\(\int \frac {\sqrt {a+b x} (d e+c f+2 d f x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [12]

Optimal result

Integrand size = 49, antiderivative size = 472 \[ \int \frac {\sqrt {a+b x} (d e+c f+2 d f x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{h \sqrt {a+b x}}-\frac {2 \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}} E\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{h \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {2 d (b g-a h)^{3/2} \sqrt {\frac {(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt {\frac {(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x) \operatorname {EllipticPi}\left (\frac {f (b g-a h)}{(b e-a f) h},\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {g+h x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{\sqrt {b e-a f} h^2 \sqrt {a+b x} \sqrt {c+d x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.102, Rules used = {1609, 171, 551, 182, 435} \[ \int \frac {\sqrt {a+b x} (d e+c f+2 d f x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 d (e+f x) (b g-a h)^{3/2} \sqrt {\frac {(a+b x) (f g-e h)}{(e+f x) (b g-a h)}} \sqrt {\frac {(c+d x) (f g-e h)}{(e+f x) (d g-c h)}} \operatorname {EllipticPi}\left (\frac {f (b g-a h)}{(b e-a f) h},\arcsin \left (\frac {\sqrt {b e-a f} \sqrt {g+h x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right ),\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{h^2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {b e-a f}}-\frac {2 \sqrt {c+d x} \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}} E\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{h \sqrt {g+h x} \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}}}+\frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{h \sqrt {a+b x}} \]

[In]

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

[Out]

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

Rule 171

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

Rule 182

Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[-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]*Sqrt[(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]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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
]

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 1609

Int[(Sqrt[(a_.) + (b_.)*(x_)]*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g
_.) + (h_.)*(x_)]), x_Symbol] :> Simp[b*B*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(d*f*h*Sqrt[a + b*x])), x
] + (-Dist[B*((b*g - a*h)/(2*f*h)), Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g + h*x]), x], x] + Di
st[B*(b*e - a*f)*((b*g - a*h)/(2*d*f*h)), Int[Sqrt[c + d*x]/((a + b*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x],
x]) /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && EqQ[2*A*d*f - B*(d*e + c*f), 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{h \sqrt {a+b x}}-\frac {(d (b g-a h)) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}} \, dx}{h}+\frac {((b e-a f) (b g-a h)) \int \frac {\sqrt {c+d x}}{(a+b x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx}{h} \\ & = \frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{h \sqrt {a+b x}}-\frac {\left (2 d (b g-a h) \sqrt {\frac {(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt {\frac {(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\left (h-f x^2\right ) \sqrt {1+\frac {(-b e+a f) x^2}{b g-a h}} \sqrt {1+\frac {(-d e+c f) x^2}{d g-c h}}} \, dx,x,\frac {\sqrt {g+h x}}{\sqrt {e+f x}}\right )}{h \sqrt {a+b x} \sqrt {c+d x}}-\frac {\left (2 (b g-a h) \sqrt {c+d x} \sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {(b c-a d) x^2}{d e-c f}}}{\sqrt {1-\frac {(b g-a h) x^2}{f g-e h}}} \, dx,x,\frac {\sqrt {e+f x}}{\sqrt {a+b x}}\right )}{h \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}} \\ & = \frac {2 b \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{h \sqrt {a+b x}}-\frac {2 \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)}} E\left (\sin ^{-1}\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 )}{h \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}}-\frac {2 d (b g-a h)^{3/2} \sqrt {\frac {(f g-e h) (a+b x)}{(b g-a h) (e+f x)}} \sqrt {\frac {(f g-e h) (c+d x)}{(d g-c h) (e+f x)}} (e+f x) \Pi \left (\frac {f (b g-a h)}{(b e-a f) h};\sin ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {g+h x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right )|\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{\sqrt {b e-a f} h^2 \sqrt {a+b x} \sqrt {c+d x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 36.16 (sec) , antiderivative size = 443, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+b x} (d e+c f+2 d f x)}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \left (-\frac {d h (e+f x) (g+h x)}{c+d x}-\frac {(f g-e h) \sqrt {\frac {(-d e+c f) (d g-c h) (e+f x) (g+h x)}{(f g-e h)^2 (c+d x)^2}} \left ((d e-c f) h E\left (\arcsin \left (\sqrt {\frac {(-d e+c f) (g+h x)}{(f g-e h) (c+d x)}}\right )|\frac {(b c-a d) (-f g+e h)}{(d e-c f) (b g-a h)}\right )+(-d e h+c f h) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(-d e+c f) (g+h x)}{(f g-e h) (c+d x)}}\right ),\frac {(b c-a d) (-f g+e h)}{(d e-c f) (b g-a h)}\right )+f (d g-c h) \operatorname {EllipticPi}\left (\frac {d (-f g+e h)}{(d e-c f) h},\arcsin \left (\sqrt {\frac {(-d e+c f) (g+h x)}{(f g-e h) (c+d x)}}\right ),\frac {(b c-a d) (-f g+e h)}{(d e-c f) (b g-a h)}\right )\right )}{(d e-c f) \sqrt {\frac {(d g-c h) (a+b x)}{(b g-a h) (c+d x)}}}\right )}{h^2 \sqrt {e+f x} \sqrt {g+h x}} \]

[In]

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

[Out]

(-2*Sqrt[a + b*x]*Sqrt[c + d*x]*(-((d*h*(e + f*x)*(g + h*x))/(c + d*x)) - ((f*g - e*h)*Sqrt[((-(d*e) + c*f)*(d
*g - c*h)*(e + f*x)*(g + h*x))/((f*g - e*h)^2*(c + d*x)^2)]*((d*e - c*f)*h*EllipticE[ArcSin[Sqrt[((-(d*e) + c*
f)*(g + h*x))/((f*g - e*h)*(c + d*x))]], ((b*c - a*d)*(-(f*g) + e*h))/((d*e - c*f)*(b*g - a*h))] + (-(d*e*h) +
c*f*h)*EllipticF[ArcSin[Sqrt[((-(d*e) + c*f)*(g + h*x))/((f*g - e*h)*(c + d*x))]], ((b*c - a*d)*(-(f*g) + e*h
))/((d*e - c*f)*(b*g - a*h))] + f*(d*g - c*h)*EllipticPi[(d*(-(f*g) + e*h))/((d*e - c*f)*h), ArcSin[Sqrt[((-(d
*e) + c*f)*(g + h*x))/((f*g - e*h)*(c + d*x))]], ((b*c - a*d)*(-(f*g) + e*h))/((d*e - c*f)*(b*g - a*h))]))/((d
*e - c*f)*Sqrt[((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])))/(h^2*Sqrt[e + f*x]*Sqrt[g + h*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1559\) vs. \(2(426)=852\).

Time = 5.17 (sec) , antiderivative size = 1560, normalized size of antiderivative = 3.31

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1560\)
default	\(\text {Expression too large to display}\)	\(13180\)

[In]

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

[Out]

((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*(a*c*f+a*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*(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)*(-c/d*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))+(c/d-a/b)*EllipticPi(((-g/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c/d))^(1/2),
(-g/h+a/b)/(-g/h+c/d),((e/f-c/d)*(g/h-a/b)/(-a/b+e/f)/(-c/d+g/h))^(1/2)))+2*b*d*f*((x+a/b)*(x+e/f)*(x+g/h)+(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)*((a*c/b/d-g/h*a/b+g/h*c/d+c^2/d^2)/(-g/h+c/d)/(-c/d+a/b)*Elliptic
F(((-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))+(-a/b+e/f)*
EllipticE(((-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))/(-c
/d+a/b)+(a*d*f*h+b*c*f*h+b*d*e*h+b*d*f*g)/b/d/f/h/(-g/h+c/d)*EllipticPi(((-g/h+c/d)*(x+a/b)/(-g/h+a/b)/(x+c/d)
)^(1/2),(g/h-a/b)/(-c/d+g/h),((e/f-c/d)*(g/h-a/b)/(-a/b+e/f)/(-c/d+g/h))^(1/2))))/(b*d*f*h*(x+a/b)*(x+c/d)*(x+
e/f)*(x+g/h))^(1/2))

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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