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

Optimal. Leaf size=704 \[ -\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^3 C d^2 f^2 h^3-15 a^2 b d^2 f^2 h^2 (B h+C g)+5 a b^2 d f h (6 B d f g h-C (c h (f g-e h)+d g (e h+2 f g)))+b^3 \left (-\left (5 B d f h (c h (f g-e h)+d g (e h+2 f g))-C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 b^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (2 C (a d f h-2 b (c f h+d e h+d f g))+5 b B d f h)}{15 d^2 f^2 h^2}+\frac{2 b^2 C (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}+\frac{2 b \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) (3 d f h (5 a d f h (2 b B-a C)-b C (2 a (c f h+d e h+d f g)+3 b (c e h+c f g+d e g)))-2 b (c f h+d e h+d f g) (2 C (a d f h-2 b (c f h+d e h+d f g))+5 b B d f h))}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}} \]

[Out]

(2*b^2*(5*b*B*d*f*h + 2*C*(a*d*f*h - 2*b*(d*f*g + d*e*h + c*f*h)))*Sqrt[c + d*x]
*Sqrt[e + f*x]*Sqrt[g + h*x])/(15*d^2*f^2*h^2) + (2*b^2*C*(a + b*x)*Sqrt[c + d*x
]*Sqrt[e + f*x]*Sqrt[g + h*x])/(5*d*f*h) + (2*b*Sqrt[-(d*e) + c*f]*(3*d*f*h*(5*a
*(2*b*B - a*C)*d*f*h - b*C*(3*b*(d*e*g + c*f*g + c*e*h) + 2*a*(d*f*g + d*e*h + c
*f*h))) - 2*b*(d*f*g + d*e*h + c*f*h)*(5*b*B*d*f*h + 2*C*(a*d*f*h - 2*b*(d*f*g +
d*e*h + c*f*h))))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSi
n[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])
/(15*d^3*f^(5/2)*h^3*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) - (2*Sqrt[-(
d*e) + c*f]*(15*a^3*C*d^2*f^2*h^3 - 15*a^2*b*d^2*f^2*h^2*(C*g + B*h) + 5*a*b^2*d
*f*h*(6*B*d*f*g*h - C*(c*h*(f*g - e*h) + d*g*(2*f*g + e*h))) - b^3*(5*B*d*f*h*(c
*h*(f*g - e*h) + d*g*(2*f*g + e*h)) - C*(4*c^2*f*h^2*(f*g - e*h) + c*d*h*(3*f^2*
g^2 + e*f*g*h - 4*e^2*h^2) + d^2*g*(8*f^2*g^2 + 3*e*f*g*h + 4*e^2*h^2))))*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))])/(15*d^
3*f^(5/2)*h^3*Sqrt[e + f*x]*Sqrt[g + h*x])

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Rubi [A]  time = 5.15379, antiderivative size = 702, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 58, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121 \[ -\frac{2 \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) \left (15 a^3 C d^2 f^2 h^3-15 a^2 b d^2 f^2 h^2 (B h+C g)+5 a b^2 d f h (6 B d f g h-c C h (f g-e h)-C d g (e h+2 f g))+b^3 \left (-\left (5 B d f h (c h (f g-e h)+d g (e h+2 f g))-C \left (4 c^2 f h^2 (f g-e h)+c d h \left (-4 e^2 h^2+e f g h+3 f^2 g^2\right )+d^2 g \left (4 e^2 h^2+3 e f g h+8 f^2 g^2\right )\right )\right )\right )\right )}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 b^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (2 a C d f h+5 b B d f h-4 b C (c f h+d e h+d f g))}{15 d^2 f^2 h^2}+\frac{2 b^2 C (a+b x) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{5 d f h}-\frac{2 b \sqrt{g+h x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right ) (2 b (c f h+d e h+d f g) (2 a C d f h+5 b B d f h-4 b C (c f h+d e h+d f g))-3 d f h (5 a d f h (2 b B-a C)-b C (2 a (c f h+d e h+d f g)+3 b (c e h+c f g+d e g))))}{15 d^3 f^{5/2} h^3 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(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*(5*b*B*d*f*h + 2*a*C*d*f*h - 4*b*C*(d*f*g + d*e*h + c*f*h))*Sqrt[c + d*x]
*Sqrt[e + f*x]*Sqrt[g + h*x])/(15*d^2*f^2*h^2) + (2*b^2*C*(a + b*x)*Sqrt[c + d*x
]*Sqrt[e + f*x]*Sqrt[g + h*x])/(5*d*f*h) - (2*b*Sqrt[-(d*e) + c*f]*(2*b*(d*f*g +
d*e*h + c*f*h)*(5*b*B*d*f*h + 2*a*C*d*f*h - 4*b*C*(d*f*g + d*e*h + c*f*h)) - 3*
d*f*h*(5*a*(2*b*B - a*C)*d*f*h - b*C*(3*b*(d*e*g + c*f*g + c*e*h) + 2*a*(d*f*g +
d*e*h + c*f*h))))*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[g + h*x]*EllipticE[ArcSi
n[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])
/(15*d^3*f^(5/2)*h^3*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) - (2*Sqrt[-(
d*e) + c*f]*(15*a^3*C*d^2*f^2*h^3 - 15*a^2*b*d^2*f^2*h^2*(C*g + B*h) + 5*a*b^2*d
*f*h*(6*B*d*f*g*h - c*C*h*(f*g - e*h) - C*d*g*(2*f*g + e*h)) - b^3*(5*B*d*f*h*(c
*h*(f*g - e*h) + d*g*(2*f*g + e*h)) - C*(4*c^2*f*h^2*(f*g - e*h) + c*d*h*(3*f^2*
g^2 + e*f*g*h - 4*e^2*h^2) + d^2*g*(8*f^2*g^2 + 3*e*f*g*h + 4*e^2*h^2))))*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))])/(15*d^
3*f^(5/2)*h^3*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)*(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 = 18.9488, size = 12665, normalized size = 17.99 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(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]

Result too large to show

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Maple [B]  time = 0.091, size = 8421, normalized size = 12. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(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]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)*(b*x + a)/(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)*(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. \[{\rm integral}\left (\frac{C b^{3} x^{3} - C a^{3} + B a^{2} b +{\left (C a b^{2} + B b^{3}\right )} x^{2} -{\left (C a^{2} b - 2 \, B a b^{2}\right )} x}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((C*b^3*x^3 - C*a^3 + B*a^2*b + (C*a*b^2 + B*b^3)*x^2 - (C*a^2*b - 2*B*a
*b^2)*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)*(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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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b\right )}{\left (b x + a\right )}}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)*(b*x + a)/(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)*(b*x + a)/(sqrt(d*x + c)*sqrt(f*
x + e)*sqrt(h*x + g)), x)