3.3 \(\int \frac {A+B x}{\sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\)

Optimal. Leaf size=284 \[ \frac {2 B \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 (B g-A h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right ),\frac {h (d e-c f)}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}} \]

[Out]

2*B*EllipticE(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)*(h*x+g)^(1/2)/d/h/f^(1/2)/(f*x+e)^(1/2)/(d*(h*x+g)/(-c*h+d*g))^(1/2)-2*(-A*h+B*g)*Ellipt
icF(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)/d/h/f^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)

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Rubi [A]  time = 0.17, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {158, 114, 113, 121, 120} \[ \frac {2 B \sqrt {g+h x} \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {\frac {d (g+h x)}{d g-c h}}}-\frac {2 (B g-A h) \sqrt {c f-d e} \sqrt {\frac {d (e+f x)}{d e-c f}} \sqrt {\frac {d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {c f-d e}}\right )|\frac {(d e-c f) h}{f (d g-c h)}\right )}{d \sqrt {f} h \sqrt {e+f x} \sqrt {g+h x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*B*Sqrt[-(d*e) + c*f]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])
/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(d*Sqrt[f]*h*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c
*h)]) - (2*Sqrt[-(d*e) + c*f]*(B*g - A*h)*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*Elli
pticF[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])/(d*Sqrt[f]*h*Sqrt[
e + f*x]*Sqrt[g + h*x])

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*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))])/b, 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 114

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 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 158

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

Rubi steps

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

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Mathematica [C]  time = 1.95, size = 319, normalized size = 1.12 \[ -\frac {2 \left (i d h (c+d x)^{3/2} (B e-A f) \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {d e}{f}-c}}{\sqrt {c+d x}}\right ),\frac {d f g-c f h}{d e h-c f h}\right )-B d^2 (e+f x) (g+h x) \sqrt {\frac {d e}{f}-c}-i B h (c+d x)^{3/2} (d e-c f) \sqrt {\frac {d (e+f x)}{f (c+d x)}} \sqrt {\frac {d (g+h x)}{h (c+d x)}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {d e}{f}-c}}{\sqrt {c+d x}}\right )|\frac {d f g-c f h}{d e h-c f h}\right )\right )}{d^2 f h \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \sqrt {\frac {d e}{f}-c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-(B*d^2*Sqrt[-c + (d*e)/f]*(e + f*x)*(g + h*x)) - I*B*(d*e - c*f)*h*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f
*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g -
c*f*h)/(d*e*h - c*f*h)] + I*d*(B*e - A*f)*h*(c + d*x)^(3/2)*Sqrt[(d*(e + f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*
x))/(h*(c + d*x))]*EllipticF[I*ArcSinh[Sqrt[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)]))/(
d^2*Sqrt[-c + (d*e)/f]*f*h*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])

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fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x + A\right )} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}}{d f h x^{3} + c e g + {\left (d f g + {\left (d e + c f\right )} h\right )} x^{2} + {\left (c e h + {\left (d e + c f\right )} g\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*x + A)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)/(d*f*h*x^3 + c*e*g + (d*f*g + (d*e + c*f)*h)*x^2
+ (c*e*h + (d*e + c*f)*g)*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*x + A)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)

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maple [B]  time = 0.03, size = 559, normalized size = 1.97 \[ \frac {2 \left (A c d f h \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-A \,d^{2} e h \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-B \,c^{2} f h \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+B c d e h \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+B c d f g \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-B c d f g \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )-B \,d^{2} e g \EllipticE \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )+B \,d^{2} e g \EllipticF \left (\sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}, \sqrt {\frac {\left (c f -d e \right ) h}{\left (c h -d g \right ) f}}\right )\right ) \sqrt {-\frac {\left (f x +e \right ) d}{c f -d e}}\, \sqrt {-\frac {\left (h x +g \right ) d}{c h -d g}}\, \sqrt {\frac {\left (d x +c \right ) f}{c f -d e}}\, \sqrt {d x +c}\, \sqrt {f x +e}\, \sqrt {h x +g}}{\left (d f h \,x^{3}+c f h \,x^{2}+d e h \,x^{2}+d f g \,x^{2}+c e h x +c f g x +d e g x +c e g \right ) d^{2} f h} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(A*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*c*d*f*h-A*EllipticF(((d*x+c)/(c*f-
d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*d^2*e*h-B*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h
-d*g)/f*h)^(1/2))*c*d*f*g+B*EllipticF(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*d^2*e*g-B*E
llipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*c^2*f*h+B*EllipticE(((d*x+c)/(c*f-d*e)*f
)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*c*d*e*h+B*EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/
f*h)^(1/2))*c*d*f*g-B*EllipticE(((d*x+c)/(c*f-d*e)*f)^(1/2),((c*f-d*e)/(c*h-d*g)/f*h)^(1/2))*d^2*e*g)*(-(f*x+e
)/(c*f-d*e)*d)^(1/2)*(-(h*x+g)/(c*h-d*g)*d)^(1/2)*((d*x+c)/(c*f-d*e)*f)^(1/2)/h/f/d^2*(d*x+c)^(1/2)*(f*x+e)^(1
/2)*(h*x+g)^(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((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 \[ \int \frac {A+B\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,\sqrt {c+d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((A + B*x)/(sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x)), x)

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