3.139 \(\int \frac{\sqrt{a+b x} \left (A+C x^2\right )}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)

Optimal. Leaf size=951 \[ -\frac{\sqrt{d g-c h} \sqrt{f g-e h} (a d f h-3 b (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) C}{4 b d^2 f^2 h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{(b e-a f) \sqrt{b g-a h} (a d f h+b (c f h+3 d (f g+e h))) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right ) C}{4 b^2 d f^2 h^2 \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} C}{2 d f h}+\frac{(a d f h-3 b (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} C}{4 b d f^2 h^2 \sqrt{c+d x}}-\frac{\sqrt{c h-d g} \left (-\left (8 A d^2 f^2 h^2+C \left (\left (3 f^2 g^2+2 e f h g+3 e^2 h^2\right ) d^2+2 c f h (f g+e h) d+3 c^2 f^2 h^2\right )\right ) b^2+2 a C d f h (d f g+d e h+c f h) b+a^2 C d^2 f^2 h^2\right ) (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{4 b^2 d^2 \sqrt{b c-a d} f^2 h^3 \sqrt{c+d x} \sqrt{e+f x}} \]

[Out]

(C*(a*d*f*h - 3*b*(d*f*g + d*e*h + c*f*h))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g +
h*x])/(4*b*d*f^2*h^2*Sqrt[c + d*x]) + (C*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*
x]*Sqrt[g + h*x])/(2*d*f*h) - (C*Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*(a*d*f*h - 3*b*
(d*f*g + d*e*h + c*f*h))*Sqrt[a + b*x]*Sqrt[-(((d*e - c*f)*(g + h*x))/((f*g - e*
h)*(c + d*x)))]*EllipticE[ArcSin[(Sqrt[d*g - c*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h
]*Sqrt[c + d*x])], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))])/(4*b*d^
2*f^2*h^2*Sqrt[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) +
(C*(b*e - a*f)*Sqrt[b*g - a*h]*(a*d*f*h + b*(c*f*h + 3*d*(f*g + e*h)))*Sqrt[((b
*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*EllipticF[ArcSin[(Sq
rt[b*g - a*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h]*Sqrt[a + b*x])], -(((b*c - a*d)*(f
*g - e*h))/((d*e - c*f)*(b*g - a*h)))])/(4*b^2*d*f^2*h^2*Sqrt[f*g - e*h]*Sqrt[c
+ d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]) - (Sqrt[-(d*g)
+ c*h]*(a^2*C*d^2*f^2*h^2 + 2*a*b*C*d*f*h*(d*f*g + d*e*h + c*f*h) - b^2*(8*A*d^2
*f^2*h^2 + C*(3*c^2*f^2*h^2 + 2*c*d*f*h*(f*g + e*h) + d^2*(3*f^2*g^2 + 2*e*f*g*h
+ 3*e^2*h^2))))*(a + b*x)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]
*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*EllipticPi[-((b*(d*g - c*
h))/((b*c - a*d)*h)), ArcSin[(Sqrt[b*c - a*d]*Sqrt[g + h*x])/(Sqrt[-(d*g) + c*h]
*Sqrt[a + b*x])], ((b*e - a*f)*(d*g - c*h))/((b*c - a*d)*(f*g - e*h))])/(4*b^2*d
^2*Sqrt[b*c - a*d]*f^2*h^3*Sqrt[c + d*x]*Sqrt[e + f*x])

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Rubi [A]  time = 6.86803, antiderivative size = 950, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\sqrt{d g-c h} \sqrt{f g-e h} (a d f h-3 b (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) C}{4 b d^2 f^2 h^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{(b e-a f) \sqrt{b g-a h} (b c f h+a d f h+3 b d (f g+e h)) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right ) C}{4 b^2 d f^2 h^2 \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} C}{2 d f h}+\frac{(a d f h-3 b (d f g+d e h+c f h)) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x} C}{4 b d f^2 h^2 \sqrt{c+d x}}-\frac{\sqrt{c h-d g} \left (-\left (8 A d^2 f^2 h^2+C \left (\left (3 f^2 g^2+2 e f h g+3 e^2 h^2\right ) d^2+2 c f h (f g+e h) d+3 c^2 f^2 h^2\right )\right ) b^2+2 a C d f h (d f g+d e h+c f h) b+a^2 C d^2 f^2 h^2\right ) (a+b x) \sqrt{\frac{(b g-a h) (c+d x)}{(d g-c h) (a+b x)}} \sqrt{\frac{(b g-a h) (e+f x)}{(f g-e h) (a+b x)}} \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{4 b^2 d^2 \sqrt{b c-a d} f^2 h^3 \sqrt{c+d x} \sqrt{e+f x}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(Sqrt[a + b*x]*(A + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(C*(a*d*f*h - 3*b*(d*f*g + d*e*h + c*f*h))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g +
h*x])/(4*b*d*f^2*h^2*Sqrt[c + d*x]) + (C*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*
x]*Sqrt[g + h*x])/(2*d*f*h) - (C*Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*(a*d*f*h - 3*b*
(d*f*g + d*e*h + c*f*h))*Sqrt[a + b*x]*Sqrt[-(((d*e - c*f)*(g + h*x))/((f*g - e*
h)*(c + d*x)))]*EllipticE[ArcSin[(Sqrt[d*g - c*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h
]*Sqrt[c + d*x])], ((b*c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))])/(4*b*d^
2*f^2*h^2*Sqrt[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) +
(C*(b*e - a*f)*Sqrt[b*g - a*h]*(b*c*f*h + a*d*f*h + 3*b*d*(f*g + e*h))*Sqrt[((b
*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*EllipticF[ArcSin[(Sq
rt[b*g - a*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h]*Sqrt[a + b*x])], -(((b*c - a*d)*(f
*g - e*h))/((d*e - c*f)*(b*g - a*h)))])/(4*b^2*d*f^2*h^2*Sqrt[f*g - e*h]*Sqrt[c
+ d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a + b*x)))]) - (Sqrt[-(d*g)
+ c*h]*(a^2*C*d^2*f^2*h^2 + 2*a*b*C*d*f*h*(d*f*g + d*e*h + c*f*h) - b^2*(8*A*d^2
*f^2*h^2 + C*(3*c^2*f^2*h^2 + 2*c*d*f*h*(f*g + e*h) + d^2*(3*f^2*g^2 + 2*e*f*g*h
+ 3*e^2*h^2))))*(a + b*x)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]
*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*EllipticPi[-((b*(d*g - c*
h))/((b*c - a*d)*h)), ArcSin[(Sqrt[b*c - a*d]*Sqrt[g + h*x])/(Sqrt[-(d*g) + c*h]
*Sqrt[a + b*x])], ((b*e - a*f)*(d*g - c*h))/((b*c - a*d)*(f*g - e*h))])/(4*b^2*d
^2*Sqrt[b*c - a*d]*f^2*h^3*Sqrt[c + d*x]*Sqrt[e + f*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)**(1/2)*(C*x**2+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 [B]  time = 21.5105, size = 16659, normalized size = 17.52 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(Sqrt[a + b*x]*(A + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

Result too large to show

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Maple [B]  time = 0.128, size = 42545, normalized size = 44.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (C x^{2} + A\right )} \sqrt{b x + a}}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + A)*sqrt(b*x + a)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="maxima")

[Out]

integrate((C*x^2 + A)*sqrt(b*x + a)/(sqrt(d*x + c)*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((C*x^2 + A)*sqrt(b*x + a)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="fricas")

[Out]

Timed out

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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)**(1/2)*(C*x**2+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. \[ \int \frac{{\left (C x^{2} + A\right )} \sqrt{b x + a}}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + A)*sqrt(b*x + a)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")

[Out]

integrate((C*x^2 + A)*sqrt(b*x + a)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),
x)