\(\int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [71]

Optimal result

Integrand size = 35, antiderivative size = 393 \[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (d e-c f) (d g-c h) \sqrt {c+d x}}-\frac {2 d \sqrt {h} \sqrt {-f g+e h} \sqrt {c+d x} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {-f g+e h}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{(b c-a d) (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}} \sqrt {g+h x}}-\frac {2 b \sqrt {-d e+c f} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {-d e+c f}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{(b c-a d)^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \]

[Out]

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*Ellipti
cE(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)

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {185, 106, 21, 115, 114, 175, 552, 551} \[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 d \sqrt {h} \sqrt {c+d x} \sqrt {e h-f g} \sqrt {\frac {f (g+h x)}{f g-e h}} E\left (\arcsin \left (\frac {\sqrt {h} \sqrt {e+f x}}{\sqrt {e h-f g}}\right )|-\frac {d (f g-e h)}{(d e-c f) h}\right )}{\sqrt {g+h x} (b c-a d) (d e-c f) (d g-c h) \sqrt {-\frac {f (c+d x)}{d e-c f}}}-\frac {2 b \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticPi}\left (-\frac {b (d e-c f)}{(b c-a d) f},\arcsin \left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {(d e-c f) h}{f (d g-c h)}\right )}{\sqrt {f} \sqrt {e+f x} \sqrt {g+h x} (b c-a d)^2}+\frac {2 d^2 \sqrt {e+f x} \sqrt {g+h x}}{\sqrt {c+d x} (b c-a d) (d e-c f) (d g-c h)} \]

[In]

Int[1/((a + b*x)*(c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(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]) - (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])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 106

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[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]

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 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) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}+\frac {b}{(b c-a d) (a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}\right ) \, dx \\ & = \frac {b \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b c-a d}-\frac {d \int \frac {1}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{b c-a d} \\ & = \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 b) \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 c-a d}+\frac {(2 d) \int \frac {-\frac {1}{2} c f h-\frac {1}{2} d f h x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{(b c-a d) (d e-c f) (d g-c h)} \\ & = \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 {(d f h) \int \frac {\sqrt {c+d x}}{\sqrt {e+f x} \sqrt {g+h x}} \, dx}{(b c-a d) (d e-c f) (d g-c h)}-\frac {\left (2 b \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 c-a d) \sqrt {e+f x}} \\ & = \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 {\left (2 b \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 c-a d) \sqrt {e+f x} \sqrt {g+h x}}-\frac {\left (d f h \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 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 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 (\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 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}} \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 c-a d)^2 \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.22 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 i (c+d x) \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \left ((b c-a d) h E\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )+(b d g-2 b c h+a d h) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )+b (-d g+c h) \operatorname {EllipticPi}\left (-\frac {b c f-a d f}{b d e-b c f},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {d e}{f}}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )\right )}{(b c-a d)^2 \sqrt {-c+\frac {d e}{f}} (-d g+c h) \sqrt {e+f x} \sqrt {g+h x}} \]

[In]

Integrate[1/((a + b*x)*(c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

((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)*Elli
pticF[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)*Ellipti
cPi[-((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])

Maple [B] (verified)

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	\(\text {Expression too large to display}\)	\(1978\)

[In]

int(1/(b*x+a)/(d*x+c)^(3/2)/(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*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)*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*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)))

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {1}{\left (a + b x\right ) \left (c + d x\right )^{\frac {3}{2}} \sqrt {e + f x} \sqrt {g + h x}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {1}{{\left (b x + a\right )} {\left (d x + c\right )}^{\frac {3}{2}} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {1}{{\left (b x + a\right )} {\left (d x + c\right )}^{\frac {3}{2}} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b x) (c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {1}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\left (a+b\,x\right )\,{\left (c+d\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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