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

Optimal. Leaf size=436 \[ \frac{2 \sqrt{g+h x} (b B-2 a C) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt{c+d x} \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac{2 C (a+b x) \sqrt{c h-d g} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{h \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d}} \]

[Out]

(2*(b*B - 2*a*C)*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g +
h*x]*EllipticF[ArcSin[(Sqrt[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)))])/(Sqrt[b*g - a*h
]*Sqrt[f*g - e*h]*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a +
b*x)))]) + (2*C*Sqrt[-(d*g) + c*h]*(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))]*Ellipt
icPi[-((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))])/(Sqrt[b*c - a*d]*h*Sqrt[c + d*x]*Sqrt[e + f*x])

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Rubi [A]  time = 2.0255, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 62, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.113 \[ \frac{2 \sqrt{g+h x} (b B-2 a C) \sqrt{\frac{(c+d x) (b e-a f)}{(a+b x) (d e-c f)}} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt{c+d x} \sqrt{b g-a h} \sqrt{f g-e h} \sqrt{-\frac{(g+h x) (b e-a f)}{(a+b x) (f g-e h)}}}+\frac{2 C (a+b x) \sqrt{c h-d g} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} \Pi \left (-\frac{b (d g-c h)}{(b c-a d) h};\sin ^{-1}\left (\frac{\sqrt{b c-a d} \sqrt{g+h x}}{\sqrt{c h-d g} \sqrt{a+b x}}\right )|\frac{(b e-a f) (d g-c h)}{(b c-a d) (f g-e h)}\right )}{h \sqrt{c+d x} \sqrt{e+f x} \sqrt{b c-a d}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*B - 2*a*C)*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g +
h*x]*EllipticF[ArcSin[(Sqrt[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)))])/(Sqrt[b*g - a*h
]*Sqrt[f*g - e*h]*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x))/((f*g - e*h)*(a +
b*x)))]) + (2*C*Sqrt[-(d*g) + c*h]*(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))]*Ellipt
icPi[-((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))])/(Sqrt[b*c - a*d]*h*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((C*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(b*x+a)**(3/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 10.3425, size = 583, normalized size = 1.34 \[ \frac{2 (a+b x)^{3/2} \sqrt{\frac{(c+d x) (b g-a h)}{(a+b x) (d g-c h)}} \left (-\frac{b B (g+h x) \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} F\left (\sin ^{-1}\left (\sqrt{\frac{(a f-b e) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac{(a d-b c) (e h-f g)}{(b e-a f) (d g-c h)}\right )}{(a+b x) (b g-a h) \sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}}-\frac{2 a C (g+h x) \sqrt{\frac{(e+f x) (b g-a h)}{(a+b x) (f g-e h)}} F\left (\sin ^{-1}\left (\sqrt{\frac{(a f-b e) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac{(a d-b c) (e h-f g)}{(b e-a f) (d g-c h)}\right )}{(a+b x) (a h-b g) \sqrt{\frac{(g+h x) (a f-b e)}{(a+b x) (f g-e h)}}}+\frac{C (e h-f g) \sqrt{-\frac{(e+f x) (g+h x) (b e-a f) (b g-a h)}{(a+b x)^2 (f g-e h)^2}} \Pi \left (\frac{b (e h-f g)}{(b e-a f) h};\sin ^{-1}\left (\sqrt{\frac{(a f-b e) (g+h x)}{(f g-e h) (a+b x)}}\right )|\frac{(a d-b c) (e h-f g)}{(b e-a f) (d g-c h)}\right )}{h (b e-a f)}\right )}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^(3/2)*Sqrt[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]*(-((b*B
*Sqrt[((b*g - a*h)*(e + f*x))/((f*g - e*h)*(a + b*x))]*(g + h*x)*EllipticF[ArcSi
n[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(
f*g) + e*h))/((b*e - a*f)*(d*g - c*h))])/((b*g - a*h)*(a + b*x)*Sqrt[((-(b*e) +
a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))])) - (2*a*C*Sqrt[((b*g - a*h)*(e + f*x))
/((f*g - e*h)*(a + b*x))]*(g + h*x)*EllipticF[ArcSin[Sqrt[((-(b*e) + a*f)*(g + h
*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c) + a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*
g - c*h))])/((-(b*g) + a*h)*(a + b*x)*Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*
h)*(a + b*x))]) + (C*(-(f*g) + e*h)*Sqrt[-(((b*e - a*f)*(b*g - a*h)*(e + f*x)*(g
+ h*x))/((f*g - e*h)^2*(a + b*x)^2))]*EllipticPi[(b*(-(f*g) + e*h))/((b*e - a*f
)*h), ArcSin[Sqrt[((-(b*e) + a*f)*(g + h*x))/((f*g - e*h)*(a + b*x))]], ((-(b*c)
+ a*d)*(-(f*g) + e*h))/((b*e - a*f)*(d*g - c*h))])/((b*e - a*f)*h)))/(Sqrt[c +
d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])

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Maple [B]  time = 0.116, size = 3003, normalized size = 6.9 \[ \text{output too large to display} \]

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)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2),x)

[Out]

2/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)/(h*x+g)^(1/2)/h/f*((a*f-b*e)*(h*x+g)
/(a*h-b*g)/(f*x+e))^(1/2)*((e*h-f*g)*(d*x+c)/(c*h-d*g)/(f*x+e))^(1/2)*((e*h-f*g)
*(b*x+a)/(a*h-b*g)/(f*x+e))^(1/2)*(C*EllipticF(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x
+e))^(1/2),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*x^2*a*b*f^3*g*h+C*El
lipticF(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),((c*f-d*e)*(a*h-b*g)/(c*h-d*
g)/(a*f-b*e))^(1/2))*x^2*b^2*e*f^2*g*h+C*EllipticPi(((a*f-b*e)*(h*x+g)/(a*h-b*g)
/(f*x+e))^(1/2),(a*h-b*g)*f/h/(a*f-b*e),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e)
)^(1/2))*x^2*a*b*e*f^2*h^2-C*EllipticPi(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1
/2),(a*h-b*g)*f/h/(a*f-b*e),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*x^2
*a*b*f^3*g*h-C*EllipticPi(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),(a*h-b*g)*
f/h/(a*f-b*e),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*x^2*b^2*e*f^2*g*h
+2*B*EllipticF(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),((c*f-d*e)*(a*h-b*g)/
(c*h-d*g)/(a*f-b*e))^(1/2))*x*a*b*e*f^2*h^2-2*B*EllipticF(((a*f-b*e)*(h*x+g)/(a*
h-b*g)/(f*x+e))^(1/2),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*x*b^2*e*f
^2*g*h-2*C*EllipticF(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),((c*f-d*e)*(a*h
-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*x*a*b*e^2*f*h^2+2*C*EllipticF(((a*f-b*e)*(h*x+
g)/(a*h-b*g)/(f*x+e))^(1/2),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*x*b
^2*e^2*f*g*h+2*C*EllipticPi(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),(a*h-b*g
)*f/h/(a*f-b*e),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*x*a*b*e^2*f*h^2
-2*C*EllipticPi(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),(a*h-b*g)*f/h/(a*f-b
*e),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*x*b^2*e^2*f*g*h+C*EllipticF
(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f
-b*e))^(1/2))*a*b*e^2*f*g*h-C*EllipticPi(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(
1/2),(a*h-b*g)*f/h/(a*f-b*e),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*a*
b*e^2*f*g*h-C*EllipticF(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),((c*f-d*e)*(
a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*x^2*a*b*e*f^2*h^2+C*EllipticPi(((a*f-b*e)*(
h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),(a*h-b*g)*f/h/(a*f-b*e),((c*f-d*e)*(a*h-b*g)/(c*
h-d*g)/(a*f-b*e))^(1/2))*x^2*b^2*f^3*g^2-C*EllipticF(((a*f-b*e)*(h*x+g)/(a*h-b*g
)/(f*x+e))^(1/2),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*a^2*e^2*f*h^2-
C*EllipticF(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),((c*f-d*e)*(a*h-b*g)/(c*
h-d*g)/(a*f-b*e))^(1/2))*a*b*e^3*h^2+C*EllipticF(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f
*x+e))^(1/2),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*b^2*e^3*g*h+C*Elli
pticPi(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),(a*h-b*g)*f/h/(a*f-b*e),((c*f
-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*a*b*e^3*h^2-C*EllipticPi(((a*f-b*e)*
(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),(a*h-b*g)*f/h/(a*f-b*e),((c*f-d*e)*(a*h-b*g)/(c
*h-d*g)/(a*f-b*e))^(1/2))*b^2*e^3*g*h+C*EllipticPi(((a*f-b*e)*(h*x+g)/(a*h-b*g)/
(f*x+e))^(1/2),(a*h-b*g)*f/h/(a*f-b*e),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))
^(1/2))*b^2*e^2*f*g^2-B*EllipticF(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),((
c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*x^2*b^2*f^3*g*h-2*C*EllipticF(((a
*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e
))^(1/2))*x*a^2*e*f^2*h^2+2*C*EllipticPi(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(
1/2),(a*h-b*g)*f/h/(a*f-b*e),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*x*
b^2*e*f^2*g^2+B*EllipticF(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),((c*f-d*e)
*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*a*b*e^2*f*h^2-B*EllipticF(((a*f-b*e)*(h*x
+g)/(a*h-b*g)/(f*x+e))^(1/2),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*b^
2*e^2*f*g*h+B*EllipticF(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),((c*f-d*e)*(
a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*x^2*a*b*f^3*h^2+2*C*EllipticF(((a*f-b*e)*(h
*x+g)/(a*h-b*g)/(f*x+e))^(1/2),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*
x*a*b*e*f^2*g*h-2*C*EllipticPi(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),(a*h-
b*g)*f/h/(a*f-b*e),((c*f-d*e)*(a*h-b*g)/(c*h-d*g)/(a*f-b*e))^(1/2))*x*a*b*e*f^2*
g*h-C*EllipticF(((a*f-b*e)*(h*x+g)/(a*h-b*g)/(f*x+e))^(1/2),((c*f-d*e)*(a*h-b*g)
/(c*h-d*g)/(a*f-b*e))^(1/2))*x^2*a^2*f^3*h^2)/(a*f-b*e)/(e*h-f*g)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

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)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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