Integrand size = 49, antiderivative size = 625 \[ \int \frac {d e+c f+2 d f x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 d (b d e+b c f-2 a d f) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {c+d x}}-\frac {2 b (b d e+b c f-2 a d f) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}-\frac {2 (b d e+b c f-2 a d f) \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{(b c-a d) (b e-a f) (b g-a h) \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 d (d e-c f) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{(b c-a d) \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}} \]
2*d*(-2*a*d*f+b*c*f+b*d*e)*(b*x+a)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d +b*c)/(-a*f+b*e)/(-a*h+b*g)/(d*x+c)^(1/2)-2*b*(-2*a*d*f+b*c*f+b*d*e)*(d*x+ c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(-a*h+b*g)/(b*x +a)^(1/2)-2*d*(-c*f+d*e)*EllipticF((-a*h+b*g)^(1/2)*(f*x+e)^(1/2)/(-e*h+f* g)^(1/2)/(b*x+a)^(1/2),(-(-a*d+b*c)*(-e*h+f*g)/(-c*f+d*e)/(-a*h+b*g))^(1/2 ))*((-a*f+b*e)*(d*x+c)/(-c*f+d*e)/(b*x+a))^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)/ (-a*h+b*g)^(1/2)/(-e*h+f*g)^(1/2)/(d*x+c)^(1/2)/(-(-a*f+b*e)*(h*x+g)/(-e*h +f*g)/(b*x+a))^(1/2)-2*(-2*a*d*f+b*c*f+b*d*e)*EllipticE((-c*h+d*g)^(1/2)*( f*x+e)^(1/2)/(-e*h+f*g)^(1/2)/(d*x+c)^(1/2),((-a*d+b*c)*(-e*h+f*g)/(-a*f+b *e)/(-c*h+d*g))^(1/2))*(-c*h+d*g)^(1/2)*(-e*h+f*g)^(1/2)*(b*x+a)^(1/2)*(-( -c*f+d*e)*(h*x+g)/(-e*h+f*g)/(d*x+c))^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(-a*h+b* g)/((-c*f+d*e)*(b*x+a)/(-a*f+b*e)/(d*x+c))^(1/2)/(h*x+g)^(1/2)
Time = 25.78 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.55 \[ \int \frac {d e+c f+2 d f x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 (b e-a f) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} (e+f x)^{3/2} (g+h x)^{3/2} \left ((b d e+b c f-2 a d f) (d g-c h) E\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )-d (d e-c f) (b g-a h) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ),\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )\right )}{(b c-a d) (f g-e h)^3 (a+b x)^{5/2} \sqrt {c+d x} \left (-\frac {(b e-a f) (b g-a h) (e+f x) (g+h x)}{(f g-e h)^2 (a+b x)^2}\right )^{3/2}} \]
Integrate[(d*e + c*f + 2*d*f*x)/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f* x]*Sqrt[g + h*x]),x]
(2*(b*e - a*f)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*(e + f*x)^(3/2)*(g + h*x)^(3/2)*((b*d*e + b*c*f - 2*a*d*f)*(d*g - c*h)*Elliptic E[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))] - d*(d*e - c*f)*(b*g - a*h) *EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))] ], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))]))/((b*c - a*d)*(f* g - e*h)^3*(a + b*x)^(5/2)*Sqrt[c + d*x]*(-(((b*e - a*f)*(b*g - a*h)*(e + f*x)*(g + h*x))/((f*g - e*h)^2*(a + b*x)^2)))^(3/2))
Time = 1.49 (sec) , antiderivative size = 595, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2102, 2105, 27, 188, 194, 321, 327}
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.
| \(\displaystyle \int \frac {c f+d e+2 d f x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) |
| \(\Big \downarrow \) 2102 |
| \(\displaystyle \frac {\int \frac {-d f (d e+c f) h a^2-b \left (e (f g-e h) d^2+c f^2 g d-c^2 f^2 h\right ) a+2 b d f (b d e+b c f-2 a d f) h x^2+2 b^2 c d e f g+(b d e+b c f-2 a d f) (a d f h+b (d f g+d e h+c f h)) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 2105 |
| \(\displaystyle \frac {\frac {\int -\frac {2 b d^2 f (b e-a f) (d e-c f) h (b g-a h)}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 b d f h}+(d e-c f) (d g-c h) (-2 a d f+b c f+b d e) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {c+d x}}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(d e-c f) (d g-c h) (-2 a d f+b c f+b d e) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx-d (b e-a f) (b g-a h) (d e-c f) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {c+d x}}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 188 |
| \(\displaystyle \frac {(d e-c f) (d g-c h) (-2 a d f+b c f+b d e) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx-\frac {2 d \sqrt {g+h x} (b e-a f) (b g-a h) (d e-c f) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{\sqrt {c+d x} (f g-e h) \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {c+d x}}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 194 |
| \(\displaystyle \frac {-\frac {2 \sqrt {a+b x} (d g-c h) (-2 a d f+b c f+b d e) \sqrt {-\frac {(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}}}{\sqrt {g+h x} \sqrt {\frac {(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}-\frac {2 d \sqrt {g+h x} (b e-a f) (b g-a h) (d e-c f) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{\sqrt {c+d x} (f g-e h) \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {c+d x}}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {-\frac {2 \sqrt {a+b x} (d g-c h) (-2 a d f+b c f+b d e) \sqrt {-\frac {(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}}}{\sqrt {g+h x} \sqrt {\frac {(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}-\frac {2 d \sqrt {g+h x} (b e-a f) \sqrt {b g-a h} (d e-c f) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {c+d x} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {c+d x}}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {-\frac {2 d \sqrt {g+h x} (b e-a f) \sqrt {b g-a h} (d e-c f) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {c+d x} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}-\frac {2 \sqrt {a+b x} \sqrt {d g-c h} \sqrt {f g-e h} (-2 a d f+b c f+b d e) \sqrt {-\frac {(g+h x) (d e-c f)}{(c+d x) (f g-e h)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{\sqrt {g+h x} \sqrt {\frac {(a+b x) (d e-c f)}{(c+d x) (b e-a f)}}}+\frac {2 d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {c+d x}}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}\) |
(-2*b*(b*d*e + b*c*f - 2*a*d*f)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]) /((b*c - a*d)*(b*e - a*f)*(b*g - a*h)*Sqrt[a + b*x]) + ((2*d*(b*d*e + b*c* f - 2*a*d*f)*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/Sqrt[c + d*x] - (2 *(b*d*e + b*c*f - 2*a*d*f)*Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*Sqrt[a + b*x]*S qrt[-(((d*e - c*f)*(g + h*x))/((f*g - e*h)*(c + d*x)))]*EllipticE[ArcSin[( Sqrt[d*g - c*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h]*Sqrt[c + d*x])], ((b*c - a *d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))])/(Sqrt[((d*e - c*f)*(a + b*x)) /((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - (2*d*(b*e - a*f)*(d*e - c*f)*Sq rt[b*g - a*h]*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*EllipticF[ArcSin[(Sqrt[b*g - a*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h]* Sqrt[a + b*x])], -(((b*c - a*d)*(f*g - e*h))/((d*e - c*f)*(b*g - a*h)))])/ (Sqrt[f*g - e*h]*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h) *(a + b*x)))]))/((b*c - a*d)*(b*e - a*f)*(b*g - a*h))
3.1.14.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.) *(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[2*Sqrt[g + h*x]*(Sqrt[(b*e - a*f)*((c + d*x)/((d*e - c*f)*(a + b*x)))]/((f*g - e*h)*Sqrt[c + d*x]*Sqrt[( -(b*e - a*f))*((g + h*x)/((f*g - e*h)*(a + b*x)))])) Subst[Int[1/(Sqrt[1 + (b*c - a*d)*(x^2/(d*e - c*f))]*Sqrt[1 - (b*g - a*h)*(x^2/(f*g - e*h))]), x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.) *(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2*Sqrt[c + d*x]*(Sqrt[(-(b*e - a*f))*((g + h*x)/((f*g - e*h)*(a + b*x)))]/((b*e - a*f)*Sqrt[g + h*x]*Sq rt[(b*e - a*f)*((c + d*x)/((d*e - c*f)*(a + b*x)))])) Subst[Int[Sqrt[1 + (b*c - a*d)*(x^2/(d*e - c*f))]/Sqrt[1 - (b*g - a*h)*(x^2/(f*g - e*h))], x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x _)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[( A*b^2 - a*b*B)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x] /((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))), x] - Simp[1/(2*(m + 1)*(b* c - a*d)*(b*e - a*f)*(b*g - a*h)) Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*S qrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)*(d *f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) - b*B*(a*(d* e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)))*x + d*f*h*(2*m + 5)*(A*b^2 - a*b*B)* x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && IntegerQ[2*m ] && LtQ[m, -1]
Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_. ) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbo l] :> Simp[C*Sqrt[a + b*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(b*f*h*Sqrt[c + d*x ])), x] + (Simp[1/(2*b*d*f*h) Int[(1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[2*A*b*d*f*h - C*(b*d*e*g + a*c*f*h) + (2*b*B*d* f*h - C*(a*d*f*h + b*(d*f*g + d*e*h + c*f*h)))*x, x], x], x] + Simp[C*(d*e - c*f)*((d*g - c*h)/(2*b*d*f*h)) Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[ e + f*x]*Sqrt[g + h*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C} , x]
Leaf count of result is larger than twice the leaf count of optimal. \(2297\) vs. \(2(571)=1142\).
Time = 7.75 (sec) , antiderivative size = 2298, normalized size of antiderivative = 3.68
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2298\) |
| default | \(\text {Expression too large to display}\) | \(21256\) |
int((2*d*f*x+c*f+d*e)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1 /2),x,method=_RETURNVERBOSE)
((b*x+a)*(d*x+c)*(f*x+e)*(h*x+g))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e )^(1/2)/(h*x+g)^(1/2)*(-2*(b*d*f*h*x^3+b*c*f*h*x^2+b*d*e*h*x^2+b*d*f*g*x^2 +b*c*e*h*x+b*c*f*g*x+b*d*e*g*x+b*c*e*g)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h -a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)*(2*a*d*f-b*c*f -b*d*e)/((x+a/b)*(b*d*f*h*x^3+b*c*f*h*x^2+b*d*e*h*x^2+b*d*f*g*x^2+b*c*e*h* x+b*c*f*g*x+b*d*e*g*x+b*c*e*g))^(1/2)+2*(2/b*d*f-1/b*(a^2*d*f*h-a*b*c*f*h- a*b*d*e*h-a*b*d*f*g+b^2*c*e*h+b^2*c*f*g+b^2*d*e*g)*(2*a*d*f-b*c*f-b*d*e)/( a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^ 2*d*e*g-b^3*c*e*g)+(b*c*e*h+b*c*f*g+b*d*e*g)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b* d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)*(2*a*d*f- b*c*f-b*d*e))*(g/h-a/b)*((-g/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c/d))^(1/2)*(x+c /d)^2*((-c/d+a/b)*(x+e/f)/(-e/f+a/b)/(x+c/d))^(1/2)*((-c/d+a/b)*(x+g/h)/(- g/h+a/b)/(x+c/d))^(1/2)/(-g/h+c/d)/(-c/d+a/b)/(b*d*f*h*(x+a/b)*(x+c/d)*(x+ e/f)*(x+g/h))^(1/2)*EllipticF(((-g/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c/d))^(1/2 ),((e/f-c/d)*(g/h-a/b)/(-a/b+e/f)/(-c/d+g/h))^(1/2))+2*((a*d*f*h-b*c*f*h-b *d*e*h-b*d*f*g)*(2*a*d*f-b*c*f-b*d*e)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a ^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)+(2*b*c*f*h+2*b*d *e*h+2*b*d*f*g)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h +a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)*(2*a*d*f-b*c*f-b*d*e))*(g/h-a/b)*((-g/ h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c/d))^(1/2)*(x+c/d)^2*((-c/d+a/b)*(x+e/f)/...
\[ \int \frac {d e+c f+2 d f x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {2 \, d f x + d e + c f}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((2*d*f*x+c*f+d*e)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x +g)^(1/2),x, algorithm="fricas")
integral((2*d*f*x + d*e + c*f)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*s qrt(h*x + g)/(b^2*d*f*h*x^5 + a^2*c*e*g + (b^2*d*f*g + (b^2*d*e + (b^2*c + 2*a*b*d)*f)*h)*x^4 + ((b^2*d*e + (b^2*c + 2*a*b*d)*f)*g + ((b^2*c + 2*a*b *d)*e + (2*a*b*c + a^2*d)*f)*h)*x^3 + (((b^2*c + 2*a*b*d)*e + (2*a*b*c + a ^2*d)*f)*g + (a^2*c*f + (2*a*b*c + a^2*d)*e)*h)*x^2 + (a^2*c*e*h + (a^2*c* f + (2*a*b*c + a^2*d)*e)*g)*x), x)
\[ \int \frac {d e+c f+2 d f x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {c f + d e + 2 d f x}{\left (a + b x\right )^{\frac {3}{2}} \sqrt {c + d x} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]
Integral((c*f + d*e + 2*d*f*x)/((a + b*x)**(3/2)*sqrt(c + d*x)*sqrt(e + f* x)*sqrt(g + h*x)), x)
\[ \int \frac {d e+c f+2 d f x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {2 \, d f x + d e + c f}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((2*d*f*x+c*f+d*e)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x +g)^(1/2),x, algorithm="maxima")
integrate((2*d*f*x + d*e + c*f)/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)
\[ \int \frac {d e+c f+2 d f x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {2 \, d f x + d e + c f}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((2*d*f*x+c*f+d*e)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x +g)^(1/2),x, algorithm="giac")
integrate((2*d*f*x + d*e + c*f)/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)
Timed out. \[ \int \frac {d e+c f+2 d f x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {c\,f+d\,e+2\,d\,f\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{3/2}\,\sqrt {c+d\,x}} \,d x \]
int((c*f + d*e + 2*d*f*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)^(3/2) *(c + d*x)^(1/2)),x)