3.114 \(\int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)
Optimal. Leaf size=410 \[ \frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) \left (-3 a^2 C d f h^2+3 a b B d f h^2+b^2 (-(3 B d f g h-C (c h (f g-e h)+d g (e h+2 f g))))\right )}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 b^2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} (3 B d f h-2 C (c f h+d e h+d f g)) E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 b^2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h} \]
[Out]
(2*b^2*C*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*d*f*h) + (2*b^2*Sqrt[-(d*
e) + c*f]*(3*B*d*f*h - 2*C*(d*f*g + d*e*h + c*f*h))*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))])/(3*d^2*f^(3/2)*h^2*Sqrt[e + f*x]*Sqrt[(d*(g +
h*x))/(d*g - c*h)]) + (2*Sqrt[-(d*e) + c*f]*(3*a*b*B*d*f*h^2 - 3*a^2*C*d*f*h^2 -
b^2*(3*B*d*f*g*h - C*(c*h*(f*g - e*h) + d*g*(2*f*g + e*h))))*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))])/(3*d^2*f^(3/2)*h^2
*Sqrt[e + f*x]*Sqrt[g + h*x])
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Rubi [A] time = 1.71677, antiderivative size = 410, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 53, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.113
\[ \frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) \left (-3 a^2 C d f h^2+3 a b B d f h^2+b^2 (-(3 B d f g h-c C h (f g-e h)-C d g (e h+2 f g)))\right )}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 b^2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} (3 B d f h-2 C (c f h+d e h+d f g)) E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 b^2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h} \]
Antiderivative was successfully verified.
[In] Int[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
(2*b^2*C*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*d*f*h) + (2*b^2*Sqrt[-(d*
e) + c*f]*(3*B*d*f*h - 2*C*(d*f*g + d*e*h + c*f*h))*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))])/(3*d^2*f^(3/2)*h^2*Sqrt[e + f*x]*Sqrt[(d*(g +
h*x))/(d*g - c*h)]) + (2*Sqrt[-(d*e) + c*f]*(3*a*b*B*d*f*h^2 - 3*a^2*C*d*f*h^2 -
b^2*(3*B*d*f*g*h - c*C*h*(f*g - e*h) - C*d*g*(2*f*g + e*h)))*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))])/(3*d^2*f^(3/2)*h^2
*Sqrt[e + f*x]*Sqrt[g + h*x])
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((C*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
[Out]
Timed out
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Mathematica [C] time = 11.8062, size = 442, normalized size = 1.08 \[ \frac{\sqrt{c+d x} \left (\frac{2 i d h \sqrt{c+d x} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right ) \left (-3 a^2 C d f^2 h+3 a b B d f^2 h+b^2 (-3 B d e f h+c C f (e h-f g)+C d e (2 e h+f g))\right )}{\sqrt{\frac{d e}{f}-c}}+\frac{2 b^2 d^2 (e+f x) (g+h x) (3 B d f h-2 C (c f h+d e h+d f g))}{c+d x}+2 i b^2 f h \sqrt{c+d x} \sqrt{\frac{d e}{f}-c} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} (3 B d f h-2 C (c f h+d e h+d f g)) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right )+2 b^2 C d^2 f h (e+f x) (g+h x)\right )}{3 d^3 f^2 h^2 \sqrt{e+f x} \sqrt{g+h x}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
(Sqrt[c + d*x]*(2*b^2*C*d^2*f*h*(e + f*x)*(g + h*x) + (2*b^2*d^2*(3*B*d*f*h - 2*
C*(d*f*g + d*e*h + c*f*h))*(e + f*x)*(g + h*x))/(c + d*x) + (2*I)*b^2*Sqrt[-c +
(d*e)/f]*f*h*(3*B*d*f*h - 2*C*(d*f*g + d*e*h + c*f*h))*Sqrt[c + d*x]*Sqrt[(d*(e
+ f*x))/(f*(c + d*x))]*Sqrt[(d*(g + h*x))/(h*(c + d*x))]*EllipticE[I*ArcSinh[Sqr
t[-c + (d*e)/f]/Sqrt[c + d*x]], (d*f*g - c*f*h)/(d*e*h - c*f*h)] + ((2*I)*d*h*(3
*a*b*B*d*f^2*h - 3*a^2*C*d*f^2*h + b^2*(-3*B*d*e*f*h + c*C*f*(-(f*g) + e*h) + C*
d*e*(f*g + 2*e*h)))*Sqrt[c + d*x]*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)])/Sqrt[-c + (d*e)/f]))/(3*d^3*f^2*h^2*Sqrt[e + f*x]
*Sqrt[g + h*x])
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Maple [B] time = 0.045, size = 2825, normalized size = 6.9 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)
[Out]
-2/3*(-C*x*b^2*d^3*e*f*g*h+C*((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))*b^2*c^2*d*e*f*h^2-C*((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))*b^2*c^2*d*f^2*g*h-C*x*b^2*c*d^2*e*f*h
^2+3*B*((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)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/
2))*b^2*c^2*d*f^2*h^2+3*C*((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))*a^2*c*d^2*f^2*h^2-3*C*((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))*a^2*d^3*e*f*h^2-C*((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))*b^2*c*d^2*e^2*h^2-2*C*(
(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))*b^2*
c*d^2*f^2*g^2+C*((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))*b^2*d^3*e^2*g*h+2*C*((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))*b^2*d^3*e*f*g^2+2*C*((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)*EllipticE(((d*x+c)*f/(c*
f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*e^2*h^2+2*C*((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)*Ellip
ticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*f^2*
g^2-2*C*((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)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1
/2))*b^2*d^3*e^2*g*h-2*C*((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)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*
h/f/(c*h-d*g))^(1/2))*b^2*d^3*e*f*g^2-3*B*((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))*a*b*c*d^2*f^2*h^2+3*B*((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))*a*b*d^3*e*f*h^2+3*B*((
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))*b^2*c
*d^2*f^2*g*h-3*B*((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))*b^2*d^3*e*f*g*h-3*B*((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)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((
c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*e*f*h^2-3*B*((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)*EllipticE(((d*x+c)*f/
(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*f^2*g*h+3*B*((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)*El
lipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*d^3*e*f
*g*h-C*x*b^2*c*d^2*f^2*g*h-C*b^2*c*d^2*e*f*g*h-2*C*((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)*EllipticE(((d*x+c)*f/(c
*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c^3*f^2*h^2+2*C*((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)*Ellipt
icE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b^2*c*d^2*e*f*g
*h-C*x^2*b^2*d^3*e*f*h^2-C*x^2*b^2*d^3*f^2*g*h-C*x^2*b^2*c*d^2*f^2*h^2-C*x^3*b^2
*d^3*f^2*h^2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/h^2/f^2/d^3/(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)
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="maxima")
[Out]
integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/(sqrt(d*x + c)*sqrt(f*x + e)*sqr
t(h*x + g)), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="fricas")
[Out]
integral((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt
(h*x + g)), x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right ) \left (B b - C a + C b x\right )}{\sqrt{c + d x} \sqrt{e + f x} \sqrt{g + h x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
[Out]
Integral((a + b*x)*(B*b - C*a + C*b*x)/(sqrt(c + d*x)*sqrt(e + f*x)*sqrt(g + h*x
)), x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")
[Out]
integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/(sqrt(d*x + c)*sqrt(f*x + e)*sqr
t(h*x + g)), x)