∖int 1_______________ (a+bx)5∕2√ ___ c+dx√ ___ e+fx∖,dx

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3.2841 \(\int \frac{1}{(a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=437 \[ \frac{4 b \sqrt{c+d x} \sqrt{e+f x} (-2 a d f+b c f+b d e)}{3 \sqrt{a+b x} (b c-a d)^2 (b e-a f)^2}-\frac{2 b \sqrt{c+d x} \sqrt{e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}+\frac{2 \sqrt{d} \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} (-3 a d f+b c f+2 b d e) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b \sqrt{c+d x} \sqrt{e+f x} (a d-b c)^{3/2} (b e-a f)}-\frac{4 \sqrt{d} \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} (-2 a d f+b c f+b d e) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 \sqrt{c+d x} (a d-b c)^{3/2} (b e-a f)^2 \sqrt{\frac{b (e+f x)}{b e-a f}}} \]

[Out]

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

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

*e - a*f)^2*Sqrt[a + b*x]) - (4*Sqrt[d]*(b*d*e + b*c*f - 2*a*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))])/(3*(-(b*c) + a*d)^(3/2)*(b*e - a

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

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

lipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(

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

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Rubi [A] time = 1.90828, antiderivative size = 437, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{4 b \sqrt{c+d x} \sqrt{e+f x} (-2 a d f+b c f+b d e)}{3 \sqrt{a+b x} (b c-a d)^2 (b e-a f)^2}-\frac{2 b \sqrt{c+d x} \sqrt{e+f x}}{3 (a+b x)^{3/2} (b c-a d) (b e-a f)}+\frac{2 \sqrt{d} \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} (-3 a d f+b c f+2 b d e) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b \sqrt{c+d x} \sqrt{e+f x} (a d-b c)^{3/2} (b e-a f)}-\frac{4 \sqrt{d} \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} (-2 a d f+b c f+b d e) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 \sqrt{c+d x} (a d-b c)^{3/2} (b e-a f)^2 \sqrt{\frac{b (e+f x)}{b e-a f}}} \]

Antiderivative was successfully veriﬁed.

[In] Int[1/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

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

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

*e - a*f)^2*Sqrt[a + b*x]) - (4*Sqrt[d]*(b*d*e + b*c*f - 2*a*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))])/(3*(-(b*c) + a*d)^(3/2)*(b*e - a

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

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

lipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(

b*e - a*f))])/(3*b*(-(b*c) + a*d)^(3/2)*(b*e - a*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(1/(b*x+a)**(5/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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Mathematica [C] time = 6.11028, size = 449, normalized size = 1.03 \[ -\frac{2 \left (b^2 (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} ((b c-a d) (b e-a f)-2 (a+b x) (-2 a d f+b c f+b d e))+(a+b x) \left (2 b^2 (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} (-2 a d f+b c f+b d e)-i f (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} (-3 a d f+2 b c f+b d e) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )+2 i f (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} (-2 a d f+b c f+b d e) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )\right )\right )}{3 b (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{\frac{b c}{d}-a} (b c-a d)^2 (b e-a f)^2} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[1/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

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

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

b*c*f - 2*a*d*f)*(c + d*x)*(e + f*x) + (2*I)*(b*c - a*d)*f*(b*d*e + b*c*f - 2*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))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)

/(b*c*f - a*d*f)] - I*(b*c - a*d)*f*(b*d*e + 2*b*c*f - 3*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)])))/(

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

*Sqrt[e + f*x])

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Maple [B] time = 0.109, size = 4067, normalized size = 9.3 \[ \text{output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int(1/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x)

[Out]

2/3*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(-x^2*a*b^3*c*d*f^2-4*x^3*a*b^3*d^2*f^2-4*Ellipt

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

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

)*x*a^2*b^2*d^2*e*f*(d*(b*x+a)/(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)+6*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/2),((a*d-b*c)*f/d

/(a*f-b*e))^(1/2))*x*a^2*b^2*c*d*f^2*(d*(b*x+a)/(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)+6*EllipticE((d*(b*x+a)/(a*d-b*c))^(1/

2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*x*a^2*b^2*d^2*e*f*(d*(b*x+a)/(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)+6*EllipticF((d*(b*x+

a)/(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*(d*(b*x+a)/

(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)-8*Ell

ipticE((d*(b*x+a)/(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*(d*(b*x+a)/(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)+6*EllipticE((d*(b*x+a)/(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*(d*(b*x+a)/(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*EllipticE((d*(b*x+a)/(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*(d*(b*x+a)/(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*EllipticE((d*(b*x+a)/(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*(d*(b*x+a)/(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*EllipticF((d*(b*x+a)/(a*d-b*c

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

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

)/(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*E

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

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

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

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

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

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

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

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

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

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

b*x+a)/(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*(d*(b*x+a

)/(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^4

*c^2*e^2+2*x^2*b^4*c^2*f^2+2*x^2*b^4*d^2*e^2+6*EllipticF((d*(b*x+a)/(a*d-b*c))^(

1/2),((a*d-b*c)*f/d/(a*f-b*e))^(1/2))*x*a*b^3*c*d*e*f*(d*(b*x+a)/(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)-8*EllipticE((d*(b*x+

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

(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)+6*Ell

ipticE((d*(b*x+a)/(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*(d*(b*x+a)/(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)-x^2*a*b^3*d^2*e*f+3*x^2*b^4*c*d*e*f-5*x*a^2*b^2*c*d*f^2-5*x*a^2*b^2*d^2*

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

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

*c)*f/d/(a*f-b*e))^(1/2))*a^4*d^2*f^2*(d*(b*x+a)/(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*a*b^3*c^2*e*f+3*a*b^3*c*d*e^2+2*x^

3*b^4*c*d*f^2+2*x^3*b^4*d^2*e*f-5*x^2*a^2*b^2*d^2*f^2+3*x*a*b^3*c^2*f^2+3*x*a*b^

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

((d*(b*x+a)/(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*(d

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

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

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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}} \sqrt{d x + c} \sqrt{f x + e}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)),x, algorithm="maxima")

[Out]

integrate(1/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)), x)

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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e}}, x\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)),x, algorithm="fricas")

[Out]

integral(1/((b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)*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(1/(b*x+a)**(5/2)/(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. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{2}} \sqrt{d x + c} \sqrt{f x + e}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate(1/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)),x, algorithm="giac")

[Out]

integrate(1/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)), x)

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