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

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

[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
])

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Rubi [A]  time = 2.3379, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ \frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}}-\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b \sqrt{f} \sqrt{e+f x} \sqrt{g+h x}} \]

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
])

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Rubi in Sympy [A]  time = 104.411, size = 240, normalized size = 0.82 \[ - \frac{2 \sqrt{\frac{d \left (- e - f x\right )}{c f - d e}} \sqrt{\frac{d \left (- g - h x\right )}{c h - d g}} \Pi \left (- \frac{b \left (c h - d g\right )}{h \left (a d - b c\right )}; \operatorname{asin}{\left (\sqrt{\frac{h}{c h - d g}} \sqrt{c + d x} \right )}\middle | \frac{f \left (c h - d g\right )}{h \left (c f - d e\right )}\right )}{b \sqrt{\frac{h}{c h - d g}} \sqrt{e + f x} \sqrt{g + h x}} + \frac{2 \sqrt{\frac{d \left (- e - f x\right )}{c f - d e}} \sqrt{\frac{d \left (- g - h x\right )}{c h - d g}} \sqrt{c f - d e} F\left (\operatorname{asin}{\left (\frac{\sqrt{f} \sqrt{c + d x}}{\sqrt{c f - d e}} \right )}\middle | \frac{h \left (c f - d e\right )}{f \left (c h - d g\right )}\right )}{b \sqrt{f} \sqrt{e + f x} \sqrt{g + h x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(1/2)/(b*x+a)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

-2*sqrt(d*(-e - f*x)/(c*f - d*e))*sqrt(d*(-g - h*x)/(c*h - d*g))*elliptic_pi(-b*
(c*h - d*g)/(h*(a*d - b*c)), asin(sqrt(h/(c*h - d*g))*sqrt(c + d*x)), f*(c*h - d
*g)/(h*(c*f - d*e)))/(b*sqrt(h/(c*h - d*g))*sqrt(e + f*x)*sqrt(g + h*x)) + 2*sqr
t(d*(-e - f*x)/(c*f - d*e))*sqrt(d*(-g - h*x)/(c*h - d*g))*sqrt(c*f - d*e)*ellip
tic_f(asin(sqrt(f)*sqrt(c + d*x)/sqrt(c*f - d*e)), h*(c*f - d*e)/(f*(c*h - d*g))
)/(b*sqrt(f)*sqrt(e + f*x)*sqrt(g + h*x))

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Mathematica [C]  time = 1.17099, size = 202, normalized size = 0.69 \[ -\frac{2 i \sqrt{c+d x} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (F\left (i \sinh ^{-1}\left (\sqrt{\frac{f (c+d x)}{d e-c f}}\right )|\frac{d e h-c f h}{d f g-c f h}\right )-\Pi \left (\frac{b (c f-d e)}{(b c-a d) f};i \sinh ^{-1}\left (\sqrt{\frac{f (c+d x)}{d e-c f}}\right )|\frac{d e h-c f h}{d f g-c f h}\right )\right )}{b \sqrt{e+f x} \sqrt{g+h x} \sqrt{\frac{f (c+d x)}{d (e+f x)}}} \]

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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Maple [A]  time = 0.069, size = 382, normalized size = 1.3 \[ 2\,{\frac{\sqrt{dx+c}\sqrt{fx+e}\sqrt{hx+g}}{fb \left ( dfh{x}^{3}+cfh{x}^{2}+deh{x}^{2}+dfg{x}^{2}+cehx+cfgx+degx+ceg \right ) }\sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}}\sqrt{-{\frac{d \left ( hx+g \right ) }{ch-dg}}}\sqrt{-{\frac{d \left ( fx+e \right ) }{cf-de}}} \left ({\it EllipticF} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) cf-{\it EllipticF} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) de-{\it EllipticPi} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},-{\frac{ \left ( cf-de \right ) b}{f \left ( ad-bc \right ) }},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) cf+{\it EllipticPi} \left ( \sqrt{{\frac{f \left ( dx+c \right ) }{cf-de}}},-{\frac{ \left ( cf-de \right ) b}{f \left ( ad-bc \right ) }},\sqrt{{\frac{ \left ( cf-de \right ) h}{f \left ( ch-dg \right ) }}} \right ) de \right ) } \]

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)/b/f*((d*x+c)*f/(c*f-d*e))^(1/2)*(-(h
*x+g)*d/(c*h-d*g))^(1/2)*(-(f*x+e)*d/(c*f-d*e))^(1/2)*(EllipticF(((d*x+c)*f/(c*f
-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*c*f-EllipticF(((d*x+c)*f/(c*f-d*e)
)^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*d*e-EllipticPi(((d*x+c)*f/(c*f-d*e))^(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)*f/(c*f-d*e))^(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., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x + c}}{{\left (b x + a\right )} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)/((b*x + a)*sqrt(f*x + e)*sqrt(h*x + g)),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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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)/((b*x + a)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="fricas")

[Out]

Timed out

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)/((b*x + a)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")

[Out]

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