Integrand size = 35, antiderivative size = 393 \[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {2 d \sqrt {h} \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{(b c-a d) (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}-\frac {2 b \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d)^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \]
2*d^2*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)/(-c*f+d*e)/(-c*h+d*g)/(d*x+c) ^(1/2)-2*b*EllipticPi(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),-b*(-c*f+d*e)/ (-a*d+b*c)/f,((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*(c*f-d*e)^(1/2)*(d*(f*x+e) /(-c*f+d*e))^(1/2)*(d*(h*x+g)/(-c*h+d*g))^(1/2)/(-a*d+b*c)^2/f^(1/2)/(f*x+ e)^(1/2)/(h*x+g)^(1/2)-2*d*EllipticE(h^(1/2)*(f*x+e)^(1/2)/(e*h-f*g)^(1/2) ,(-d*(-e*h+f*g)/(-c*f+d*e)/h)^(1/2))*h^(1/2)*(e*h-f*g)^(1/2)*(d*x+c)^(1/2) *(f*(h*x+g)/(-e*h+f*g))^(1/2)/(-a*d+b*c)/(-c*f+d*e)/(-c*h+d*g)/(-f*(d*x+c) /(-c*f+d*e))^(1/2)/(h*x+g)^(1/2)
Result contains complex when optimal does not.
Time = 23.22 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 i (c+d x) \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \left ((b c-a d) h E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )+(b d g-2 b c h+a d h) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )+b (-d g+c h) \operatorname {EllipticPi}\left (-\frac {b c f-a d f}{b d e-b c f},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )\right )}{(b c-a d)^2 \sqrt {-c+\frac {d e}{f}} (-d g+c h) \sqrt {e+f x} \sqrt {g+h x}} \]
((2*I)*(c + d*x)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*( c + d*x))]*((b*c - a*d)*h*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)] + (b*d*g - 2*b*c*h + a*d*h)*Ellipt icF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)] + b*(-(d*g) + c*h)*EllipticPi[-((b*c*f - a*d*f)/(b*d*e - b*c*f)), I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f* h)]))/((b*c - a*d)^2*Sqrt[-c + (d*e)/f]*(-(d*g) + c*h)*Sqrt[e + f*x]*Sqrt[ g + h*x])
Time = 0.74 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {197, 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.
| \(\displaystyle \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) |
| \(\Big \downarrow \) 197 |
| \(\displaystyle \int \left (\frac {b}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)}-\frac {d}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 d \sqrt {h} \sqrt {c+d x} \sqrt {e h-f g} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{\sqrt {g+h x} (b c-a d) (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}}}-\frac {2 b \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {f} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)^2}+\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{\sqrt {c+d x} (b c-a d) (d e-c f) (d g-c h)}\) |
(2*d^2*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)*(d*e - c*f)*(d*g - c*h)*S qrt[c + d*x]) - (2*d*Sqrt[h]*Sqrt[-(f*g) + e*h]*Sqrt[c + d*x]*Sqrt[(f*(g + h*x))/(f*g - e*h)]*EllipticE[ArcSin[(Sqrt[h]*Sqrt[e + f*x])/Sqrt[-(f*g) + e*h]], -((d*(f*g - e*h))/((d*e - c*f)*h))])/((b*c - a*d)*(d*e - c*f)*(d*g - c*h)*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*Sqrt[g + h*x]) - (2*b*Sqrt[-(d* e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]* EllipticPi[-((b*(d*e - c*f))/((b*c - a*d)*f)), ArcSin[(Sqrt[f]*Sqrt[c + d* x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/((b*c - a*d)^2* Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x])
3.1.71.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_))/(Sqrt[(e_.) + (f _.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Int[ExpandIntegrand[1/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), (a + b*x)^m*(c + d*x)^(n + 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IntegerQ[m] && IntegerQ[n + 1/2]
Leaf count of result is larger than twice the leaf count of optimal. \(975\) vs. \(2(353)=706\).
Time = 2.06 (sec) , antiderivative size = 976, normalized size of antiderivative = 2.48
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (-\frac {2 \left (d f h \,x^{2}+d e h x +d f g x +d e g \right ) d}{\left (c^{2} f h -c d e h -c d f g +d^{2} e g \right ) \left (a d -b c \right ) \sqrt {\left (x +\frac {c}{d}\right ) \left (d f h \,x^{2}+d e h x +d f g x +d e g \right )}}+\frac {2 \left (\frac {d \left (c f h -d e h -d f g \right )}{\left (c^{2} f h -c d e h -c d f g +d^{2} e g \right ) \left (a d -b c \right )}+\frac {\left (d e h +d f g \right ) d}{\left (c^{2} f h -c d e h -c d f g +d^{2} e g \right ) \left (a d -b c \right )}\right ) \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}+\frac {2 d^{2} f h \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \left (\left (-\frac {g}{h}+\frac {c}{d}\right ) E\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )-\frac {c F\left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{d}\right )}{\left (c^{2} f h -c d e h -c d f g +d^{2} e g \right ) \left (a d -b c \right ) \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}}-\frac {2 \left (\frac {g}{h}-\frac {e}{f}\right ) \sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {g}{h}+\frac {c}{d}}}\, \sqrt {\frac {x +\frac {e}{f}}{-\frac {g}{h}+\frac {e}{f}}}\, \Pi \left (\sqrt {\frac {x +\frac {g}{h}}{\frac {g}{h}-\frac {e}{f}}}, \frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {a}{b}}, \sqrt {\frac {-\frac {g}{h}+\frac {e}{f}}{-\frac {g}{h}+\frac {c}{d}}}\right )}{\left (a d -b c \right ) \sqrt {d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g}\, \left (-\frac {g}{h}+\frac {a}{b}\right )}\right )}{\sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}\) | \(976\) |
| default | \(\text {Expression too large to display}\) | \(1978\) |
((d*x+c)*(f*x+e)*(h*x+g))^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)* (-2*(d*f*h*x^2+d*e*h*x+d*f*g*x+d*e*g)/(c^2*f*h-c*d*e*h-c*d*f*g+d^2*e*g)*d/ (a*d-b*c)/((x+c/d)*(d*f*h*x^2+d*e*h*x+d*f*g*x+d*e*g))^(1/2)+2*(1/(c^2*f*h- c*d*e*h-c*d*f*g+d^2*e*g)*d*(c*f*h-d*e*h-d*f*g)/(a*d-b*c)+(d*e*h+d*f*g)/(c^ 2*f*h-c*d*e*h-c*d*f*g+d^2*e*g)*d/(a*d-b*c))*(g/h-e/f)*((x+g/h)/(g/h-e/f))^ (1/2)*((x+c/d)/(-g/h+c/d))^(1/2)*((x+e/f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f *h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)*EllipticF( ((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2))+2/(c^2*f*h-c*d*e* h-c*d*f*g+d^2*e*g)*d^2*f*h/(a*d-b*c)*(g/h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*( (x+c/d)/(-g/h+c/d))^(1/2)*((x+e/f)/(-g/h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+ d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e*g*x+c*e*g)^(1/2)*((-g/h+c/d)*Ellip ticE(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2))-c/d*Elliptic F(((x+g/h)/(g/h-e/f))^(1/2),((-g/h+e/f)/(-g/h+c/d))^(1/2)))-2/(a*d-b*c)*(g /h-e/f)*((x+g/h)/(g/h-e/f))^(1/2)*((x+c/d)/(-g/h+c/d))^(1/2)*((x+e/f)/(-g/ h+e/f))^(1/2)/(d*f*h*x^3+c*f*h*x^2+d*e*h*x^2+d*f*g*x^2+c*e*h*x+c*f*g*x+d*e *g*x+c*e*g)^(1/2)/(-g/h+a/b)*EllipticPi(((x+g/h)/(g/h-e/f))^(1/2),(-g/h+e/ f)/(-g/h+a/b),((-g/h+e/f)/(-g/h+c/d))^(1/2)))
Timed out. \[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {1}{\left (a + b x\right ) \left (c + d x\right )^{\frac {3}{2}} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]
\[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {1}{{\left (b x + a\right )} {\left (d x + c\right )}^{\frac {3}{2}} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
\[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {1}{{\left (b x + a\right )} {\left (d x + c\right )}^{\frac {3}{2}} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {1}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^{3/2}} \,d x \]