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

Optimal. Leaf size=721 \[ \frac {\sqrt {g+h x} (d e-c f) (-2 a f h+b e h+b f g) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {e+f x} \sqrt {b g-a h}}{\sqrt {a+b x} \sqrt {f g-e h}}\right ),-\frac {(b c-a d) (f g-e h)}{(b g-a h) (d e-c f)}\right )}{f^2 h \sqrt {c+d x} \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac {(e+f x) \sqrt {b g-a h} \sqrt {\frac {(a+b x) (f g-e h)}{(e+f x) (b g-a h)}} \sqrt {\frac {(c+d x) (f g-e h)}{(e+f x) (d g-c h)}} (a d f h-b (-c f h+d e h+d f g)) \Pi \left (\frac {f (b g-a h)}{(b e-a f) h};\sin ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {g+h x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right )|\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{f^2 h^2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {b e-a f}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}}{h \sqrt {e+f x}}-\frac {\sqrt {a+b x} \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {\frac {(g+h x) (d e-c f)}{(e+f x) (d g-c h)}} E\left (\sin ^{-1}\left (\frac {\sqrt {f g-e h} \sqrt {c+d x}}{\sqrt {d g-c h} \sqrt {e+f x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{f h \sqrt {g+h x} \sqrt {-\frac {(a+b x) (d e-c f)}{(e+f x) (b c-a d)}}} \]

[Out]

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

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Rubi [A]  time = 0.67, antiderivative size = 721, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {173, 176, 424, 170, 419, 165, 537} \[ \frac {(e+f x) \sqrt {b g-a h} \sqrt {\frac {(a+b x) (f g-e h)}{(e+f x) (b g-a h)}} \sqrt {\frac {(c+d x) (f g-e h)}{(e+f x) (d g-c h)}} (a d f h-b (-c f h+d e h+d f g)) \Pi \left (\frac {f (b g-a h)}{(b e-a f) h};\sin ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {g+h x}}{\sqrt {b g-a h} \sqrt {e+f x}}\right )|\frac {(d e-c f) (b g-a h)}{(b e-a f) (d g-c h)}\right )}{f^2 h^2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {b e-a f}}+\frac {\sqrt {g+h x} (d e-c f) (-2 a f h+b e h+b f g) \sqrt {\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} F\left (\sin ^{-1}\left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right )|-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{f^2 h \sqrt {c+d x} \sqrt {b g-a h} \sqrt {f g-e h} \sqrt {-\frac {(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac {\sqrt {a+b x} \sqrt {c+d x} \sqrt {g+h x}}{h \sqrt {e+f x}}-\frac {\sqrt {a+b x} \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {\frac {(g+h x) (d e-c f)}{(e+f x) (d g-c h)}} E\left (\sin ^{-1}\left (\frac {\sqrt {f g-e h} \sqrt {c+d x}}{\sqrt {d g-c h} \sqrt {e+f x}}\right )|\frac {(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{f h \sqrt {g+h x} \sqrt {-\frac {(a+b x) (d e-c f)}{(e+f x) (b c-a d)}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 165

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 170

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Dist[(2*Sqrt[g + h*x]*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))])/((f*g - e*h)*Sqrt[c +
d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]), Subst[Int[1/(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 173

Int[(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)])/(Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x
_Symbol] :> Simp[(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g + h*x])/(h*Sqrt[e + f*x]), x] + (-Dist[((d*e - c*f)*(f*g
- e*h))/(2*f*h), Int[Sqrt[a + b*x]/(Sqrt[c + d*x]*(e + f*x)^(3/2)*Sqrt[g + h*x]), x], x] + Dist[((d*e - c*f)*(
b*f*g + b*e*h - 2*a*f*h))/(2*f^2*h), Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] +
Dist[(a*d*f*h - b*(d*f*g + d*e*h - c*f*h))/(2*f^2*h), Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[g +
h*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 176

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 419

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

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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)
])

Rubi steps

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

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Mathematica [B]  time = 15.05, size = 6667, normalized size = 9.25 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b x + a} \sqrt {d x + c}}{\sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.16, size = 18077, normalized size = 25.07 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b x + a} \sqrt {d x + c}}{\sqrt {f x + e} \sqrt {h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {a+b\,x}\,\sqrt {c+d\,x}}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a + b x} \sqrt {c + d x}}{\sqrt {e + f x} \sqrt {g + h x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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