3.142 \(\int \frac{A+C x^2}{(a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}} \, dx\)

Optimal. Leaf size=1070 \[ -\frac{2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \left (C a^2+A b^2\right )}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac{4 \sqrt{d g-c h} \sqrt{f g-e h} \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{2 \left (-\left (3 A d^2 f h-C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a^2+3 b \left (C c^2+A d^2\right ) (f g+e h) a-b^2 \left ((3 C e g-A f h) c^2+A d (f g+e h) c+2 A d^2 e g\right )\right ) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac{4 b \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}-\frac{4 d \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}} \]

[Out]

(-4*d*(A*b^3*(d*e*g + c*f*g + c*e*h) + a^3*C*(d*f*g + d*e*h + c*f*h) + a^2*b*(3*
A*d*f*h - 2*C*(d*e*g + c*f*g + c*e*h)) - a*b^2*(2*A*d*(f*g + e*h) - c*(3*C*e*g -
2*A*f*h)))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d)^2*(b*e - a
*f)^2*(b*g - a*h)^2*Sqrt[c + d*x]) - (2*(A*b^2 + a^2*C)*Sqrt[c + d*x]*Sqrt[e + f
*x]*Sqrt[g + h*x])/(3*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)^(3/2)) + (4*
b*(A*b^3*(d*e*g + c*f*g + c*e*h) + a^3*C*(d*f*g + d*e*h + c*f*h) + a^2*b*(3*A*d*
f*h - 2*C*(d*e*g + c*f*g + c*e*h)) - a*b^2*(2*A*d*(f*g + e*h) - c*(3*C*e*g - 2*A
*f*h)))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d)^2*(b*e - a*f)^
2*(b*g - a*h)^2*Sqrt[a + b*x]) + (4*Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*(A*b^3*(d*e*
g + c*f*g + c*e*h) + a^3*C*(d*f*g + d*e*h + c*f*h) + a^2*b*(3*A*d*f*h - 2*C*(d*e
*g + c*f*g + c*e*h)) - a*b^2*(2*A*d*(f*g + e*h) - c*(3*C*e*g - 2*A*f*h)))*Sqrt[a
+ b*x]*Sqrt[-(((d*e - c*f)*(g + h*x))/((f*g - e*h)*(c + d*x)))]*EllipticE[ArcSi
n[(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))])/(3*(b*c - a*d)^2*(b*e - a*f)^2*(b*g -
a*h)^2*Sqrt[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - (2
*(3*a*b*(c^2*C + A*d^2)*(f*g + e*h) - b^2*(2*A*d^2*e*g + A*c*d*(f*g + e*h) + c^2
*(3*C*e*g - A*f*h)) - a^2*(3*A*d^2*f*h - C*(d^2*e*g - c*d*f*g - c*d*e*h - 2*c^2*
f*h)))*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*Ellip
ticF[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)))])/(3*(b*c - a*d)^2*(b*e - a
*f)*(b*g - a*h)^(3/2)*Sqrt[f*g - e*h]*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x
))/((f*g - e*h)*(a + b*x)))])

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Rubi [A]  time = 8.21987, antiderivative size = 1070, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.159 \[ -\frac{2 \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} \left (C a^2+A b^2\right )}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}+\frac{4 \sqrt{d g-c h} \sqrt{f g-e h} \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt{a+b x} \sqrt{-\frac{(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{d g-c h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{c+d x}}\right )|\frac{(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{\frac{(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt{g+h x}}-\frac{2 \left (-\left (3 A d^2 f h-C \left (-2 f h c^2-d f g c-d e h c+d^2 e g\right )\right ) a^2+3 b \left (C c^2+A d^2\right ) (f g+e h) a-b^2 \left ((3 C e g-A f h) c^2+A d (f g+e h) c+2 A d^2 e g\right )\right ) \sqrt{\frac{(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt{g+h x} F\left (\sin ^{-1}\left (\frac{\sqrt{b g-a h} \sqrt{e+f x}}{\sqrt{f g-e h} \sqrt{a+b x}}\right )|-\frac{(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt{f g-e h} \sqrt{c+d x} \sqrt{-\frac{(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}+\frac{4 b \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{a+b x}}-\frac{4 d \left (C (d f g+d e h+c f h) a^3+b (3 A d f h-2 C (d e g+c f g+c e h)) a^2-b^2 (2 A d (f g+e h)-c (3 C e g-2 A f h)) a+A b^3 (d e g+c f g+c e h)\right ) \sqrt{a+b x} \sqrt{e+f x} \sqrt{g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt{c+d x}} \]

Warning: Unable to verify antiderivative.

[In]  Int[(A + C*x^2)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]

[Out]

(-4*d*(A*b^3*(d*e*g + c*f*g + c*e*h) + a^3*C*(d*f*g + d*e*h + c*f*h) + a^2*b*(3*
A*d*f*h - 2*C*(d*e*g + c*f*g + c*e*h)) - a*b^2*(2*A*d*(f*g + e*h) - c*(3*C*e*g -
2*A*f*h)))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d)^2*(b*e - a
*f)^2*(b*g - a*h)^2*Sqrt[c + d*x]) - (2*(A*b^2 + a^2*C)*Sqrt[c + d*x]*Sqrt[e + f
*x]*Sqrt[g + h*x])/(3*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)^(3/2)) + (4*
b*(A*b^3*(d*e*g + c*f*g + c*e*h) + a^3*C*(d*f*g + d*e*h + c*f*h) + a^2*b*(3*A*d*
f*h - 2*C*(d*e*g + c*f*g + c*e*h)) - a*b^2*(2*A*d*(f*g + e*h) - c*(3*C*e*g - 2*A
*f*h)))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d)^2*(b*e - a*f)^
2*(b*g - a*h)^2*Sqrt[a + b*x]) + (4*Sqrt[d*g - c*h]*Sqrt[f*g - e*h]*(A*b^3*(d*e*
g + c*f*g + c*e*h) + a^3*C*(d*f*g + d*e*h + c*f*h) + a^2*b*(3*A*d*f*h - 2*C*(d*e
*g + c*f*g + c*e*h)) - a*b^2*(2*A*d*(f*g + e*h) - c*(3*C*e*g - 2*A*f*h)))*Sqrt[a
+ b*x]*Sqrt[-(((d*e - c*f)*(g + h*x))/((f*g - e*h)*(c + d*x)))]*EllipticE[ArcSi
n[(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))])/(3*(b*c - a*d)^2*(b*e - a*f)^2*(b*g -
a*h)^2*Sqrt[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - (2
*(3*a*b*(c^2*C + A*d^2)*(f*g + e*h) - b^2*(2*A*d^2*e*g + A*c*d*(f*g + e*h) + c^2
*(3*C*e*g - A*f*h)) - a^2*(3*A*d^2*f*h - C*(d^2*e*g - c*d*f*g - c*d*e*h - 2*c^2*
f*h)))*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*Ellip
ticF[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)))])/(3*(b*c - a*d)^2*(b*e - a
*f)*(b*g - a*h)^(3/2)*Sqrt[f*g - e*h]*Sqrt[c + d*x]*Sqrt[-(((b*e - a*f)*(g + h*x
))/((f*g - e*h)*(a + b*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)**(5/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)

[Out]

Timed out

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Mathematica [B]  time = 35.4361, size = 19544, normalized size = 18.27 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(A + C*x^2)/((a + b*x)^(5/2)*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 = 1.787, size = 72702, normalized size = 68. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((C*x^2+A)/(b*x+a)^(5/2)/(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. \[ \int \frac{C x^{2} + A}{{\left (b x + a\right )}^{\frac{5}{2}} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + A)/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="maxima")

[Out]

integrate((C*x^2 + A)/((b*x + a)^(5/2)*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. \[{\rm integral}\left (\frac{C x^{2} + A}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + A)/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="fricas")

[Out]

integral((C*x^2 + A)/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)*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((C*x**2+A)/(b*x+a)**(5/2)/(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. \[ \int \frac{C x^{2} + A}{{\left (b x + a\right )}^{\frac{5}{2}} \sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + A)/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)),x, algorithm="giac")

[Out]

integrate((C*x^2 + A)/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)
), x)