3.117 \(\int \frac{a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^3 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)

Optimal. Leaf size=680 \[ -\frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right ) \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h)}-\frac{b^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}-\frac{\sqrt{f} (b B-2 a C) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{e+f x} \sqrt{g+h x} (b c-a d) (b e-a f)}+\frac{b \sqrt{f} \sqrt{g+h x} (b B-2 a C) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{d (g+h x)}{d g-c h}}} \]

[Out]

-((b^2*(b*B - 2*a*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))) + (b*(b*B - 2*a*C)*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)]) - ((b*B - 2*a*C
)*Sqrt[f]*Sqrt[-(d*e) + c*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*c - a*d)*(b*e - a*f)*Sqrt[e + f*x]*Sqrt[g + h*x
]) - (Sqrt[-(d*e) + c*f]*(4*a^3*C*d*f*h + 2*a*b^2*B*(d*f*g + d*e*h + c*f*h) - b^
3*(B*d*e*g - c*(2*C*e*g - B*f*g - B*e*h)) - a^2*b*(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[(d*(g + h*x))/(d*g - c*h)]*Ell
ipticPi[-((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*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 = 4.86865, antiderivative size = 680, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 60, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.183 \[ -\frac{\sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} \left (4 a^3 C d f h-a^2 b (3 B d f h+2 C (c f h+d e h+d f g))+2 a b^2 B (c f h+d e h+d f g)-b^3 (B d e g-c (-B e h-B f g+2 C e g))\right ) \Pi \left (-\frac{b (d e-c f)}{(b c-a d) f};\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{f} \sqrt{e+f x} \sqrt{g+h x} (b c-a d)^2 (b e-a f) (b g-a h)}-\frac{b^2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (b B-2 a C)}{(a+b x) (b c-a d) (b e-a f) (b g-a h)}-\frac{\sqrt{f} (b B-2 a C) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{e+f x} \sqrt{g+h x} (b c-a d) (b e-a f)}+\frac{b \sqrt{f} \sqrt{g+h x} (b B-2 a C) \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) (b g-a h) \sqrt{\frac{d (g+h x)}{d g-c h}}} \]

Antiderivative was successfully verified.

[In]  Int[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^3*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

-((b^2*(b*B - 2*a*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))) + (b*(b*B - 2*a*C)*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)]) - ((b*B - 2*a*C
)*Sqrt[f]*Sqrt[-(d*e) + c*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*c - a*d)*(b*e - a*f)*Sqrt[e + f*x]*Sqrt[g + h*x
]) - (Sqrt[-(d*e) + c*f]*(4*a^3*C*d*f*h + 2*a*b^2*B*(d*f*g + d*e*h + c*f*h) - b^
3*(B*d*e*g - c*(2*C*e*g - B*f*g - B*e*h)) - a^2*b*(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[(d*(g + h*x))/(d*g - c*h)]*Ell
ipticPi[-((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*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*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(b*x+a)**3/(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 = 20.3224, size = 16821, normalized size = 24.74 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^3*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 = 0.143, size = 13405, normalized size = 19.7 \[ \text{output too large to display} \]

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

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

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)^3*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)^3*sqrt(d*x + c)*sqrt(
f*x + e)*sqrt(h*x + g)), x)