Integrand size = 28, antiderivative size = 240 \[ \int \frac {\sqrt {f+g x}}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx=\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} d-\sqrt {-a} e}}-\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {-a} e}} \]
arctanh((e*x+d)^(1/2)*(-g*(-a)^(1/2)+f*c^(1/2))^(1/2)/(g*x+f)^(1/2)/(-e*(- a)^(1/2)+d*c^(1/2))^(1/2))*(-g*(-a)^(1/2)+f*c^(1/2))^(1/2)/(-a)^(1/2)/c^(1 /2)/(-e*(-a)^(1/2)+d*c^(1/2))^(1/2)-arctanh((e*x+d)^(1/2)*(g*(-a)^(1/2)+f* c^(1/2))^(1/2)/(g*x+f)^(1/2)/(e*(-a)^(1/2)+d*c^(1/2))^(1/2))*(g*(-a)^(1/2) +f*c^(1/2))^(1/2)/(-a)^(1/2)/c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2))^(1/2)
Time = 10.30 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {f+g x}}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx=\frac {\frac {\sqrt {-\sqrt {c} f+\sqrt {-a} g} \text {arctanh}\left (\frac {\sqrt {-\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {-\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-\sqrt {c} d+\sqrt {-a} e}}-\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {\sqrt {c} d+\sqrt {-a} e}}}{\sqrt {-a} \sqrt {c}} \]
((Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g]*ArcTanh[(Sqrt[-(Sqrt[c]*f) + Sqrt[-a]*g] *Sqrt[d + e*x])/(Sqrt[-(Sqrt[c]*d) + Sqrt[-a]*e]*Sqrt[f + g*x])])/Sqrt[-(S qrt[c]*d) + Sqrt[-a]*e] - (Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*ArcTanh[(Sqrt[Sqrt [c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e])/(Sqrt[-a]*Sqrt[c])
Time = 0.47 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {661, 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.
\(\displaystyle \int \frac {\sqrt {f+g x}}{\left (a+c x^2\right ) \sqrt {d+e x}} \, dx\) |
\(\Big \downarrow \) 661 |
\(\displaystyle \int \left (\frac {\sqrt {-a} f-\frac {a g}{\sqrt {c}}}{2 a \left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}}+\frac {\frac {a g}{\sqrt {c}}+\sqrt {-a} f}{2 a \left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {c} f-\sqrt {-a} g}}{\sqrt {f+g x} \sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {c} d-\sqrt {-a} e}}-\frac {\sqrt {\sqrt {-a} g+\sqrt {c} f} \text {arctanh}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {-a} g+\sqrt {c} f}}{\sqrt {f+g x} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {-a} \sqrt {c} \sqrt {\sqrt {-a} e+\sqrt {c} d}}\) |
(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[c]* Sqrt[Sqrt[c]*d - Sqrt[-a]*e]) - (Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*ArcTanh[(Sqr t[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqr t[f + g*x])])/(Sqrt[-a]*Sqrt[c]*Sqrt[Sqrt[c]*d + Sqrt[-a]*e])
3.7.7.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*((a_.) + (c_.)*(x_) ^2)), x_Symbol] :> Int[ExpandIntegrand[1/(Sqrt[d + e*x]*Sqrt[f + g*x]), (d + e*x)^(m + 1/2)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && IGt Q[m + 1/2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1386\) vs. \(2(176)=352\).
Time = 0.40 (sec) , antiderivative size = 1387, normalized size of antiderivative = 5.78
1/2*(g*x+f)^(1/2)*(e*x+d)^(1/2)*(ln((-2*(-a*c)^(1/2)*e*g*x+c*d*g*x+c*e*f*x +2*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)*((g*x+f)*(e* x+d))^(1/2)*c-(-a*c)^(1/2)*d*g-(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x+(-a*c)^(1/2) ))*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*a*c*e^2*f-ln( (-2*(-a*c)^(1/2)*e*g*x+c*d*g*x+c*e*f*x+2*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)* e*f+a*e*g-c*d*f)/c)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*c-(-a*c)^(1/2)*d*g-(-a*c )^(1/2)*e*f+2*c*d*f)/(c*x+(-a*c)^(1/2)))*(-a*c)^(1/2)*(((-a*c)^(1/2)*d*g+( -a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*a*e^2*g+ln((-2*(-a*c)^(1/2)*e*g*x+c* d*g*x+c*e*f*x+2*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2) *((g*x+f)*(e*x+d))^(1/2)*c-(-a*c)^(1/2)*d*g-(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x +(-a*c)^(1/2)))*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)* c^2*d^2*f-ln((-2*(-a*c)^(1/2)*e*g*x+c*d*g*x+c*e*f*x+2*(-((-a*c)^(1/2)*d*g+ (-a*c)^(1/2)*e*f+a*e*g-c*d*f)/c)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*c-(-a*c)^(1 /2)*d*g-(-a*c)^(1/2)*e*f+2*c*d*f)/(c*x+(-a*c)^(1/2)))*(-a*c)^(1/2)*(((-a*c )^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*c*d^2*g-ln((2*(-a*c)^(1 /2)*e*g*x+c*d*g*x+c*e*f*x+2*(((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d* f)/c)^(1/2)*((g*x+f)*(e*x+d))^(1/2)*c+(-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+2* c*d*f)/(c*x-(-a*c)^(1/2)))*(-((-a*c)^(1/2)*d*g+(-a*c)^(1/2)*e*f+a*e*g-c*d* f)/c)^(1/2)*a*c*e^2*f-ln((2*(-a*c)^(1/2)*e*g*x+c*d*g*x+c*e*f*x+2*(((-a*c)^ (1/2)*d*g+(-a*c)^(1/2)*e*f-a*e*g+c*d*f)/c)^(1/2)*((g*x+f)*(e*x+d))^(1/2...
Leaf count of result is larger than twice the leaf count of optimal. 1921 vs. \(2 (176) = 352\).
Time = 6.81 (sec) , antiderivative size = 1921, normalized size of antiderivative = 8.00 \[ \int \frac {\sqrt {f+g x}}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx=\text {Too large to display} \]
-1/4*sqrt(-(c*d*f + a*e*g + (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(e^2*f^2 - 2*d*e *f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2))*log(-(e^2*f^2 - d^2*g^2 + 2*(c*d*e*f - c*d^2*g - (a*c^2*d^2*e + a^2*c*e^3)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2 *d^2*e^2 + a^3*c*e^4)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(-(c*d*f + a*e*g + (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2)) + 2*(e^2*f*g - d*e*g^2)*x + (2*(c^2*d^3 + a*c*d*e^2)*f + ((c^2*d^2*e + a*c*e^3)*f + (c^2 *d^3 + a*c*d*e^2)*g)*x)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/x) + 1/4*sqrt(-(c*d*f + a*e*g + (a*c^2*d ^2 + a^2*c*e^2)*sqrt(-(e^2*f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c ^2*d^2*e^2 + a^3*c*e^4)))/(a*c^2*d^2 + a^2*c*e^2))*log(-(e^2*f^2 - d^2*g^2 - 2*(c*d*e*f - c*d^2*g - (a*c^2*d^2*e + a^2*c*e^3)*sqrt(-(e^2*f^2 - 2*d*e *f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))*sqrt(e*x + d )*sqrt(g*x + f)*sqrt(-(c*d*f + a*e*g + (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(e^2* f^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/( a*c^2*d^2 + a^2*c*e^2)) + 2*(e^2*f*g - d*e*g^2)*x + (2*(c^2*d^3 + a*c*d*e^ 2)*f + ((c^2*d^2*e + a*c*e^3)*f + (c^2*d^3 + a*c*d*e^2)*g)*x)*sqrt(-(e^2*f ^2 - 2*d*e*f*g + d^2*g^2)/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/x) - 1/4*sqrt(-(c*d*f + a*e*g - (a*c^2*d^2 + a^2*c*e^2)*sqrt(-(e^2*f^2 - ...
\[ \int \frac {\sqrt {f+g x}}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx=\int \frac {\sqrt {f + g x}}{\left (a + c x^{2}\right ) \sqrt {d + e x}}\, dx \]
\[ \int \frac {\sqrt {f+g x}}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx=\int { \frac {\sqrt {g x + f}}{{\left (c x^{2} + a\right )} \sqrt {e x + d}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {f+g x}}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt {f+g x}}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx=\text {Hanged} \]