3.138 \(\int \frac{(a+b x)^{3/2} \left (A+C x^2\right )}{\sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)

Optimal. Leaf size=1405 \[ \text{result too large to display} \]

[Out]

((C*(3*a*d*f*h - 5*b*(d*f*g + d*e*h + c*f*h))*(a*d*f*h - 3*b*(d*f*g + d*e*h + c*
f*h)) + 8*b*d*f*h*(3*A*b*d*f*h - C*(2*b*(d*e*g + c*f*g + c*e*h) + a*(d*f*g + d*e
*h + c*f*h))))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(24*b*d^2*f^3*h^3*Sqrt
[c + d*x]) + (C*(3*a*d*f*h - 5*b*(d*f*g + d*e*h + c*f*h))*Sqrt[a + b*x]*Sqrt[c +
d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(12*d^2*f^2*h^2) + (C*(a + b*x)^(3/2)*Sqrt[c
+ d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*d*f*h) - (Sqrt[d*g - c*h]*Sqrt[f*g - e*h]
*(C*(3*a*d*f*h - 5*b*(d*f*g + d*e*h + c*f*h))*(a*d*f*h - 3*b*(d*f*g + d*e*h + c*
f*h)) + 8*b*d*f*h*(3*A*b*d*f*h - C*(2*b*(d*e*g + c*f*g + c*e*h) + a*(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))])/(24*b*d^3*f^3*h^3
*Sqrt[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) + ((b*e -
a*f)*Sqrt[b*g - a*h]*(3*a^2*C*d^2*f^2*h^2 + 6*a*b*C*d*f*h*(c*f*h + 2*d*(f*g + e*
h)) - b^2*(24*A*d^2*f^2*h^2 + C*(5*c^2*f^2*h^2 + 4*c*d*f*h*(f*g + e*h) + d^2*(15
*f^2*g^2 + 14*e*f*g*h + 15*e^2*h^2))))*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])/(Sqr
t[f*g - e*h]*Sqrt[a + b*x])], -(((b*c - a*d)*(f*g - e*h))/((d*e - c*f)*(b*g - a*
h)))])/(24*b^2*d^2*f^3*h^3*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^3*C*d^3*f^3*h^3 + 3*
a^2*b*C*d^2*f^2*h^2*(d*f*g + d*e*h + c*f*h) - 3*a*b^2*d*f*h*(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
))) + b^3*(8*A*d^2*f^2*h^2*(d*f*g + d*e*h + c*f*h) + C*(5*c^3*f^3*h^3 + 3*c^2*d*
f^2*h^2*(f*g + e*h) + c*d^2*f*h*(3*f^2*g^2 + 2*e*f*g*h + 3*e^2*h^2) + d^3*(5*f^3
*g^3 + 3*e*f^2*g^2*h + 3*e^2*f*g*h^2 + 5*e^3*h^3))))*(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))])/(8*b^2*d^3*Sqrt[b*c - a*d]*f^3*h^4*Sqrt[c + d*x]*Sqr
t[e + f*x])

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Rubi [A]  time = 24.5378, antiderivative size = 1388, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a+b x)^{3/2}}{3 d f h}-\frac{\sqrt{c h-d g} \left (\left (8 A d^2 f^2 (d f g+d e h+c f h) h^2+C \left (\left (5 f^3 g^3+3 e f^2 h g^2+3 e^2 f h^2 g+5 e^3 h^3\right ) d^3+c f h \left (3 f^2 g^2+2 e f h g+3 e^2 h^2\right ) d^2+3 c^2 f^2 h^2 (f g+e h) d+5 c^3 f^3 h^3\right )\right ) b^3-3 a d f h \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+3 a^2 C d^2 f^2 h^2 (d f g+d e h+c f h) b+a^3 C d^3 f^3 h^3\right ) \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 ) (a+b x)}{8 b^2 d^3 \sqrt{b c-a d} f^3 h^4 \sqrt{c+d x} \sqrt{e+f x}}-\frac{\sqrt{d g-c h} \sqrt{f g-e h} \left (24 A d^2 f^2 h^2 b^2+15 C (d f g+d e h+c f h)^2 b^2-16 C d f h (d e g+c f g+c e h) b^2-22 a C d f h (d f g+d e h+c f h) b+3 a^2 C d^2 f^2 h^2\right ) \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 ) \sqrt{a+b x}}{24 b d^3 f^3 h^3 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}+\frac{C (3 a d f h-5 b (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \sqrt{a+b x}}{12 d^2 f^2 h^2}+\frac{\left (24 A b f h d^2+\frac{3 a^2 C f h d^2}{b}-16 b C (d e g+c f g+c e h) d-22 a C (d f g+d e h+c f h) d+\frac{15 b C (d f g+d e h+c f h)^2}{f h}\right ) \sqrt{e+f x} \sqrt{g+h x} \sqrt{a+b x}}{24 d^2 f^2 h^2 \sqrt{c+d x}}+\frac{(b e-a f) \sqrt{b g-a h} \left (-\left (24 A d^2 f^2 h^2+C \left (\left (15 f^2 g^2+14 e f h g+15 e^2 h^2\right ) d^2+4 c f h (f g+e h) d+5 c^2 f^2 h^2\right )\right ) b^2+6 a C d f h (c f h+2 d (f g+e h)) b+3 a^2 C d^2 f^2 h^2\right ) \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 )}{24 b^2 d^2 f^3 h^3 \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)}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

((24*A*b*d^2*f*h + (3*a^2*C*d^2*f*h)/b - 16*b*C*d*(d*e*g + c*f*g + c*e*h) - 22*a
*C*d*(d*f*g + d*e*h + c*f*h) + (15*b*C*(d*f*g + d*e*h + c*f*h)^2)/(f*h))*Sqrt[a
+ b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(24*d^2*f^2*h^2*Sqrt[c + d*x]) + (C*(3*a*d*f
*h - 5*b*(d*f*g + d*e*h + c*f*h))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt
[g + h*x])/(12*d^2*f^2*h^2) + (C*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqr
t[g + h*x])/(3*d*f*h) - (Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*(24*A*b^2*d^2*f^2*h^2 +
3*a^2*C*d^2*f^2*h^2 - 16*b^2*C*d*f*h*(d*e*g + c*f*g + c*e*h) - 22*a*b*C*d*f*h*(
d*f*g + d*e*h + c*f*h) + 15*b^2*C*(d*f*g + d*e*h + c*f*h)^2)*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))])/(24*b*d^3*f^3*h^3*Sqrt[((d*e - c*f)*(a + b*x))/((b*
e - a*f)*(c + d*x))]*Sqrt[g + h*x]) + ((b*e - a*f)*Sqrt[b*g - a*h]*(3*a^2*C*d^2*
f^2*h^2 + 6*a*b*C*d*f*h*(c*f*h + 2*d*(f*g + e*h)) - b^2*(24*A*d^2*f^2*h^2 + C*(5
*c^2*f^2*h^2 + 4*c*d*f*h*(f*g + e*h) + d^2*(15*f^2*g^2 + 14*e*f*g*h + 15*e^2*h^2
))))*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*Ellipti
cF[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)))])/(24*b^2*d^2*f^3*h^3*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^3*C*d^3*f^3*h^3 + 3*a^2*b*C*d^2*f^2*h^2*(d*f*g + d*e*h
+ c*f*h) - 3*a*b^2*d*f*h*(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))) + b^3*(8*A*d^2*f^2*h^2*(d*f*g
+ d*e*h + c*f*h) + C*(5*c^3*f^3*h^3 + 3*c^2*d*f^2*h^2*(f*g + e*h) + c*d^2*f*h*(3
*f^2*g^2 + 2*e*f*g*h + 3*e^2*h^2) + d^3*(5*f^3*g^3 + 3*e*f^2*g^2*h + 3*e^2*f*g*h
^2 + 5*e^3*h^3))))*(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))])/(8*b^2
*d^3*Sqrt[b*c - a*d]*f^3*h^4*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)**(3/2)*(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 = 35.8123, size = 38310, normalized size = 27.27 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[((a + b*x)^(3/2)*(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 = 0.276, size = 89498, normalized size = 63.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + A)*(b*x + a)^(3/2)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="maxima")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + A)*(b*x + a)^(3/2)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")

[Out]

integrate((C*x^2 + A)*(b*x + a)^(3/2)/(sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)
), x)