Integrand size = 28, antiderivative size = 437 \[ \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{3 (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac {4 b (b d e+b c f-2 a d f) \sqrt {c+d x} \sqrt {e+f x}}{3 (b c-a d)^2 (b e-a f)^2 \sqrt {a+b x}}-\frac {4 \sqrt {d} (b d e+b c f-2 a d f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 (-b c+a d)^{3/2} (b e-a f)^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \sqrt {d} (2 b d e+b c f-3 a d f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b (-b c+a d)^{3/2} (b e-a f) \sqrt {c+d x} \sqrt {e+f x}} \]
-2/3*b*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^(3/2)+4/3 *b*(-2*a*d*f+b*c*f+b*d*e)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/(-a*d+b*c)^2/(-a*f+b *e)^2/(b*x+a)^(1/2)-4/3*(-2*a*d*f+b*c*f+b*d*e)*EllipticE(d^(1/2)*(b*x+a)^( 1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/2))*d^(1/2)*(b*(d*x+c) /(-a*d+b*c))^(1/2)*(f*x+e)^(1/2)/(a*d-b*c)^(3/2)/(-a*f+b*e)^2/(d*x+c)^(1/2 )/(b*(f*x+e)/(-a*f+b*e))^(1/2)+2/3*(-3*a*d*f+b*c*f+2*b*d*e)*EllipticF(d^(1 /2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/2))*d^(1/ 2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(1/2)/b/(a*d-b*c)^( 3/2)/(-a*f+b*e)/(d*x+c)^(1/2)/(f*x+e)^(1/2)
Result contains complex when optimal does not.
Time = 23.39 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 \left (b^2 \sqrt {-a+\frac {b c}{d}} (c+d x) (e+f x) ((b c-a d) (b e-a f)-2 (b d e+b c f-2 a d f) (a+b x))+(a+b x) \left (2 b^2 \sqrt {-a+\frac {b c}{d}} (b d e+b c f-2 a d f) (c+d x) (e+f x)+2 i (b c-a d) f (b d e+b c f-2 a d f) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )-i (b c-a d) f (b d e+2 b c f-3 a d f) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )\right )}{3 b \sqrt {-a+\frac {b c}{d}} (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \]
(-2*(b^2*Sqrt[-a + (b*c)/d]*(c + d*x)*(e + f*x)*((b*c - a*d)*(b*e - a*f) - 2*(b*d*e + b*c*f - 2*a*d*f)*(a + b*x)) + (a + b*x)*(2*b^2*Sqrt[-a + (b*c) /d]*(b*d*e + b*c*f - 2*a*d*f)*(c + d*x)*(e + f*x) + (2*I)*(b*c - a*d)*f*(b *d*e + b*c*f - 2*a*d*f)*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]* Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/S qrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)] - I*(b*c - a*d)*f*(b*d*e + 2*b*c*f - 3*a*d*f)*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt [(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[ a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])))/(3*b*Sqrt[-a + (b*c)/d]*(b* c - a*d)^2*(b*e - a*f)^2*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])
Time = 0.62 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {115, 27, 25, 169, 27, 176, 124, 123, 131, 131, 130}
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.
| \(\displaystyle \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle -\frac {2 \int \frac {2 b d e+2 b c f-3 a d f+b d f x}{2 (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 (b c-a d) (b e-a f)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int -\frac {3 a d f-b d x f-2 b (d e+c f)}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 (b c-a d) (b e-a f)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 a d f-b d x f-2 b (d e+c f)}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 (b c-a d) (b e-a f)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {\frac {4 b \sqrt {c+d x} \sqrt {e+f x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {2 \int \frac {d f \left (-3 d f a^2+b (d e+c f) a+b^2 c e+2 b (b d e+b c f-2 a d f) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {4 b \sqrt {c+d x} \sqrt {e+f x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {d f \int \frac {-3 d f a^2+b (d e+c f) a+b^2 c e+2 b (b d e+b c f-2 a d f) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {\frac {4 b \sqrt {c+d x} \sqrt {e+f x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {d f \left (\frac {2 b (-2 a d f+b c f+b d e) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}}dx}{f}-\frac {(b e-a f) (-3 a d f+b c f+2 b d e) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}\right )}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {\frac {4 b \sqrt {c+d x} \sqrt {e+f x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {d f \left (\frac {2 b \sqrt {e+f x} \sqrt {\frac {b (c+d x)}{b c-a d}} (-2 a d f+b c f+b d e) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}dx}{f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {(b e-a f) (-3 a d f+b c f+2 b d e) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}\right )}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {\frac {4 b \sqrt {c+d x} \sqrt {e+f x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {d f \left (\frac {4 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} (-2 a d f+b c f+b d e) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{\sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {(b e-a f) (-3 a d f+b c f+2 b d e) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}\right )}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\frac {4 b \sqrt {c+d x} \sqrt {e+f x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {d f \left (\frac {4 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} (-2 a d f+b c f+b d e) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{\sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {(b e-a f) \sqrt {\frac {b (c+d x)}{b c-a d}} (-3 a d f+b c f+2 b d e) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}}dx}{f \sqrt {c+d x}}\right )}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\frac {4 b \sqrt {c+d x} \sqrt {e+f x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {d f \left (\frac {4 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} (-2 a d f+b c f+b d e) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{\sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {(b e-a f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} (-3 a d f+b c f+2 b d e) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{f \sqrt {c+d x} \sqrt {e+f x}}\right )}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 130 |
| \(\displaystyle \frac {\frac {4 b \sqrt {c+d x} \sqrt {e+f x} (-2 a d f+b c f+b d e)}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {d f \left (\frac {4 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} (-2 a d f+b c f+b d e) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{\sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {a d-b c} (b e-a f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} (-3 a d f+b c f+2 b d e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {e+f x}}\right )}{(b c-a d) (b e-a f)}}{3 (b c-a d) (b e-a f)}-\frac {2 b \sqrt {c+d x} \sqrt {e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
(-2*b*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(3 /2)) + ((4*b*(b*d*e + b*c*f - 2*a*d*f)*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]) - (d*f*((4*Sqrt[-(b*c) + a*d]*(b*d*e + b *c*f - 2*a*d*f)*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[Ar cSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(Sqrt[d]*f*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) - (2*S qrt[-(b*c) + a*d]*(b*e - a*f)*(2*b*d*e + b*c*f - 3*a*d*f)*Sqrt[(b*(c + d*x ))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]* Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b*S qrt[d]*f*Sqrt[c + d*x]*Sqrt[e + f*x])))/((b*c - a*d)*(b*e - a*f)))/(3*(b*c - a*d)*(b*e - a*f))
3.29.58.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ [b/(b*e - a*f), 0] && SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f *x] && (PosQ[-(b*c - a*d)/d] || NegQ[-(b*e - a*f)/f])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[Sqrt[b*((c + d*x)/(b*c - a*d))]/Sqrt[c + d*x] Int[1/(Sq rt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e + f*x]), x ], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[(b*c - a*d)/b, 0] && Simpler Q[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Leaf count of result is larger than twice the leaf count of optimal. \(877\) vs. \(2(383)=766\).
Time = 2.64 (sec) , antiderivative size = 878, normalized size of antiderivative = 2.01
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b x +a \right ) \left (d x +c \right ) \left (f x +e \right )}\, \left (-\frac {2 \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}{3 \left (a^{2} d f -a c f b -a b d e +b^{2} c e \right ) b \left (x +\frac {a}{b}\right )^{2}}-\frac {4 \left (b d f \,x^{2}+b c f x +b d e x +b c e \right ) \left (2 a d f -b c f -b d e \right )}{3 \left (a^{2} d f -a c f b -a b d e +b^{2} c e \right )^{2} \sqrt {\left (x +\frac {a}{b}\right ) \left (b d f \,x^{2}+b c f x +b d e x +b c e \right )}}+\frac {2 \left (-\frac {d f}{3 \left (a^{2} d f -a c f b -a b d e +b^{2} c e \right )}+\frac {2 \left (a d f -b c f -b d e \right ) \left (2 a d f -b c f -b d e \right )}{3 \left (a^{2} d f -a c f b -a b d e +b^{2} c e \right )^{2}}+\frac {2 \left (b c f +b d e \right ) \left (2 a d f -b c f -b d e \right )}{3 \left (a^{2} d f -a c f b -a b d e +b^{2} c e \right )^{2}}\right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, F\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{\sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}+\frac {4 b d f \left (2 a d f -b c f -b d e \right ) \left (\frac {e}{f}-\frac {c}{d}\right ) \sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {e}{f}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {e}{f}+\frac {c}{d}}}\, \left (\left (-\frac {e}{f}+\frac {a}{b}\right ) E\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )-\frac {a F\left (\sqrt {\frac {x +\frac {e}{f}}{\frac {e}{f}-\frac {c}{d}}}, \sqrt {\frac {-\frac {e}{f}+\frac {c}{d}}{-\frac {e}{f}+\frac {a}{b}}}\right )}{b}\right )}{3 \left (a^{2} d f -a c f b -a b d e +b^{2} c e \right )^{2} \sqrt {b d f \,x^{3}+a d f \,x^{2}+b c f \,x^{2}+b d e \,x^{2}+a c f x +a d e x +b c e x +a c e}}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {f x +e}}\) | \(878\) |
| default | \(\text {Expression too large to display}\) | \(3386\) |
((b*x+a)*(d*x+c)*(f*x+e))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)* (-2/3/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b*(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b *d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)/(x+a/b)^2-4/3*(b*d*f*x^2+b*c *f*x+b*d*e*x+b*c*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)^2*(2*a*d*f-b*c*f-b*d *e)/((x+a/b)*(b*d*f*x^2+b*c*f*x+b*d*e*x+b*c*e))^(1/2)+2*(-1/3/(a^2*d*f-a*b *c*f-a*b*d*e+b^2*c*e)*d*f+2/3*(a*d*f-b*c*f-b*d*e)*(2*a*d*f-b*c*f-b*d*e)/(a ^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)^2+2/3*(b*c*f+b*d*e)/(a^2*d*f-a*b*c*f-a*b*d *e+b^2*c*e)^2*(2*a*d*f-b*c*f-b*d*e))*(e/f-c/d)*((x+e/f)/(e/f-c/d))^(1/2)*( (x+a/b)/(-e/f+a/b))^(1/2)*((x+c/d)/(-e/f+c/d))^(1/2)/(b*d*f*x^3+a*d*f*x^2+ b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)*EllipticF(((x+e/f )/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f+a/b))^(1/2))+4/3*b*d*f*(2*a*d*f-b*c*f -b*d*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)^2*(e/f-c/d)*((x+e/f)/(e/f-c/d))^ (1/2)*((x+a/b)/(-e/f+a/b))^(1/2)*((x+c/d)/(-e/f+c/d))^(1/2)/(b*d*f*x^3+a*d *f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)*((-e/f+a/b )*EllipticE(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f+a/b))^(1/2))-a/b*E llipticF(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f+a/b))^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 1403, normalized size of antiderivative = 3.21 \[ \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Too large to display} \]
-2/9*(3*((b^4*c*d - 3*a*b^3*d^2)*e*f - (3*a*b^3*c*d - 5*a^2*b^2*d^2)*f^2 - 2*(b^4*d^2*e*f + (b^4*c*d - 2*a*b^3*d^2)*f^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e) - (2*a^2*b^2*d^2*e^2 + (a^2*b^2*c*d - 5*a^3*b*d^2)*e*f + (2*a^2*b^2*c^2 - 5*a^3*b*c*d + 5*a^4*d^2)*f^2 + (2*b^4*d^2*e^2 + (b^4*c*d - 5*a*b^3*d^2)*e*f + (2*b^4*c^2 - 5*a*b^3*c*d + 5*a^2*b^2*d^2)*f^2)*x^2 + 2*(2*a*b^3*d^2*e^2 + (a*b^3*c*d - 5*a^2*b^2*d^2)*e*f + (2*a*b^3*c^2 - 5*a ^2*b^2*c*d + 5*a^3*b*d^2)*f^2)*x)*sqrt(b*d*f)*weierstrassPInverse(4/3*(b^2 *d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2)/(b ^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b ^3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e + ( b*c + a*d)*f)/(b*d*f)) - 6*(a^2*b^2*d^2*e*f + (a^2*b^2*c*d - 2*a^3*b*d^2)* f^2 + (b^4*d^2*e*f + (b^4*c*d - 2*a*b^3*d^2)*f^2)*x^2 + 2*(a*b^3*d^2*e*f + (a*b^3*c*d - 2*a^2*b^2*d^2)*f^2)*x)*sqrt(b*d*f)*weierstrassZeta(4/3*(b^2* d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2)/(b^ 2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^ 3*c^2*d - 4*a*b^2*c*d^2 + a^2*b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3)/(b^3*d^3*f^3), weierstrassPInverse(4/3*(b^ 2*d^2*e^2 - (b^2*c*d + a*b*d^2)*e*f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2)/( b^2*d^2*f^2), -4/27*(2*b^3*d^3*e^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - ...
\[ \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \sqrt {c + d x} \sqrt {e + f x}}\, dx \]
\[ \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e}} \,d x } \]
\[ \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {1}{\sqrt {e+f\,x}\,{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \]