Integrand size = 24, antiderivative size = 49 \[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=-\frac {\text {arctanh}\left (\frac {x^{3/2} (2 a+b x)}{2 \sqrt {a} \sqrt {a x^3+b x^4+c x^5}}\right )}{\sqrt {a}} \] Output:
-arctanh(1/2*x^(3/2)*(b*x+2*a)/a^(1/2)/(c*x^5+b*x^4+a*x^3)^(1/2))/a^(1/2)
Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.47 \[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=\frac {2 x^{3/2} \sqrt {a+x (b+c x)} \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {x^3 (a+x (b+c x))}} \] Input:
Integrate[Sqrt[x]/Sqrt[x^3*(a + b*x + c*x^2)],x]
Output:
(2*x^(3/2)*Sqrt[a + x*(b + c*x)]*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x) ])/Sqrt[a]])/(Sqrt[a]*Sqrt[x^3*(a + x*(b + c*x))])
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2035, 2094, 1960, 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.
\(\displaystyle \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle 2 \int \frac {x}{\sqrt {x^3 \left (c x^2+b x+a\right )}}d\sqrt {x}\) |
\(\Big \downarrow \) 2094 |
\(\displaystyle 2 \int \frac {x}{\sqrt {c x^5+b x^4+a x^3}}d\sqrt {x}\) |
\(\Big \downarrow \) 1960 |
\(\displaystyle -2 \int \frac {1}{4 a-x}d\frac {x^{3/2} (2 a+b x)}{\sqrt {c x^5+b x^4+a x^3}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {x^{3/2} (2 a+b x)}{2 \sqrt {a} \sqrt {a x^3+b x^4+c x^5}}\right )}{\sqrt {a}}\) |
Input:
Int[Sqrt[x]/Sqrt[x^3*(a + b*x + c*x^2)],x]
Output:
-(ArcTanh[(x^(3/2)*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x^3 + b*x^4 + c*x^5])]/S qrt[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[(x_)^(m_.)/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)] , x_Symbol] :> Simp[-2/(n - q) Subst[Int[1/(4*a - x^2), x], x, x^(m + 1)* ((2*a + b*x^(n - q))/Sqrt[a*x^q + b*x^n + c*x^r])], x] /; FreeQ[{a, b, c, m , n, q, r}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && NeQ[b^2 - 4*a*c, 0] && EqQ[m, q/2 - 1]
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_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[(d*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{d, m, p}, x] && GeneralizedTrinomialQ[u, x] && !General izedTrinomialMatchQ[u, x]
Time = 0.10 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.35
method | result | size |
default | \(-\frac {x^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{\sqrt {x^{3} \left (c \,x^{2}+b x +a \right )}\, \sqrt {a}}\) | \(66\) |
Input:
int(x^(1/2)/(x^3*(c*x^2+b*x+a))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/(x^3*(c*x^2+b*x+a))^(1/2)*x^(3/2)*(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 = 0.18 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.84 \[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=\left [\frac {\log \left (\frac {8 \, a b x^{3} + {\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a^{2} x^{2} - 4 \, \sqrt {c x^{5} + b x^{4} + a x^{3}} {\left (b x + 2 \, a\right )} \sqrt {a} \sqrt {x}}{x^{4}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {c x^{5} + b x^{4} + a x^{3}} {\left (b x + 2 \, a\right )} \sqrt {-a} \sqrt {x}}{2 \, {\left (a c x^{4} + a b x^{3} + a^{2} x^{2}\right )}}\right )}{a}\right ] \] Input:
integrate(x^(1/2)/(x^3*(c*x^2+b*x+a))^(1/2),x, algorithm="fricas")
Output:
[1/2*log((8*a*b*x^3 + (b^2 + 4*a*c)*x^4 + 8*a^2*x^2 - 4*sqrt(c*x^5 + b*x^4 + a*x^3)*(b*x + 2*a)*sqrt(a)*sqrt(x))/x^4)/sqrt(a), sqrt(-a)*arctan(1/2*s qrt(c*x^5 + b*x^4 + a*x^3)*(b*x + 2*a)*sqrt(-a)*sqrt(x)/(a*c*x^4 + a*b*x^3 + a^2*x^2))/a]
Timed out. \[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=\text {Timed out} \] Input:
integrate(x**(1/2)/(x**3*(c*x**2+b*x+a))**(1/2),x)
Output:
Timed out
\[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=\int { \frac {\sqrt {x}}{\sqrt {{\left (c x^{2} + b x + a\right )} x^{3}}} \,d x } \] Input:
integrate(x^(1/2)/(x^3*(c*x^2+b*x+a))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x)/sqrt((c*x^2 + b*x + a)*x^3), x)
Time = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.20 \[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=-\frac {2 \, \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-a}} + \frac {2 \, \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} \mathrm {sgn}\left (x\right )} \] Input:
integrate(x^(1/2)/(x^3*(c*x^2+b*x+a))^(1/2),x, algorithm="giac")
Output:
-2*arctan(sqrt(a)/sqrt(-a))*sgn(x)/sqrt(-a) + 2*arctan(-(sqrt(c)*x - sqrt( c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*sgn(x))
Timed out. \[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx=\int \frac {\sqrt {x}}{\sqrt {x^3\,\left (c\,x^2+b\,x+a\right )}} \,d x \] Input:
int(x^(1/2)/(x^3*(a + b*x + c*x^2))^(1/2),x)
Output:
int(x^(1/2)/(x^3*(a + b*x + c*x^2))^(1/2), x)
Time = 0.22 (sec) , antiderivative size = 693, normalized size of antiderivative = 14.14 \[ \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x+c x^2\right )}} \, dx =\text {Too large to display} \] Input:
int(x^(1/2)/(x^3*(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 + ...