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

Optimal. Leaf size=293 \[ \frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {h (d e-c f)}{f (d g-c h)}\right )}{b \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \]

[Out]

2*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/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)-2*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/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)

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Rubi [A]  time = 0.50, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {175, 121, 120, 169, 538, 537} \[ \frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}}-\frac {2 \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \Pi \left (-\frac {b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt {f} \sqrt {e+f x} \sqrt {g+h x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*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*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x]
) - (2*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*Sqrt[f]*Sqrt[e + f*x]*Sqrt[g + h*x])

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*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))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &
& SimplerQ[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 121

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 169

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 175

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

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)
])

Rule 538

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

Rubi steps

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

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Mathematica [C]  time = 1.77, size = 202, normalized size = 0.69 \[ -\frac {2 i \sqrt {c+d x} \sqrt {\frac {d (g+h x)}{d g-c h}} \left (\operatorname {EllipticF}\left (i \sinh ^{-1}\left (\sqrt {\frac {f (c+d x)}{d e-c f}}\right ),\frac {d e h-c f h}{d f g-c f h}\right )-\Pi \left (\frac {b (c f-d e)}{(b c-a d) f};i \sinh ^{-1}\left (\sqrt {\frac {f (c+d x)}{d e-c f}}\right )|\frac {d e h-c f h}{d f g-c f h}\right )\right )}{b \sqrt {e+f x} \sqrt {g+h x} \sqrt {\frac {f (c+d x)}{d (e+f x)}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

Timed out

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

[Out]

Timed out

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maple [A]  time = 0.05, size = 382, normalized size = 1.30 \[ \frac {2 \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}\, \sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}\, \sqrt {-\frac {\left (h x +g \right ) d}{c h -d g}}\, \sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}\, \left (c f \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-c f \EllipticPi \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, -\frac {\left (c f -d e \right ) b}{\left (a d -b c \right ) f}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-d e \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+d e \EllipticPi \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, -\frac {\left (c f -d e \right ) b}{\left (a d -b c \right ) f}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )\right )}{\left (d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g \right ) b f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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