∖int A+Bx+Cx2 _____________________________________(a+bx)3∕2 √ ___ c+dx√ ___ e+fx∖,dx

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

Optimal. Leaf size=422 \[ -\frac{2 \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{b^2 \sqrt{d} f \sqrt{c+d x} \sqrt{a d-b c} (b e-a f) \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{b \sqrt{a+b x} (b c-a d) (b e-a f)}-\frac{2 \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} (a C (d e-c f)-b (A d f-B c f+c C e)) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{b^2 \sqrt{d} f \sqrt{c+d x} \sqrt{e+f x} \sqrt{a d-b c}} \]

[Out]

(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(b*(b*c - a*d)*(b*e - a

*f)*Sqrt[a + b*x]) - (2*(2*a^2*C*d*f + b^2*(c*C*e + A*d*f) - a*b*(C*d*e + c*C*f

+ B*d*f))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d

]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b^2*Sqr

t[d]*Sqrt[-(b*c) + a*d]*f*(b*e - a*f)*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*

f)]) - (2*(a*C*(d*e - c*f) - b*(c*C*e - B*c*f + A*d*f))*Sqrt[(b*(c + d*x))/(b*c

- a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])

/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b^2*Sqrt[d]*Sqrt[-(b*c)

+ a*d]*f*Sqrt[c + d*x]*Sqrt[e + f*x])

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Rubi [A] time = 2.04927, antiderivative size = 422, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{2 \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{b^2 \sqrt{d} f \sqrt{c+d x} \sqrt{a d-b c} (b e-a f) \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{b \sqrt{a+b x} (b c-a d) (b e-a f)}-\frac{2 \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} (a C (d e-c f)-b (A d f-B c f+c C e)) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{b^2 \sqrt{d} f \sqrt{c+d x} \sqrt{e+f x} \sqrt{a d-b c}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(b*(b*c - a*d)*(b*e - a

*f)*Sqrt[a + b*x]) - (2*(2*a^2*C*d*f + b^2*(c*C*e + A*d*f) - a*b*(C*d*e + c*C*f

+ B*d*f))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d

]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b^2*Sqr

t[d]*Sqrt[-(b*c) + a*d]*f*(b*e - a*f)*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*

f)]) - (2*(a*C*(d*e - c*f) - b*(c*C*e - B*c*f + A*d*f))*Sqrt[(b*(c + d*x))/(b*c

- a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])

/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b^2*Sqrt[d]*Sqrt[-(b*c)

+ a*d]*f*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*x**2+B*x+A)/(b*x+a)**(3/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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Mathematica [C] time = 9.64346, size = 477, normalized size = 1.13 \[ \frac{2 \left (\frac{b^2 (c+d x) (e+f x) \left (2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right )}{d f}+\frac{i (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left (2 a^2 C d f-a b (B d f+c C f+C d e)+b^2 (A d f+c C e)\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )}{d \sqrt{\frac{b c}{d}-a}}+b^2 (c+d x) (e+f x) \left (-\left (a (a C-b B)+A b^2\right )\right )+\frac{i b (a+b x)^{3/2} (a d-b c) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} (a C (d e-c f)+b (A d f-B d e+c C e)) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )}{d \sqrt{\frac{b c}{d}-a}}\right )}{b^3 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x} (b c-a d) (b e-a f)} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(2*(-(b^2*(A*b^2 + a*(-(b*B) + a*C))*(c + d*x)*(e + f*x)) + (b^2*(2*a^2*C*d*f +

b^2*(c*C*e + A*d*f) - a*b*(C*d*e + c*C*f + B*d*f))*(c + d*x)*(e + f*x))/(d*f) +

(I*(b*c - a*d)*(2*a^2*C*d*f + b^2*(c*C*e + A*d*f) - a*b*(C*d*e + c*C*f + B*d*f))

*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*

x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c

*f - a*d*f)])/(Sqrt[-a + (b*c)/d]*d) + (I*b*(-(b*c) + a*d)*(a*C*(d*e - c*f) + b*

(c*C*e - B*d*e + A*d*f))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[

(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x

]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])/(Sqrt[-a + (b*c)/d]*d)))/(b^3*(b*c - a*d)*

(b*e - a*f)*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])

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Maple [B] time = 0.059, size = 3979, normalized size = 9.4 \[ \text{output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int((C*x^2+B*x+A)/(b*x+a)^(3/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x)

[Out]

-2*(-B*x^2*a*b^3*d^2*f^2-B*a*b^3*c*d*e*f+C*a^2*b^2*c*d*e*f+A*b^4*c*d*e*f+A*x^2*b

^4*d^2*f^2-A*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/

2))*a*b^3*c*d*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*

x+c)*b/(a*d-b*c))^(1/2)-A*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(

a*f-b*e))^(1/2))*a*b^3*d^2*e*f*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e)

)^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+A*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a

*d-b*c)*f/d/(a*f-b*e))^(1/2))*b^4*c*d*e*f*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*

b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+A*EllipticF(((b*x+a)*d/(a*d-b*c)

)^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*c*d*f^2*((b*x+a)*d/(a*d-b*c))^(1/

2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+A*EllipticF(((b*x+a

)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*d^2*e*f*((b*x+a)*d/(

a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-A*Elli

pticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*b^4*c*d*e*f*(

(b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(

1/2)+B*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^

2*b^2*c*d*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)

*b/(a*d-b*c))^(1/2)+B*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-

b*e))^(1/2))*a^2*b^2*d^2*e*f*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^

(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+B*EllipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d

-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b^2*c*d*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e

)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-3*C*EllipticE(((b*x+a)*d/(a*d-

b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^3*b*c*d*f^2*((b*x+a)*d/(a*d-b*c))

^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-3*C*EllipticE((

(b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^3*b*d^2*e*f*((b*x+

a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-

2*C*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3

*c^2*e*f*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a

*d-b*c))^(1/2)+2*C*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e

))^(1/2))*a^4*d^2*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(

-(d*x+c)*b/(a*d-b*c))^(1/2)+C*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f

/d/(a*f-b*e))^(1/2))*b^4*c^2*e^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*

e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-2*C*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2)

,((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3*c*d*e^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f

*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-C*EllipticF(((b*x+a)*d/(a*

d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^3*b*c*d*f^2*((b*x+a)*d/(a*d-b*c

))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+C*EllipticF((

(b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^3*b*d^2*e*f*((b*x+

a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+

2*C*EllipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a*b^3

*c*d*e^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a

*d-b*c))^(1/2)-B*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))

^(1/2))*a*b^3*c*d*e*f*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(

-(d*x+c)*b/(a*d-b*c))^(1/2)-B*EllipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f

/d/(a*f-b*e))^(1/2))*a*b^3*c*d*e*f*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-

b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+5*C*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/

2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b^2*c*d*e*f*((b*x+a)*d/(a*d-b*c))^(1/2)*

(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-C*EllipticF(((b*x+a)*d

/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b^2*c*d*e*f*((b*x+a)*d/(a

*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)-A*Ellip

ticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b^2*d^2*f^

2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c)

)^(1/2)-B*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))

*a^3*b*d^2*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c

)*b/(a*d-b*c))^(1/2)-B*EllipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f

-b*e))^(1/2))*a*b^3*c^2*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(

1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+B*EllipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-

b*c)*f/d/(a*f-b*e))^(1/2))*b^4*c^2*e*f*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(

a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+C*EllipticE(((b*x+a)*d/(a*d-b*c))^(

1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b^2*c^2*f^2*((b*x+a)*d/(a*d-b*c))^(1/2

)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+C*EllipticE(((b*x+a)

*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b^2*d^2*e^2*((b*x+a)*d/

(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*c))^(1/2)+C*Ell

ipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*a^2*b^2*c^2*

f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*

c))^(1/2)-C*EllipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2

))*a^2*b^2*d^2*e^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d

*x+c)*b/(a*d-b*c))^(1/2)+C*x^2*a^2*b^2*d^2*f^2+A*x*b^4*c*d*f^2+A*x*b^4*d^2*e*f-C

*EllipticF(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*b^4*c^2*

e^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d*x+c)*b/(a*d-b*

c))^(1/2)+A*EllipticE(((b*x+a)*d/(a*d-b*c))^(1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2

))*a^2*b^2*d^2*f^2*((b*x+a)*d/(a*d-b*c))^(1/2)*(-(f*x+e)*b/(a*f-b*e))^(1/2)*(-(d

*x+c)*b/(a*d-b*c))^(1/2)-B*x*a*b^3*c*d*f^2-B*x*a*b^3*d^2*e*f+C*x*a^2*b^2*c*d*f^2

+C*x*a^2*b^2*d^2*e*f)*(f*x+e)^(1/2)*(d*x+c)^(1/2)*(b*x+a)^(1/2)/d/f/b^3/(a*f-b*e

)/(a*d-b*c)/(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c*e*x+a*c

*e)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((C*x^2 + B*x + A)/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)), 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+B*x+A)/(b*x+a)**(3/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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