3.68 \(\int \frac{c i+d i x}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)

Optimal. Leaf size=137 \[ \frac{2 i \sqrt{c+d x} \sqrt{e h-f g} \sqrt{\frac{f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac{\sqrt{h} \sqrt{e+f x}}{\sqrt{e h-f g}}\right )|-\frac{d (f g-e h)}{(d e-c f) h}\right )}{f \sqrt{h} \sqrt{g+h x} \sqrt{-\frac{f (c+d x)}{d e-c f}}} \]

[Out]

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

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Rubi [A]  time = 0.0612999, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {21, 114, 113} \[ \frac{2 i \sqrt{c+d x} \sqrt{e h-f g} \sqrt{\frac{f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac{\sqrt{h} \sqrt{e+f x}}{\sqrt{e h-f g}}\right )|-\frac{d (f g-e h)}{(d e-c f) h}\right )}{f \sqrt{h} \sqrt{g+h x} \sqrt{-\frac{f (c+d x)}{d e-c f}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 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])

Rubi steps

\begin{align*} \int \frac{68 c+68 d x}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx &=68 \int \frac{\sqrt{c+d x}}{\sqrt{e+f x} \sqrt{g+h x}} \, dx\\ &=\frac{\left (68 \sqrt{c+d x} \sqrt{\frac{f (g+h x)}{f g-e h}}\right ) \int \frac{\sqrt{\frac{c f}{-d e+c f}+\frac{d f x}{-d e+c f}}}{\sqrt{e+f x} \sqrt{\frac{f g}{f g-e h}+\frac{f h x}{f g-e h}}} \, dx}{\sqrt{\frac{f (c+d x)}{-d e+c f}} \sqrt{g+h x}}\\ &=\frac{136 \sqrt{-f g+e h} \sqrt{c+d x} \sqrt{\frac{f (g+h x)}{f g-e h}} E\left (\sin ^{-1}\left (\frac{\sqrt{h} \sqrt{e+f x}}{\sqrt{-f g+e h}}\right )|-\frac{d (f g-e h)}{(d e-c f) h}\right )}{f \sqrt{h} \sqrt{-\frac{f (c+d x)}{d e-c f}} \sqrt{g+h x}}\\ \end{align*}

Mathematica [C]  time = 0.586411, size = 180, normalized size = 1.31 \[ -\frac{2 i i \sqrt{c+d x} \sqrt{g+h x} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{f (c+d x)}{d e-c f}}\right )|\frac{d e h-c f h}{d f g-c f h}\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{f (c+d x)}{d e-c f}}\right ),\frac{d e h-c f h}{d f g-c f h}\right )\right )}{h \sqrt{e+f x} \sqrt{\frac{f (c+d x)}{d (e+f x)}} \sqrt{\frac{d (g+h x)}{d g-c h}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*i*Sqrt[c + d*x]*Sqrt[g + h*x]*(EllipticE[I*ArcSinh[Sqrt[(f*(c + d*x))/(d*e - c*f)]], (d*e*h - c*f*h)/(
d*f*g - c*f*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)]))/(h*
Sqrt[(f*(c + d*x))/(d*(e + f*x))]*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)])

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Maple [B]  time = 0.022, size = 552, normalized size = 4. \begin{align*} 2\,{\frac{i\sqrt{dx+c}\sqrt{fx+e}\sqrt{hx+g}}{dfh \left ( dfh{x}^{3}+cfh{x}^{2}+deh{x}^{2}+dfg{x}^{2}+cehx+cfgx+degx+ceg \right ) } \left ({\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ){c}^{2}fh-{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) cdeh-{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) cdfg+{\it EllipticF} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ){d}^{2}eg-{\it EllipticE} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ){c}^{2}fh+{\it EllipticE} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) cdeh+{\it EllipticE} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) cdfg-{\it EllipticE} \left ( \sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ){d}^{2}eg \right ) \sqrt{-{\frac{ \left ( fx+e \right ) d}{cf-de}}}\sqrt{-{\frac{ \left ( hx+g \right ) d}{ch-dg}}}\sqrt{{\frac{ \left ( dx+c \right ) f}{cf-de}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*i*(EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*c^2*f*h-EllipticF(((d*x+c)*f/(c*f-
d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*c*d*e*h-EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h
-d*g))^(1/2))*c*d*f*g+EllipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*d^2*e*g-EllipticE
(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*c^2*f*h+EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((
c*f-d*e)*h/f/(c*h-d*g))^(1/2))*c*d*e*h+EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*
c*d*f*g-EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*d^2*e*g)/d*(-(f*x+e)*d/(c*f-d*e
))^(1/2)*(-(h*x+g)*d/(c*h-d*g))^(1/2)*((d*x+c)*f/(c*f-d*e))^(1/2)/h/f*(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., size = 0, normalized size = 0. \begin{align*} \int \frac{d i x + c i}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g} i}{f h x^{2} + e g +{\left (f g + e h\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} i \int \frac{\sqrt{c + d x}}{\sqrt{e + f x} \sqrt{g + h x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{d i x + c i}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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