\(\int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [59]

Optimal result

Integrand size = 35, antiderivative size = 449 \[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 d \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{b f \sqrt {h} \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}+\frac {2 (b c-a d) \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b c-a d) \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \]

[Out]

2*(-a*d+b*c)*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))*(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)/b^2/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)-2*(-a*d+b*
c)*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)/b^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))*(e*h-f*g)^(1/2
)*(d*x+c)^(1/2)*(f*(h*x+g)/(-e*h+f*g))^(1/2)/b/f/h^(1/2)/(-f*(d*x+c)/(-c*f+d*e))^(1/2)/(h*x+g)^(1/2)

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {185, 122, 121, 175, 552, 551, 115, 114} \[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 (b c-a d) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b c-a d) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}+\frac {2 d \sqrt {c+d x} \sqrt {e h-f g} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{b f \sqrt {h} \sqrt {g+h x} \sqrt {-\frac {f (c+d x)}{d e-c f}}} \]

[In]

Int[(c + d*x)^(3/2)/((a + b*x)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(2*d*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*f*Sqrt[h]*Sqrt[-((f*(c + d*x))/(d*e - c*f))]*Sqr
t[g + h*x]) + (2*(b*c - a*d)*Sqrt[-(d*e) + c*f]*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))])/(b^2*Sqrt[f]
*Sqrt[e + f*x]*Sqrt[g + h*x]) - (2*(b*c - a*d)*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^2*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> 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] &&  !LtQ[-(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])

Rule 115

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[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] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]

Rule 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> 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] && Si
mplerQ[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])

Rule 122

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[b*((c
+ d*x)/(b*c - a*d))]/Sqrt[c + d*x], Int[1/(Sqrt[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] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 175

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d
*g - c*h)/d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] &&  !SimplerQ[e
+ f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]

Rule 185

Int[(((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_))/(Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)])
, x_Symbol] :> 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]

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d/c)*x^2]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d (b c-a d)}{b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}+\frac {(b c-a d)^2}{b^2 (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}+\frac {d \sqrt {c+d x}}{b \sqrt {e+f x} \sqrt {g+h x}}\right ) \, dx \\ & = \frac {d \int \frac {\sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}} \, dx}{b}+\frac {(d (b c-a d)) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b^2}+\frac {(b c-a d)^2 \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b^2} \\ & = -\frac {\left (2 (b c-a d)^2\right ) \text {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {e-\frac {c f}{d}+\frac {f x^2}{d}} \sqrt {g-\frac {c h}{d}+\frac {h x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{b^2}+\frac {\left (d (b c-a d) \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {g+h x}} \, dx}{b^2 \sqrt {e+f x}}+\frac {\left (d \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}}\right ) \int \frac {\sqrt {\frac {c f}{-d e+c f}+\frac {d f x}{-d e+c f}}}{\sqrt {e+f x} \sqrt {\frac {f g}{f g-e h}+\frac {f h x}{f g-e h}}} \, dx}{b \sqrt {\frac {f (c+d x)}{-d e+c f}} \sqrt {g+h x}} \\ & = \frac {2 d \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\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 f \sqrt {h} \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}-\frac {\left (2 (b c-a d)^2 \sqrt {\frac {d (e+f x)}{d e-c f}}\right ) \text {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {1+\frac {f x^2}{d \left (e-\frac {c f}{d}\right )}} \sqrt {g-\frac {c h}{d}+\frac {h x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{b^2 \sqrt {e+f x}}+\frac {\left (d (b c-a d) \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}}\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {\frac {d e}{d e-c f}+\frac {d f x}{d e-c f}} \sqrt {\frac {d g}{d g-c h}+\frac {d h x}{d g-c h}}} \, dx}{b^2 \sqrt {e+f x} \sqrt {g+h x}} \\ & = \frac {2 d \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\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 f \sqrt {h} \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}+\frac {2 (b c-a d) \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}} F\left (\sin ^{-1}\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^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {\left (2 (b c-a d)^2 \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}}\right ) \text {Subst}\left (\int \frac {1}{\left (b c-a d-b x^2\right ) \sqrt {1+\frac {f x^2}{d \left (e-\frac {c f}{d}\right )}} \sqrt {1+\frac {h x^2}{d \left (g-\frac {c h}{d}\right )}}} \, dx,x,\sqrt {c+d x}\right )}{b^2 \sqrt {e+f x} \sqrt {g+h x}} \\ & = \frac {2 d \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\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 f \sqrt {h} \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}+\frac {2 (b c-a d) \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}} F\left (\sin ^{-1}\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^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 (b c-a d) \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}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\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^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.30 (sec) , antiderivative size = 1176, normalized size of antiderivative = 2.62 \[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 \left (b^2 d^2 e^2 f \sqrt {-e+\frac {c f}{d}} g-b^2 c d e f^2 \sqrt {-e+\frac {c f}{d}} g-a b d^2 e f^2 \sqrt {-e+\frac {c f}{d}} g+a b c d f^3 \sqrt {-e+\frac {c f}{d}} g-b^2 d^2 e^3 \sqrt {-e+\frac {c f}{d}} h+b^2 c d e^2 f \sqrt {-e+\frac {c f}{d}} h+a b d^2 e^2 f \sqrt {-e+\frac {c f}{d}} h-a b c d e f^2 \sqrt {-e+\frac {c f}{d}} h-b^2 d^2 e f \sqrt {-e+\frac {c f}{d}} g (e+f x)+a b d^2 f^2 \sqrt {-e+\frac {c f}{d}} g (e+f x)+2 b^2 d^2 e^2 \sqrt {-e+\frac {c f}{d}} h (e+f x)-b^2 c d e f \sqrt {-e+\frac {c f}{d}} h (e+f x)-2 a b d^2 e f \sqrt {-e+\frac {c f}{d}} h (e+f x)+a b c d f^2 \sqrt {-e+\frac {c f}{d}} h (e+f x)-b^2 d^2 e \sqrt {-e+\frac {c f}{d}} h (e+f x)^2+a b d^2 f \sqrt {-e+\frac {c f}{d}} h (e+f x)^2+i b d (b e-a f) (d e-c f) h \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-e+\frac {c f}{d}}}{\sqrt {e+f x}}\right )|\frac {d (-f g+e h)}{(d e-c f) h}\right )+i b (-b c+a d) f (d e-c f) h \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-e+\frac {c f}{d}}}{\sqrt {e+f x}}\right ),\frac {d (-f g+e h)}{(d e-c f) h}\right )-i b^2 c^2 f^2 h \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} \operatorname {EllipticPi}\left (\frac {b d e-a d f}{b d e-b c f},i \text {arcsinh}\left (\frac {\sqrt {-e+\frac {c f}{d}}}{\sqrt {e+f x}}\right ),\frac {d (-f g+e h)}{(d e-c f) h}\right )+2 i a b c d f^2 h \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} \operatorname {EllipticPi}\left (\frac {b d e-a d f}{b d e-b c f},i \text {arcsinh}\left (\frac {\sqrt {-e+\frac {c f}{d}}}{\sqrt {e+f x}}\right ),\frac {d (-f g+e h)}{(d e-c f) h}\right )-i a^2 d^2 f^2 h \sqrt {\frac {f (c+d x)}{d (e+f x)}} (e+f x)^{3/2} \sqrt {\frac {f (g+h x)}{h (e+f x)}} \operatorname {EllipticPi}\left (\frac {b d e-a d f}{b d e-b c f},i \text {arcsinh}\left (\frac {\sqrt {-e+\frac {c f}{d}}}{\sqrt {e+f x}}\right ),\frac {d (-f g+e h)}{(d e-c f) h}\right )\right )}{b^2 f^2 (-b e+a f) \sqrt {-e+\frac {c f}{d}} h \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \]

[In]

Integrate[(c + d*x)^(3/2)/((a + b*x)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(2*(b^2*d^2*e^2*f*Sqrt[-e + (c*f)/d]*g - b^2*c*d*e*f^2*Sqrt[-e + (c*f)/d]*g - a*b*d^2*e*f^2*Sqrt[-e + (c*f)/d]
*g + a*b*c*d*f^3*Sqrt[-e + (c*f)/d]*g - b^2*d^2*e^3*Sqrt[-e + (c*f)/d]*h + b^2*c*d*e^2*f*Sqrt[-e + (c*f)/d]*h
+ a*b*d^2*e^2*f*Sqrt[-e + (c*f)/d]*h - a*b*c*d*e*f^2*Sqrt[-e + (c*f)/d]*h - b^2*d^2*e*f*Sqrt[-e + (c*f)/d]*g*(
e + f*x) + a*b*d^2*f^2*Sqrt[-e + (c*f)/d]*g*(e + f*x) + 2*b^2*d^2*e^2*Sqrt[-e + (c*f)/d]*h*(e + f*x) - b^2*c*d
*e*f*Sqrt[-e + (c*f)/d]*h*(e + f*x) - 2*a*b*d^2*e*f*Sqrt[-e + (c*f)/d]*h*(e + f*x) + a*b*c*d*f^2*Sqrt[-e + (c*
f)/d]*h*(e + f*x) - b^2*d^2*e*Sqrt[-e + (c*f)/d]*h*(e + f*x)^2 + a*b*d^2*f*Sqrt[-e + (c*f)/d]*h*(e + f*x)^2 +
I*b*d*(b*e - a*f)*(d*e - c*f)*h*Sqrt[(f*(c + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*Sqrt[(f*(g + h*x))/(h*(e + f
*x))]*EllipticE[I*ArcSinh[Sqrt[-e + (c*f)/d]/Sqrt[e + f*x]], (d*(-(f*g) + e*h))/((d*e - c*f)*h)] + I*b*(-(b*c)
+ a*d)*f*(d*e - c*f)*h*Sqrt[(f*(c + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*Sqrt[(f*(g + h*x))/(h*(e + f*x))]*El
lipticF[I*ArcSinh[Sqrt[-e + (c*f)/d]/Sqrt[e + f*x]], (d*(-(f*g) + e*h))/((d*e - c*f)*h)] - I*b^2*c^2*f^2*h*Sqr
t[(f*(c + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*Sqrt[(f*(g + h*x))/(h*(e + f*x))]*EllipticPi[(b*d*e - a*d*f)/(b
*d*e - b*c*f), I*ArcSinh[Sqrt[-e + (c*f)/d]/Sqrt[e + f*x]], (d*(-(f*g) + e*h))/((d*e - c*f)*h)] + (2*I)*a*b*c*
d*f^2*h*Sqrt[(f*(c + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*Sqrt[(f*(g + h*x))/(h*(e + f*x))]*EllipticPi[(b*d*e
- a*d*f)/(b*d*e - b*c*f), I*ArcSinh[Sqrt[-e + (c*f)/d]/Sqrt[e + f*x]], (d*(-(f*g) + e*h))/((d*e - c*f)*h)] - I
*a^2*d^2*f^2*h*Sqrt[(f*(c + d*x))/(d*(e + f*x))]*(e + f*x)^(3/2)*Sqrt[(f*(g + h*x))/(h*(e + f*x))]*EllipticPi[
(b*d*e - a*d*f)/(b*d*e - b*c*f), I*ArcSinh[Sqrt[-e + (c*f)/d]/Sqrt[e + f*x]], (d*(-(f*g) + e*h))/((d*e - c*f)*
h)]))/(b^2*f^2*(-(b*e) + a*f)*Sqrt[-e + (c*f)/d]*h*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])

Maple [A] (verified)

Time = 1.48 (sec) , antiderivative size = 769, normalized size of antiderivative = 1.71

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (f x +e \right ) \left (h x +g \right )}\, \left (-\frac {2 d \left (a d -2 b c \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 )}{b^{2} \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} \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 )}{b \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 (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\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}}}\, \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 )}{b^{3} \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}}\)	\(769\)
default	\(\text {Expression too large to display}\)	\(1551\)

[In]

int((d*x+c)^(3/2)/(b*x+a)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x,method=_RETURNVERBOSE)

[Out]

((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*(a*d-2*b*c)/b^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*d^2/b*(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),((-g/h+e/f)/(-g/h+c/d))^(1/2)))+2*
(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^3*(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)*EllipticP
i(((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)))

Fricas [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]

[In]

integrate((d*x+c)^(3/2)/(b*x+a)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\left (c + d x\right )^{\frac {3}{2}}}{\left (a + b x\right ) \sqrt {e + f x} \sqrt {g + h x}}\, dx \]

[In]

integrate((d*x+c)**(3/2)/(b*x+a)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Integral((c + d*x)**(3/2)/((a + b*x)*sqrt(e + f*x)*sqrt(g + h*x)), x)

Maxima [F]

\[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{{\left (b x + a\right )} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

integrate((d*x+c)^(3/2)/(b*x+a)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")

[Out]

integrate((d*x + c)^(3/2)/((b*x + a)*sqrt(f*x + e)*sqrt(h*x + g)), x)

Giac [F]

\[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {3}{2}}}{{\left (b x + a\right )} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

integrate((d*x+c)^(3/2)/(b*x+a)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")

[Out]

integrate((d*x + c)^(3/2)/((b*x + a)*sqrt(f*x + e)*sqrt(h*x + g)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{3/2}}{(a+b x) \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\left (a+b\,x\right )} \,d x \]

[In]

int((c + d*x)^(3/2)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)),x)

[Out]

int((c + d*x)^(3/2)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)), x)