\(\int \frac {\sqrt {c+d x}}{(a+b x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) [100]
Optimal result
Integrand size = 37, antiderivative size = 161 \[
\int \frac {\sqrt {c+d x}}{(a+b x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \sqrt {c+d x} E\left (\arctan \left (\frac {\sqrt {-b e+a f} \sqrt {g+h x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right )|\frac {(-b c+a d) (f g-e h)}{(-b e+a f) (d g-c h)}\right )}{\sqrt {-b e+a f} \sqrt {b g-a h} \sqrt {a+b x} \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}}}
\]
[Out]
-2*(1/(1+(a*f-b*e)*(h*x+g)/(-a*h+b*g)/(f*x+e)))^(1/2)*(1+(a*f-b*e)*(h*x+g)/(-a*h+b*g)/(f*x+e))^(1/2)*EllipticE
((a*f-b*e)^(1/2)*(h*x+g)^(1/2)/(-a*h+b*g)^(1/2)/(f*x+e)^(1/2)/(1+(a*f-b*e)*(h*x+g)/(-a*h+b*g)/(f*x+e))^(1/2),(
(a*d-b*c)*(-e*h+f*g)/(a*f-b*e)/(-c*h+d*g))^(1/2))*(d*x+c)^(1/2)/(a*f-b*e)^(1/2)/(-a*h+b*g)^(1/2)/(b*x+a)^(1/2)
/((-a*h+b*g)*(d*x+c)/(-c*h+d*g)/(b*x+a))^(1/2)
Rubi [A] (verified)
Time = 0.06 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.29,
number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {182, 435}
\[
\int \frac {\sqrt {c+d x}}{(a+b x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=-\frac {2 \sqrt {c+d x} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}} E\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {g+h x} (b e-a f) \sqrt {b g-a h} \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}}}
\]
[In]
Int[Sqrt[c + d*x]/((a + b*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
(-2*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[(S
qrt[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)))])/((b*e - a*f)*Sqrt[b*g - a*h]*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[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
]
Rubi steps \begin{align*}
\text {integral}& = -\frac {\left (2 \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 )}{(b e-a f) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}} \\ & = -\frac {2 \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 )}{(b e-a f) \sqrt {b g-a h} \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x}} \\
\end{align*}
Mathematica [A] (verified)
Time = 23.63 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.39
\[
\int \frac {\sqrt {c+d x}}{(a+b x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 (f g-e h) \sqrt {a+b x} \sqrt {c+d x} \sqrt {\frac {(-b e+a f) (b g-a h) (e+f x) (g+h x)}{(f g-e h)^2 (a+b x)^2}} E\left (\arcsin \left (\sqrt {\frac {(-b e+a f) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{(b e-a f) (b g-a h) \sqrt {\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt {e+f x} \sqrt {g+h x}}
\]
[In]
Integrate[Sqrt[c + d*x]/((a + b*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
(2*(f*g - e*h)*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[((-(b*e) + a*f)*(b*g - a*h)*(e + f*x)*(g + h*x))/((f*g - e*h)^
2*(a + b*x)^2)]*EllipticE[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*e - a*f)*(b*g - a*h)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*
x))]*Sqrt[e + f*x]*Sqrt[g + h*x])
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(1947\) vs. \(2(251)=502\).
Time = 3.96 (sec) , antiderivative size = 1948, normalized size of antiderivative =
12.10
| | |
| method | result | size |
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| elliptic |
\(\text {Expression too large to display}\) |
\(1948\) |
| default |
\(\text {Expression too large to display}\) |
\(4561\) |
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[In]
int((d*x+c)^(1/2)/(b*x+a)^(3/2)/(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*(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^2*f*h-a*b*e*h-a*b*f*g+b^2*e*g
)/((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*(d
/b-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)/(a^2*f*h-a*b*e*h-a*b*f*g+b^2*e*
g)+(b*c*e*h+b*c*f*g+b*d*e*g)/(a^2*f*h-a*b*e*h-a*b*f*g+b^2*e*g))*(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)/(a^2
*f*h-a*b*e*h-a*b*f*g+b^2*e*g)+(2*b*c*f*h+2*b*d*e*h+2*b*d*f*g)/(a^2*f*h-a*b*e*h-a*b*f*g+b^2*e*g))*(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*h/(a^2*f*h-a*b*e*h-a*b*f*g+b^2*e*g)*((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)*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))+(-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]
\[
\int \frac {\sqrt {c+d x}}{(a+b x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {f x + e} \sqrt {h x + g}} \,d x }
\]
[In]
integrate((d*x+c)^(1/2)/(b*x+a)^(3/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")
[Out]
integral(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)/(b^2*f*h*x^4 + a^2*e*g + (b^2*f*g + (b^2*e +
2*a*b*f)*h)*x^3 + ((b^2*e + 2*a*b*f)*g + (2*a*b*e + a^2*f)*h)*x^2 + (a^2*e*h + (2*a*b*e + a^2*f)*g)*x), x)
Sympy [F]
\[
\int \frac {\sqrt {c+d x}}{(a+b x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\sqrt {c + d x}}{\left (a + b x\right )^{\frac {3}{2}} \sqrt {e + f x} \sqrt {g + h x}}\, dx
\]
[In]
integrate((d*x+c)**(1/2)/(b*x+a)**(3/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
[Out]
Integral(sqrt(c + d*x)/((a + b*x)**(3/2)*sqrt(e + f*x)*sqrt(g + h*x)), x)
Maxima [F]
\[
\int \frac {\sqrt {c+d x}}{(a+b x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {f x + e} \sqrt {h x + g}} \,d x }
\]
[In]
integrate((d*x+c)^(1/2)/(b*x+a)^(3/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(d*x + c)/((b*x + a)^(3/2)*sqrt(f*x + e)*sqrt(h*x + g)), x)
Giac [F]
\[
\int \frac {\sqrt {c+d x}}{(a+b x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {\sqrt {d x + c}}{{\left (b x + a\right )}^{\frac {3}{2}} \sqrt {f x + e} \sqrt {h x + g}} \,d x }
\]
[In]
integrate((d*x+c)^(1/2)/(b*x+a)^(3/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(d*x + c)/((b*x + a)^(3/2)*sqrt(f*x + e)*sqrt(h*x + g)), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {\sqrt {c+d x}}{(a+b x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {\sqrt {c+d\,x}}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{3/2}} \,d x
\]
[In]
int((c + d*x)^(1/2)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)^(3/2)),x)
[Out]
int((c + d*x)^(1/2)/((e + f*x)^(1/2)*(g + h*x)^(1/2)*(a + b*x)^(3/2)), x)