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

Optimal. Leaf size=738 \[ -\frac{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}+\frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^2 d \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d) (b e-a f)}+\frac{\sqrt{f} \sqrt{g+h x} \left (\frac{a^2 C}{b}+A b\right ) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h)} \]

[Out]

-(((A*b^2 + a^2*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)*(b*e
- a*f)*(b*g - a*h)*(a + b*x))) + ((A*b + (a^2*C)/b)*Sqrt[f]*Sqrt[-(d*e) + c*f]*S
qrt[(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))])/((b*c - a*d)*(b*e -
a*f)*(b*g - a*h)*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) + (Sqrt[-(d*e)
+ c*f]*(a^2*C*d*f - 2*a*b*C*(d*e + c*f) + b^2*(2*c*C*e - A*d*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^2*d*(b*c - a
*d)*Sqrt[f]*(b*e - a*f)*Sqrt[e + f*x]*Sqrt[g + h*x]) - (Sqrt[-(d*e) + c*f]*(a^4*
C*d*f*h - A*b^4*(d*e*g + c*f*g + c*e*h) - 2*a^3*b*C*(d*f*g + d*e*h + c*f*h) - 2*
a*b^3*(2*c*C*e*g - A*d*f*g - A*d*e*h - A*c*f*h) - 3*a^2*b^2*(A*d*f*h - C*(d*e*g
+ c*f*g + c*e*h)))*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^2*(b*c - a*d)^2*Sqrt[
f]*(b*e - a*f)*(b*g - a*h)*Sqrt[e + f*x]*Sqrt[g + h*x])

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Rubi [A]  time = 5.53412, antiderivative size = 738, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \left (a^2 C+A b^2\right )}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}+\frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (a^2 C d f-2 a b C (c f+d e)+b^2 (2 c C e-A d f)\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^2 d \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d) (b e-a f)}+\frac{\sqrt{f} \sqrt{g+h x} \left (\frac{a^2 C}{b}+A b\right ) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{d (g+h x)}{d g-c h}}}-\frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (a^4 C d f h-2 a^3 b C (c f h+d e h+d f g)-3 a^2 b^2 (A d f h-C (c e h+c f g+d e g))-2 a b^3 (-A c f h-A d e h-A d f g+2 c C e g)-A b^4 (c e h+c f g+d e g)\right ) \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{b^2 \sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h)} \]

Antiderivative was successfully verified.

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

[Out]

-(((A*b^2 + a^2*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)*(b*e
- a*f)*(b*g - a*h)*(a + b*x))) + ((A*b + (a^2*C)/b)*Sqrt[f]*Sqrt[-(d*e) + c*f]*S
qrt[(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))])/((b*c - a*d)*(b*e -
a*f)*(b*g - a*h)*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) + (Sqrt[-(d*e)
+ c*f]*(a^2*C*d*f - 2*a*b*C*(d*e + c*f) + b^2*(2*c*C*e - A*d*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^2*d*(b*c - a
*d)*Sqrt[f]*(b*e - a*f)*Sqrt[e + f*x]*Sqrt[g + h*x]) - (Sqrt[-(d*e) + c*f]*(a^4*
C*d*f*h - A*b^4*(d*e*g + c*f*g + c*e*h) - 2*a^3*b*C*(d*f*g + d*e*h + c*f*h) - 2*
a*b^3*(2*c*C*e*g - A*d*f*g - A*d*e*h - A*c*f*h) - 3*a^2*b^2*(A*d*f*h - C*(d*e*g
+ c*f*g + c*e*h)))*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^2*(b*c - a*d)^2*Sqrt[
f]*(b*e - a*f)*(b*g - a*h)*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*x**2+A)/(b*x+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 = 21.1554, size = 17743, normalized size = 24.04 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(A + C*x^2)/((a + b*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 = 0.131, size = 17460, normalized size = 23.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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