Integrand size = 35, antiderivative size = 875 \[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}+\frac {2 b d^2 \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {4 d^2 (d f g+d e h-2 c f h) \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt {c+d x}}+\frac {4 d \sqrt {f} (d f g+d e h-2 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (-d e+c f)^{3/2} (d g-c h)^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 b d \sqrt {h} \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}-\frac {2 \sqrt {f} (2 d f g+d e h-3 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (-d e+c f)^{3/2} (d g-c h) \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 b^2 \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d)^3 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \]
2/3*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)^(3/2)+2*b*d^2*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)^2/(-c*f+d*e)/(-c*h +d*g)/(d*x+c)^(1/2)-4/3*d^2*(-2*c*f*h+d*e*h+d*f*g)*(f*x+e)^(1/2)*(h*x+g)^( 1/2)/(-a*d+b*c)/(-c*f+d*e)^2/(-c*h+d*g)^2/(d*x+c)^(1/2)+4/3*d*(-2*c*f*h+d* e*h+d*f*g)*EllipticE(f^(1/2)*(d*x+c)^(1/2)/(c*f-d*e)^(1/2),((-c*f+d*e)*h/f /(-c*h+d*g))^(1/2))*f^(1/2)*(d*(f*x+e)/(-c*f+d*e))^(1/2)*(h*x+g)^(1/2)/(-a *d+b*c)/(c*f-d*e)^(3/2)/(-c*h+d*g)^2/(f*x+e)^(1/2)/(d*(h*x+g)/(-c*h+d*g))^ (1/2)-2/3*(-3*c*f*h+d*e*h+2*d*f*g)*EllipticF(f^(1/2)*(d*x+c)^(1/2)/(c*f-d* e)^(1/2),((-c*f+d*e)*h/f/(-c*h+d*g))^(1/2))*f^(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)/(c*f-d*e)^(3/2)/(-c*h+d*g)/ (f*x+e)^(1/2)/(h*x+g)^(1/2)-2*b^2*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)^3/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)-2*b*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)^2/(-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 = 35.51 (sec) , antiderivative size = 4180, normalized size of antiderivative = 4.78 \[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]
Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]*((2*d^2)/(3*(b*c - a*d)*(-(d*e) + c*f)*(-(d*g) + c*h)*(c + d*x)^2) + (2*d^2*(3*b*d^2*e*g - 5*b*c*d*f*g + 2 *a*d^2*f*g - 5*b*c*d*e*h + 2*a*d^2*e*h + 7*b*c^2*f*h - 4*a*c*d*f*h))/(3*(b *c - a*d)^2*(-(d*e) + c*f)^2*(-(d*g) + c*h)^2*(c + d*x))) + (2*(c + d*x)^( 3/2)*(-3*b^2*c*d^2*e*Sqrt[-c + (d*e)/f]*f*g*h + 3*a*b*d^3*e*Sqrt[-c + (d*e )/f]*f*g*h + 5*b^2*c^2*d*Sqrt[-c + (d*e)/f]*f^2*g*h - 7*a*b*c*d^2*Sqrt[-c + (d*e)/f]*f^2*g*h + 2*a^2*d^3*Sqrt[-c + (d*e)/f]*f^2*g*h + 5*b^2*c^2*d*e* Sqrt[-c + (d*e)/f]*f*h^2 - 7*a*b*c*d^2*e*Sqrt[-c + (d*e)/f]*f*h^2 + 2*a^2* d^3*e*Sqrt[-c + (d*e)/f]*f*h^2 - 7*b^2*c^3*Sqrt[-c + (d*e)/f]*f^2*h^2 + 11 *a*b*c^2*d*Sqrt[-c + (d*e)/f]*f^2*h^2 - 4*a^2*c*d^2*Sqrt[-c + (d*e)/f]*f^2 *h^2 - (3*b^2*c*d^4*e^2*Sqrt[-c + (d*e)/f]*g^2)/(c + d*x)^2 + (3*a*b*d^5*e ^2*Sqrt[-c + (d*e)/f]*g^2)/(c + d*x)^2 + (8*b^2*c^2*d^3*e*Sqrt[-c + (d*e)/ f]*f*g^2)/(c + d*x)^2 - (10*a*b*c*d^4*e*Sqrt[-c + (d*e)/f]*f*g^2)/(c + d*x )^2 + (2*a^2*d^5*e*Sqrt[-c + (d*e)/f]*f*g^2)/(c + d*x)^2 - (5*b^2*c^3*d^2* Sqrt[-c + (d*e)/f]*f^2*g^2)/(c + d*x)^2 + (7*a*b*c^2*d^3*Sqrt[-c + (d*e)/f ]*f^2*g^2)/(c + d*x)^2 - (2*a^2*c*d^4*Sqrt[-c + (d*e)/f]*f^2*g^2)/(c + d*x )^2 + (8*b^2*c^2*d^3*e^2*Sqrt[-c + (d*e)/f]*g*h)/(c + d*x)^2 - (10*a*b*c*d ^4*e^2*Sqrt[-c + (d*e)/f]*g*h)/(c + d*x)^2 + (2*a^2*d^5*e^2*Sqrt[-c + (d*e )/f]*g*h)/(c + d*x)^2 - (20*b^2*c^3*d^2*e*Sqrt[-c + (d*e)/f]*f*g*h)/(c + d *x)^2 + (28*a*b*c^2*d^3*e*Sqrt[-c + (d*e)/f]*f*g*h)/(c + d*x)^2 - (8*a^...
Time = 1.43 (sec) , antiderivative size = 875, 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.
| \(\displaystyle \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) |
| \(\Big \downarrow \) 197 |
| \(\displaystyle \int \left (\frac {b^2}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)^2}-\frac {b d}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)^2}-\frac {d}{(c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right ) b^2}{(b c-a d)^3 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 d \sqrt {h} \sqrt {e h-f g} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right ) b}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}+\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x} b}{(b c-a d)^2 (d e-c f) (d g-c h) \sqrt {c+d x}}+\frac {4 d \sqrt {f} (d f g+d e h-2 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {g+h x} E\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (c f-d e)^{3/2} (d g-c h)^2 \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 \sqrt {f} (2 d f g+d e h-3 c f h) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{3 (b c-a d) (c f-d e)^{3/2} (d g-c h) \sqrt {e+f x} \sqrt {g+h x}}-\frac {4 d^2 (d f g+d e h-2 c f h) \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f)^2 (d g-c h)^2 \sqrt {c+d x}}+\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (d e-c f) (d g-c h) (c+d x)^{3/2}}\) |
(2*d^2*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d)*(d*e - c*f)*(d*g - c*h) *(c + d*x)^(3/2)) + (2*b*d^2*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)^2*( d*e - c*f)*(d*g - c*h)*Sqrt[c + d*x]) - (4*d^2*(d*f*g + d*e*h - 2*c*f*h)*S qrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d)*(d*e - c*f)^2*(d*g - c*h)^2*Sqr t[c + d*x]) + (4*d*Sqrt[f]*(d*f*g + d*e*h - 2*c*f*h)*Sqrt[(d*(e + f*x))/(d *e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d *e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(3*(b*c - a*d)*(-(d*e) + c* f)^(3/2)*(d*g - c*h)^2*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) - (2 *b*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)^2*(d*e - c*f)*(d*g - c*h)*Sqrt[- ((f*(c + d*x))/(d*e - c*f))]*Sqrt[g + h*x]) - (2*Sqrt[f]*(2*d*f*g + d*e*h - 3*c*f*h)*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)] *EllipticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f )*h)/(f*(d*g - c*h))])/(3*(b*c - a*d)*(-(d*e) + c*f)^(3/2)*(d*g - c*h)*Sqr t[e + f*x]*Sqrt[g + h*x]) - (2*b^2*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)^3*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x])
3.1.72.3.1 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]
Time = 2.77 (sec) , antiderivative size = 1335, normalized size of antiderivative = 1.53
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1335\) |
| default | \(\text {Expression too large to display}\) | \(16647\) |
((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)* (-2/3/(c^2*f*h-c*d*e*h-c*d*f*g+d^2*e*g)/(a*d-b*c)*(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)/(x+c/d)^2-2/3*(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)^2*d*(4*a*c*d *f*h-2*a*d^2*e*h-2*a*d^2*f*g-7*b*c^2*f*h+5*b*c*d*e*h+5*b*c*d*f*g-3*b*d^2*e *g)/(a*d-b*c)^2/((x+c/d)*(d*f*h*x^2+d*e*h*x+d*f*g*x+d*e*g))^(1/2)+2*(-1/3* d*f*h/(c^2*f*h-c*d*e*h-c*d*f*g+d^2*e*g)/(a*d-b*c)+1/3*d*(c*f*h-d*e*h-d*f*g )*(4*a*c*d*f*h-2*a*d^2*e*h-2*a*d^2*f*g-7*b*c^2*f*h+5*b*c*d*e*h+5*b*c*d*f*g -3*b*d^2*e*g)/(c^2*f*h-c*d*e*h-c*d*f*g+d^2*e*g)^2/(a*d-b*c)^2+1/3*(d*e*h+d *f*g)/(c^2*f*h-c*d*e*h-c*d*f*g+d^2*e*g)^2*d*(4*a*c*d*f*h-2*a*d^2*e*h-2*a*d ^2*f*g-7*b*c^2*f*h+5*b*c*d*e*h+5*b*c*d*f*g-3*b*d^2*e*g)/(a*d-b*c)^2)*(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))+2/3*f*h*d^2*(4*a*c*d*f*h-2*a*d^2*e*h-2*a*d^2*f*g-7*b*c^2*f*h+5*b*c *d*e*h+5*b*c*d*f*g-3*b*d^2*e*g)/(c^2*f*h-c*d*e*h-c*d*f*g+d^2*e*g)^2/(a*d-b *c)^2*(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)*EllipticE(((x+g/h)/(g/h-e/f))^(1/2), ((-g/h+e/f)/(-g/h+c/d))^(1/2))-c/d*EllipticF(((x+g/h)/(g/h-e/f))^(1/2),...
Timed out. \[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {1}{\left (a + b x\right ) \left (c + d x\right )^{\frac {5}{2}} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]
\[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {1}{{\left (b x + a\right )} {\left (d x + c\right )}^{\frac {5}{2}} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
\[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {1}{{\left (b x + a\right )} {\left (d x + c\right )}^{\frac {5}{2}} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b x) (c+d x)^{5/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {1}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^{5/2}} \,d x \]