Output:
[1/2*log((8*a*b*x^2 + (b^2 + 4*a*c)*x^3 + 8*a^2*x - 4*sqrt(c*x^3 + b*x^2 + a*x)*(b*x + 2*a)*sqrt(a)*sqrt(x))/x^3)/sqrt(a), sqrt(-a)*arctan(1/2*sqrt( c*x^3 + b*x^2 + a*x)*(b*x + 2*a)*sqrt(-a)*sqrt(x)/(a*c*x^3 + a*b*x^2 + a^2 *x))/a]
Timed out. \[ \int \frac {1}{\sqrt {x} \sqrt {x \left (a+b x+c x^2\right )}} \, dx=\text {Timed out} \] Input:
integrate(1/x**(1/2)/(x*(c*x**2+b*x+a))**(1/2),x)
Output:
Timed out
\[ \int \frac {1}{\sqrt {x} \sqrt {x \left (a+b x+c x^2\right )}} \, dx=\int { \frac {1}{\sqrt {{\left (c x^{2} + b x + a\right )} x} \sqrt {x}} \,d x } \] Input:
integrate(1/x^(1/2)/(x*(c*x^2+b*x+a))^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt((c*x^2 + b*x + a)*x)*sqrt(x)), x)
Time = 0.15 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {x} \sqrt {x \left (a+b x+c x^2\right )}} \, dx=\frac {2 \, \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} \] Input:
integrate(1/x^(1/2)/(x*(c*x^2+b*x+a))^(1/2),x, algorithm="giac")
Output:
2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/sqrt(-a)
Timed out. \[ \int \frac {1}{\sqrt {x} \sqrt {x \left (a+b x+c x^2\right )}} \, dx=\int \frac {1}{\sqrt {x}\,\sqrt {x\,\left (c\,x^2+b\,x+a\right )}} \,d x \] Input:
int(1/(x^(1/2)*(x*(a + b*x + c*x^2))^(1/2)),x)
Output:
int(1/(x^(1/2)*(x*(a + b*x + c*x^2))^(1/2)), x)
Time = 0.24 (sec) , antiderivative size = 693, normalized size of antiderivative = 14.74 \[ \int \frac {1}{\sqrt {x} \sqrt {x \left (a+b x+c x^2\right )}} \, dx =\text {Too large to display} \] Input:
int(1/x^(1/2)/(x*(c*x^2+b*x+a))^(1/2),x)
Output:
( - 2*sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)*atan((2*sqrt(c)*sqr t(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)) *b - 4*sqrt(c)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)*atan((2*sqrt(c)*sq rt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2) )*a - sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2)*log( - sqrt(4*sqrt( c)*sqrt(a)*b + 4*a*c + b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c* x)*b + sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2)*log(sqrt(4*sqrt(c) *sqrt(a)*b + 4*a*c + b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x) *b + 2*sqrt(c)*sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2)*log( - sqrt(4*sqrt (c)*sqrt(a)*b + 4*a*c + b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c *x)*a - 2*sqrt(c)*sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2)*log(sqrt(4*sqrt (c)*sqrt(a)*b + 4*a*c + b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c *x)*a + 4*sqrt(a)*log( - sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2) + 2*sqrt (c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)*a*c - sqrt(a)*log( - sqrt(4*sqrt(c )*sqrt(a)*b + 4*a*c + b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x )*b**2 + 4*sqrt(a)*log(sqrt(4*sqrt(c)*sqrt(a)*b + 4*a*c + b**2) + 2*sqrt(c )*sqrt(a + b*x + c*x**2) + b + 2*c*x)*a*c - sqrt(a)*log(sqrt(4*sqrt(c)*sqr t(a)*b + 4*a*c + b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)*b** 2 - 4*sqrt(a)*log(4*sqrt(c)*sqrt(a + b*x + c*x**2)*b + 8*sqrt(c)*sqrt(a + b*x + c*x**2)*c*x + 4*sqrt(c)*sqrt(a)*b + 8*b*c*x + 8*c**2*x**2)*a*c + ...