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

Optimal. Leaf size=1096 \[ \frac{2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a+b x)^2}{7 d f h}+\frac{4 C (2 a d f h-3 b (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a+b x)}{35 d^2 f^2 h^2}-\frac{4 \sqrt{c f-d e} \left (3 d f h \left (C (3 b (d e g+c f g+c e h)+2 a (d f g+d e h+c f h)) (2 a d f h-3 b (d f g+d e h+c f h))+5 d f h \left (C (d f g+d e h+c f h) a^2-b (7 A d f h-3 C (d e g+c f g+c e h)) a+2 b^2 c C e g\right )\right )+(d f g+d e h+c f h) (4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a d f h-2 b (d f g+d e h+c f h))+5 b d f h (7 A b d f h-C (5 b (d e g+c f g+c e h)+2 a (d f g+d e h+c f h))))\right ) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{105 d^4 f^{7/2} h^4 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 \sqrt{c f-d e} \left (\left (35 A d^2 f^2 (c h (f g-e h)+d g (2 f g+e h)) h^2+C \left (g \left (48 f^3 g^3+16 e f^2 h g^2+17 e^2 f h^2 g+24 e^3 h^3\right ) d^3+2 c h \left (8 f^3 g^3+e f^2 h g^2+3 e^2 f h^2 g-12 e^3 h^3\right ) d^2+c^2 f h^2 \left (17 f^2 g^2+6 e f h g-23 e^2 h^2\right ) d+24 c^3 f^2 h^3 (f g-e h)\right )\right ) b^2-14 a d f h \left (15 A d^2 f^2 g h^2+C \left (g \left (8 f^2 g^2+3 e f h g+4 e^2 h^2\right ) d^2+c h \left (3 f^2 g^2+e f h g-4 e^2 h^2\right ) d+4 c^2 f h^2 (f g-e h)\right )\right ) b+35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+C (c h (f g-e h)+d g (2 f g+e h))\right )\right ) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{105 d^4 f^{7/2} h^4 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 (4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a d f h-2 b (d f g+d e h+c f h))+5 b d f h (7 A b d f h-C (5 b (d e g+c f g+c e h)+2 a (d f g+d e h+c f h)))) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{105 d^3 f^3 h^3} \]

[Out]

(2*(4*C*(2*a*d*f*h - 3*b*(d*f*g + d*e*h + c*f*h))*(a*d*f*h - 2*b*(d*f*g + d*e*h
+ c*f*h)) + 5*b*d*f*h*(7*A*b*d*f*h - C*(5*b*(d*e*g + c*f*g + c*e*h) + 2*a*(d*f*g
+ d*e*h + c*f*h))))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(105*d^3*f^3*h^3
) + (4*C*(2*a*d*f*h - 3*b*(d*f*g + d*e*h + c*f*h))*(a + b*x)*Sqrt[c + d*x]*Sqrt[
e + f*x]*Sqrt[g + h*x])/(35*d^2*f^2*h^2) + (2*C*(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e
+ f*x]*Sqrt[g + h*x])/(7*d*f*h) - (4*Sqrt[-(d*e) + c*f]*(3*d*f*h*(C*(3*b*(d*e*g
+ c*f*g + c*e*h) + 2*a*(d*f*g + d*e*h + c*f*h))*(2*a*d*f*h - 3*b*(d*f*g + d*e*h
+ c*f*h)) + 5*d*f*h*(2*b^2*c*C*e*g + a^2*C*(d*f*g + d*e*h + c*f*h) - a*b*(7*A*d
*f*h - 3*C*(d*e*g + c*f*g + c*e*h)))) + (d*f*g + d*e*h + c*f*h)*(4*C*(2*a*d*f*h
- 3*b*(d*f*g + d*e*h + c*f*h))*(a*d*f*h - 2*b*(d*f*g + d*e*h + c*f*h)) + 5*b*d*f
*h*(7*A*b*d*f*h - C*(5*b*(d*e*g + c*f*g + c*e*h) + 2*a*(d*f*g + d*e*h + c*f*h)))
))*Sqrt[(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))])/(105*d^4*f^(7/2
)*h^4*Sqrt[e + f*x]*Sqrt[(d*(g + h*x))/(d*g - c*h)]) + (2*Sqrt[-(d*e) + c*f]*(35
*a^2*d^2*f^2*h^2*(3*A*d*f*h^2 + C*(c*h*(f*g - e*h) + d*g*(2*f*g + e*h))) - 14*a*
b*d*f*h*(15*A*d^2*f^2*g*h^2 + C*(4*c^2*f*h^2*(f*g - e*h) + c*d*h*(3*f^2*g^2 + e*
f*g*h - 4*e^2*h^2) + d^2*g*(8*f^2*g^2 + 3*e*f*g*h + 4*e^2*h^2))) + b^2*(35*A*d^2
*f^2*h^2*(c*h*(f*g - e*h) + d*g*(2*f*g + e*h)) + C*(24*c^3*f^2*h^3*(f*g - e*h) +
c^2*d*f*h^2*(17*f^2*g^2 + 6*e*f*g*h - 23*e^2*h^2) + 2*c*d^2*h*(8*f^3*g^3 + e*f^
2*g^2*h + 3*e^2*f*g*h^2 - 12*e^3*h^3) + d^3*g*(48*f^3*g^3 + 16*e*f^2*g^2*h + 17*
e^2*f*g*h^2 + 24*e^3*h^3))))*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))])/(105*d^4*f^(7/2)*h^4*Sqrt[e + f*x]*Sqrt[g + h*x])

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Rubi [A]  time = 13.6873, antiderivative size = 1080, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 7, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 C \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a+b x)^2}{7 d f h}+\frac{4 C (2 a d f h-3 b (d f g+d e h+c f h)) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x} (a+b x)}{35 d^2 f^2 h^2}-\frac{4 \sqrt{c f-d e} \left ((d f g+d e h+c f h) (4 C (2 a d f h-3 b (d f g+d e h+c f h)) (a d f h-2 b (d f g+d e h+c f h))+5 b d f h (7 A b d f h-5 b C (d e g+c f g+c e h)-2 a C (d f g+d e h+c f h)))+3 d f h \left (C (3 b (d e g+c f g+c e h)+2 a (d f g+d e h+c f h)) (2 a d f h-3 b (d f g+d e h+c f h))+5 d f h \left (C (d f g+d e h+c f h) a^2-b (7 A d f h-3 C (d e g+c f g+c e h)) a+2 b^2 c C e g\right )\right )\right ) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{g+h x} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{105 d^4 f^{7/2} h^4 \sqrt{e+f x} \sqrt{\frac{d (g+h x)}{d g-c h}}}+\frac{2 \sqrt{c f-d e} \left (\left (35 A d^2 f^2 (c h (f g-e h)+d g (2 f g+e h)) h^2+C \left (g \left (48 f^3 g^3+16 e f^2 h g^2+17 e^2 f h^2 g+24 e^3 h^3\right ) d^3+2 c h \left (8 f^3 g^3+e f^2 h g^2+3 e^2 f h^2 g-12 e^3 h^3\right ) d^2+c^2 f h^2 \left (17 f^2 g^2+6 e f h g-23 e^2 h^2\right ) d+24 c^3 f^2 h^3 (f g-e h)\right )\right ) b^2-14 a d f h \left (15 A d^2 f^2 g h^2+C \left (g \left (8 f^2 g^2+3 e f h g+4 e^2 h^2\right ) d^2+c h \left (3 f^2 g^2+e f h g-4 e^2 h^2\right ) d+4 c^2 f h^2 (f g-e h)\right )\right ) b+35 a^2 d^2 f^2 h^2 \left (3 A d f h^2+c C (f g-e h) h+C d g (2 f g+e h)\right )\right ) \sqrt{\frac{d (e+f x)}{d e-c f}} \sqrt{\frac{d (g+h x)}{d g-c h}} F\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|\frac{(d e-c f) h}{f (d g-c h)}\right )}{105 d^4 f^{7/2} h^4 \sqrt{e+f x} \sqrt{g+h x}}+\frac{2 \left (8 C d f h a^2-38 b C (d f g+d e h+c f h) a+\frac{24 b^2 C (d f g+d e h+c f h)^2}{d f h}+35 A b^2 d f h-25 b^2 C (d e g+c f g+c e h)\right ) \sqrt{c+d x} \sqrt{e+f x} \sqrt{g+h x}}{105 d^2 f^2 h^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*(35*A*b^2*d*f*h + 8*a^2*C*d*f*h - 25*b^2*C*(d*e*g + c*f*g + c*e*h) - 38*a*b*C
*(d*f*g + d*e*h + c*f*h) + (24*b^2*C*(d*f*g + d*e*h + c*f*h)^2)/(d*f*h))*Sqrt[c
+ d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(105*d^2*f^2*h^2) + (4*C*(2*a*d*f*h - 3*b*(d
*f*g + d*e*h + c*f*h))*(a + b*x)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(35*
d^2*f^2*h^2) + (2*C*(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(7*d*
f*h) - (4*Sqrt[-(d*e) + c*f]*((d*f*g + d*e*h + c*f*h)*(4*C*(2*a*d*f*h - 3*b*(d*f
*g + d*e*h + c*f*h))*(a*d*f*h - 2*b*(d*f*g + d*e*h + c*f*h)) + 5*b*d*f*h*(7*A*b*
d*f*h - 5*b*C*(d*e*g + c*f*g + c*e*h) - 2*a*C*(d*f*g + d*e*h + c*f*h))) + 3*d*f*
h*(C*(3*b*(d*e*g + c*f*g + c*e*h) + 2*a*(d*f*g + d*e*h + c*f*h))*(2*a*d*f*h - 3*
b*(d*f*g + d*e*h + c*f*h)) + 5*d*f*h*(2*b^2*c*C*e*g + a^2*C*(d*f*g + d*e*h + c*f
*h) - a*b*(7*A*d*f*h - 3*C*(d*e*g + c*f*g + c*e*h)))))*Sqrt[(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))])/(105*d^4*f^(7/2)*h^4*Sqrt[e + f*x]*Sqrt[(d*
(g + h*x))/(d*g - c*h)]) + (2*Sqrt[-(d*e) + c*f]*(35*a^2*d^2*f^2*h^2*(3*A*d*f*h^
2 + c*C*h*(f*g - e*h) + C*d*g*(2*f*g + e*h)) - 14*a*b*d*f*h*(15*A*d^2*f^2*g*h^2
+ C*(4*c^2*f*h^2*(f*g - e*h) + c*d*h*(3*f^2*g^2 + e*f*g*h - 4*e^2*h^2) + d^2*g*(
8*f^2*g^2 + 3*e*f*g*h + 4*e^2*h^2))) + b^2*(35*A*d^2*f^2*h^2*(c*h*(f*g - e*h) +
d*g*(2*f*g + e*h)) + C*(24*c^3*f^2*h^3*(f*g - e*h) + c^2*d*f*h^2*(17*f^2*g^2 + 6
*e*f*g*h - 23*e^2*h^2) + 2*c*d^2*h*(8*f^3*g^3 + e*f^2*g^2*h + 3*e^2*f*g*h^2 - 12
*e^3*h^3) + d^3*g*(48*f^3*g^3 + 16*e*f^2*g^2*h + 17*e^2*f*g*h^2 + 24*e^3*h^3))))
*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Sqrt[(d*(g + h*x))/(d*g - c*h)]*EllipticF[ArcSi
n[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], ((d*e - c*f)*h)/(f*(d*g - c*h))])
/(105*d^4*f^(7/2)*h^4*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((b*x+a)**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 [C]  time = 23.9218, size = 18383, normalized size = 16.77 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^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 = 0.1, size = 12279, normalized size = 11.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{C b^{2} x^{4} + 2 \, C a b x^{3} + 2 \, A a b x + A a^{2} +{\left (C a^{2} + A b^{2}\right )} x^{2}}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((C*b^2*x^4 + 2*C*a*b*x^3 + 2*A*a*b*x + A*a^2 + (C*a^2 + A*b^2)*x^2)/(sq
rt(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((b*x+a)**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. \[ \int \frac{{\left (C x^{2} + A\right )}{\left (b x + a\right )}^{2}}{\sqrt{d x + c} \sqrt{f x + e} \sqrt{h x + g}}\,{d x} \]

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
)