Integrand size = 23, antiderivative size = 87 \[ \int \frac {\sqrt {x (a+b x)}}{c x+d x^2} \, dx=\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{d}-\frac {2 \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{\sqrt {c} d} \] Output:
2*b^(1/2)*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/d-2*(-a*d+b*c)^(1/2)*arctan h((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a*x)^(1/2))/c^(1/2)/d
Time = 0.00 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {x (a+b x)}}{c x+d x^2} \, dx=-\frac {2 \sqrt {x} \sqrt {a+b x} \left (\sqrt {-b c+a d} \arctan \left (\frac {-d \sqrt {x} \sqrt {a+b x}+\sqrt {b} (c+d x)}{\sqrt {c} \sqrt {-b c+a d}}\right )+\sqrt {b} \sqrt {c} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )\right )}{\sqrt {c} d \sqrt {x (a+b x)}} \] Input:
Integrate[Sqrt[x*(a + b*x)]/(c*x + d*x^2),x]
Output:
(-2*Sqrt[x]*Sqrt[a + b*x]*(Sqrt[-(b*c) + a*d]*ArcTan[(-(d*Sqrt[x]*Sqrt[a + b*x]) + Sqrt[b]*(c + d*x))/(Sqrt[c]*Sqrt[-(b*c) + a*d])] + Sqrt[b]*Sqrt[c ]*Log[-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x]]))/(Sqrt[c]*d*Sqrt[x*(a + b*x)])
Time = 0.70 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.68, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2026, 2048, 1260, 2009}
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.
\(\displaystyle \int \frac {\sqrt {x (a+b x)}}{c x+d x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\sqrt {x (a+b x)}}{x (c+d x)}dx\) |
\(\Big \downarrow \) 2048 |
\(\displaystyle \int \frac {\sqrt {a x+b x^2}}{x (c+d x)}dx\) |
\(\Big \downarrow \) 1260 |
\(\displaystyle \int \left (\frac {\sqrt {a x+b x^2}}{c x}-\frac {d \sqrt {a x+b x^2}}{c (c+d x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (2 b c-a d)}{\sqrt {b} c d}-\frac {\sqrt {b c-a d} \text {arctanh}\left (\frac {x (2 b c-a d)+a c}{2 \sqrt {c} \sqrt {a x+b x^2} \sqrt {b c-a d}}\right )}{\sqrt {c} d}+\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{\sqrt {b} c}\) |
Input:
Int[Sqrt[x*(a + b*x)]/(c*x + d*x^2),x]
Output:
(a*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(Sqrt[b]*c) + ((2*b*c - a*d)*Ar cTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(Sqrt[b]*c*d) - (Sqrt[b*c - a*d]*Arc Tanh[(a*c + (2*b*c - a*d)*x)/(2*Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[a*x + b*x^2]) ])/(Sqrt[c]*d)
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x + c*x^2)^p , (d + e*x)^m*(f + g*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ [c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + n + 2*p + 1, 0] && ILtQ[m, 0] && ILtQ [n, 0]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))*((c_) + (d_.)*(x_)^(n_.)))^(p_) , x_Symbol] :> Int[u*(a*c*e + (b*c + a*d)*e*x^n + b*d*e*x^(2*n))^p, x] /; F reeQ[{a, b, c, d, e, n, p}, x]
Time = 0.54 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89
method | result | size |
pseudoelliptic | \(-\frac {2 \left (-\sqrt {b}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )+\frac {\left (a d -b c \right ) \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )}{\sqrt {c \left (a d -b c \right )}}\right )}{d}\) | \(77\) |
default | \(\frac {\sqrt {b \,x^{2}+a x}+\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 \sqrt {b}}}{c}-\frac {\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}+\frac {\left (a d -2 b c \right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}\right )}{2 d \sqrt {b}}+\frac {c \left (a d -b c \right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}}{c}\) | \(332\) |
Input:
int((x*(b*x+a))^(1/2)/(d*x^2+c*x),x,method=_RETURNVERBOSE)
Output:
-2/d*(-b^(1/2)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))+(a*d-b*c)/(c*(a*d-b*c) )^(1/2)*arctan((x*(b*x+a))^(1/2)/x*c/(c*(a*d-b*c))^(1/2)))
Time = 0.12 (sec) , antiderivative size = 396, normalized size of antiderivative = 4.55 \[ \int \frac {\sqrt {x (a+b x)}}{c x+d x^2} \, dx=\left [\frac {\sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + \sqrt {\frac {b c - a d}{c}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x - 2 \, \sqrt {b x^{2} + a x} c \sqrt {\frac {b c - a d}{c}}}{d x + c}\right )}{d}, -\frac {2 \, \sqrt {-\frac {b c - a d}{c}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} c \sqrt {-\frac {b c - a d}{c}}}{{\left (b c - a d\right )} x}\right ) - \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{d}, -\frac {2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) - \sqrt {\frac {b c - a d}{c}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x - 2 \, \sqrt {b x^{2} + a x} c \sqrt {\frac {b c - a d}{c}}}{d x + c}\right )}{d}, -\frac {2 \, {\left (\sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + \sqrt {-\frac {b c - a d}{c}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} c \sqrt {-\frac {b c - a d}{c}}}{{\left (b c - a d\right )} x}\right )\right )}}{d}\right ] \] Input:
integrate((x*(b*x+a))^(1/2)/(d*x^2+c*x),x, algorithm="fricas")
Output:
[(sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) + sqrt((b*c - a*d)/ c)*log((a*c + (2*b*c - a*d)*x - 2*sqrt(b*x^2 + a*x)*c*sqrt((b*c - a*d)/c)) /(d*x + c)))/d, -(2*sqrt(-(b*c - a*d)/c)*arctan(-sqrt(b*x^2 + a*x)*c*sqrt( -(b*c - a*d)/c)/((b*c - a*d)*x)) - sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)))/d, -(2*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a) ) - sqrt((b*c - a*d)/c)*log((a*c + (2*b*c - a*d)*x - 2*sqrt(b*x^2 + a*x)*c *sqrt((b*c - a*d)/c))/(d*x + c)))/d, -2*(sqrt(-b)*arctan(sqrt(b*x^2 + a*x) *sqrt(-b)/(b*x + a)) + sqrt(-(b*c - a*d)/c)*arctan(-sqrt(b*x^2 + a*x)*c*sq rt(-(b*c - a*d)/c)/((b*c - a*d)*x)))/d]
\[ \int \frac {\sqrt {x (a+b x)}}{c x+d x^2} \, dx=\int \frac {\sqrt {x \left (a + b x\right )}}{x \left (c + d x\right )}\, dx \] Input:
integrate((x*(b*x+a))**(1/2)/(d*x**2+c*x),x)
Output:
Integral(sqrt(x*(a + b*x))/(x*(c + d*x)), x)
Exception generated. \[ \int \frac {\sqrt {x (a+b x)}}{c x+d x^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((x*(b*x+a))^(1/2)/(d*x^2+c*x),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(a*d-b*c>0)', see `assume?` for m ore detail
Exception generated. \[ \int \frac {\sqrt {x (a+b x)}}{c x+d x^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((x*(b*x+a))^(1/2)/(d*x^2+c*x),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\sqrt {x (a+b x)}}{c x+d x^2} \, dx=\int \frac {\sqrt {x\,\left (a+b\,x\right )}}{d\,x^2+c\,x} \,d x \] Input:
int((x*(a + b*x))^(1/2)/(c*x + d*x^2),x)
Output:
int((x*(a + b*x))^(1/2)/(c*x + d*x^2), x)
Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt {x (a+b x)}}{c x+d x^2} \, dx=\frac {-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right )-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right )+2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) c}{c d} \] Input:
int((x*(b*x+a))^(1/2)/(d*x^2+c*x),x)
Output:
(2*( - sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b* x) - sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b))) - sqrt(c)*sqrt(a*d - b*c) *atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/ (sqrt(c)*sqrt(b))) + sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a) )*c))/(c*d)