3.124 \(\int \frac{(a+b x) (A+B x)}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)
Optimal. Leaf size=402 \[ -\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) (3 a d f h (B g-A h)+b (3 A d f g h-B (c h (f g-e h)+d g (e h+2 f g))))}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) (3 d f h (a B+A b)-2 b B (c f h+d e h+d f g))}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 b B \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h} \]
[Out]
(2*b*B*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*d*f*h) + (2*Sqrt[-(d*e) + c
*f]*(3*(A*b + a*B)*d*f*h - 2*b*B*(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*d*f*h*(B*g - A*h) + b*(3*
A*d*f*g*h - B*(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])/S
qrt[-(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 = 2.27807, antiderivative size = 402, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 7, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184
\[ -\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) (3 a d f h (B g-A h)+b (3 A d f g h-B c h (f g-e h)-B d g (e h+2 f g)))}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) (3 d f h (a B+A b)-2 b B (c f h+d e h+d f g))}{3 d^2 f^{3/2} h^2 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 b B \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 d f h} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(A + B*x))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
(2*b*B*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*d*f*h) + (2*Sqrt[-(d*e) + c
*f]*(3*(A*b + a*B)*d*f*h - 2*b*B*(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*d*f*h*(B*g - A*h) + b*(3*
A*d*f*g*h - B*c*h*(f*g - e*h) - B*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])/S
qrt[-(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((b*x+a)*(B*x+A)/(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 = 9.62189, size = 450, normalized size = 1.12 \[ \frac{\sqrt{c+d x} \left (-\frac{2 d^2 (e+f x) (g+h x) (-3 a B d f h-3 A b d f h+2 b B (c f h+d e h+d f g))}{c+d x}+\frac{2 i d h \sqrt{c+d x} \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right ) (3 a d f h (A f-B e)+b (-3 A d e f h+B c f (e h-f g)+B d e (2 e h+f g)))}{\sqrt{\frac{d e}{f}-c}}+\frac{2 i h \sqrt{c+d x} (d e-c f) \sqrt{\frac{d (e+f x)}{f (c+d x)}} \sqrt{\frac{d (g+h x)}{h (c+d x)}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{d e}{f}-c}}{\sqrt{c+d x}}\right )|\frac{d f g-c f h}{d e h-c f h}\right ) (3 a B d f h+3 A b d f h-2 b B (c f h+d e h+d f g))}{\sqrt{\frac{d e}{f}-c}}+2 b B d^2 f h (e+f x) (g+h x)\right )}{3 d^3 f^2 h^2 \sqrt{e+f x} \sqrt{g+h x}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(A + B*x))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
[Out]
(Sqrt[c + d*x]*(2*b*B*d^2*f*h*(e + f*x)*(g + h*x) - (2*d^2*(-3*A*b*d*f*h - 3*a*B
*d*f*h + 2*b*B*(d*f*g + d*e*h + c*f*h))*(e + f*x)*(g + h*x))/(c + d*x) + ((2*I)*
(d*e - c*f)*h*(3*A*b*d*f*h + 3*a*B*d*f*h - 2*b*B*(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))]*Elli
pticE[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] + ((2*I)*d*h*(3*a*d*f*(-(B*e) + A*f)*h + b*(-3*A*d*e*f*h
+ B*c*f*(-(f*g) + e*h) + B*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 = 3224, normalized size = 8. \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)
[Out]
2/3*(3*A*((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*c*d^2*f^2*h^2-3*A*((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*d^3*e*f*h^2-3*A*((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*c^2*d*f^2*h^2+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*c*d^2*e^2*h^2+2*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)*E
llipticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b*c*d^2*f^
2*g^2-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*d^3*e^2*g*h-2*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*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)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2
),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a*c^2*d*f^2*h^2-2*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*c*d^2*e^2*h^2-2*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*c*d^2*f^2
*g^2+2*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*d^3*e^2*g*h+2*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*d^3*e*f*g^2-3*A*((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*c*d^2*f^2*g*h+3*A*((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*d^3*e*f*g*h+3*A*((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)*Ell
ipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b*c*d^2*e*f*
h^2+3*A*((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*c*d^2*f^2*g*h-3*A*((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*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)*EllipticF(((d*x+c)*f/(c*f-d*e))^(1
/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a*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))*a*d^3*e*f*g*h-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)*Elli
pticF(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*b*c^2*d*e*f*h
^2+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*c^2*d*f^2*g*h-2*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*c*d^2*e*f*g*h+B*b*c*d^2*e*f*g*h+B*x*b*c*d^2*f^2*g*h+2*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*c^3*
f^2*h^2+B*x*b*d^3*e*f*g*h+B*x*b*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))*a*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)*Ellipti
cE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))*a*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)*EllipticE(((d*x+c)*f/(c*f-d*e))^(1/2),((c*f-d*e)*h/f/(c*h-d*g))^(1/2))
*a*d^3*e*f*g*h+B*x^2*b*c*d^2*f^2*h^2+B*x^2*b*d^3*e*f*h^2+B*x^2*b*d^3*f^2*g*h+B*x
^3*b*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)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (b x + a\right )}}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="maxima")
[Out]
integrate((B*x + A)*(b*x + a)/(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. \[{\rm integral}\left (\frac{B b x^{2} + A a +{\left (B a + A b\right )} x}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="fricas")
[Out]
integral((B*b*x^2 + A*a + (B*a + A*b)*x)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x +
g)), x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(B*x+A)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
[Out]
Timed out
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (b x + a\right )}}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")
[Out]
integrate((B*x + A)*(b*x + a)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)