3.2840 \(\int \frac{1}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=204 \[ \frac{2 \sqrt{f} \sqrt{a+b x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|-\frac{b (d e-c f)}{(b c-a d) f}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) \sqrt{-\frac{d (a+b x)}{b c-a d}}}-\frac{2 b \sqrt{c+d x} \sqrt{e+f x}}{\sqrt{a+b x} (b c-a d) (b e-a f)} \]

[Out]

(-2*b*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]) + (2*
Sqrt[f]*Sqrt[-(d*e) + c*f]*Sqrt[a + b*x]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Ellipti
cE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], -((b*(d*e - c*f))/((b*c -
a*d)*f))])/((b*c - a*d)*(b*e - a*f)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Sqrt[e +
f*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.591382, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{2 \sqrt{f} \sqrt{a+b x} \sqrt{c f-d e} \sqrt{\frac{d (e+f x)}{d e-c f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{c+d x}}{\sqrt{c f-d e}}\right )|-\frac{b (d e-c f)}{(b c-a d) f}\right )}{\sqrt{e+f x} (b c-a d) (b e-a f) \sqrt{-\frac{d (a+b x)}{b c-a d}}}-\frac{2 b \sqrt{c+d x} \sqrt{e+f x}}{\sqrt{a+b x} (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

(-2*b*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]) + (2*
Sqrt[f]*Sqrt[-(d*e) + c*f]*Sqrt[a + b*x]*Sqrt[(d*(e + f*x))/(d*e - c*f)]*Ellipti
cE[ArcSin[(Sqrt[f]*Sqrt[c + d*x])/Sqrt[-(d*e) + c*f]], -((b*(d*e - c*f))/((b*c -
a*d)*f))])/((b*c - a*d)*(b*e - a*f)*Sqrt[-((d*(a + b*x))/(b*c - a*d))]*Sqrt[e +
f*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 74.3414, size = 167, normalized size = 0.82 \[ - \frac{2 b \sqrt{c + d x} \sqrt{e + f x}}{\sqrt{a + b x} \left (a d - b c\right ) \left (a f - b e\right )} + \frac{2 \sqrt{f} \sqrt{\frac{d \left (- e - f x\right )}{c f - d e}} \sqrt{a + b x} \sqrt{c f - d e} E\left (\operatorname{asin}{\left (\frac{\sqrt{f} \sqrt{c + d x}}{\sqrt{c f - d e}} \right )}\middle | \frac{b \left (- c f + d e\right )}{f \left (a d - b c\right )}\right )}{\sqrt{\frac{d \left (a + b x\right )}{a d - b c}} \sqrt{e + f x} \left (a d - b c\right ) \left (a f - b e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x+a)**(3/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

-2*b*sqrt(c + d*x)*sqrt(e + f*x)/(sqrt(a + b*x)*(a*d - b*c)*(a*f - b*e)) + 2*sqr
t(f)*sqrt(d*(-e - f*x)/(c*f - d*e))*sqrt(a + b*x)*sqrt(c*f - d*e)*elliptic_e(asi
n(sqrt(f)*sqrt(c + d*x)/sqrt(c*f - d*e)), b*(-c*f + d*e)/(f*(a*d - b*c)))/(sqrt(
d*(a + b*x)/(a*d - b*c))*sqrt(e + f*x)*(a*d - b*c)*(a*f - b*e))

_______________________________________________________________________________________

Mathematica [C]  time = 1.75697, size = 201, normalized size = 0.99 \[ \frac{2 b \sqrt{c+d x} \sqrt{e+f x} \left (-1-\frac{i \sqrt{\frac{d (a+b x)}{b (c+d x)}} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{d (a+b x)}{b c-a d}}\right )|\frac{b c f-a d f}{b d e-a d f}\right )-F\left (i \sinh ^{-1}\left (\sqrt{\frac{d (a+b x)}{b c-a d}}\right )|\frac{b c f-a d f}{b d e-a d f}\right )\right )}{\sqrt{\frac{b (e+f x)}{b e-a f}}}\right )}{\sqrt{a+b x} (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

(2*b*Sqrt[c + d*x]*Sqrt[e + f*x]*(-1 - (I*Sqrt[(d*(a + b*x))/(b*(c + d*x))]*(Ell
ipticE[I*ArcSinh[Sqrt[(d*(a + b*x))/(b*c - a*d)]], (b*c*f - a*d*f)/(b*d*e - a*d*
f)] - EllipticF[I*ArcSinh[Sqrt[(d*(a + b*x))/(b*c - a*d)]], (b*c*f - a*d*f)/(b*d
*e - a*d*f)]))/Sqrt[(b*(e + f*x))/(b*e - a*f)]))/((b*c - a*d)*(b*e - a*f)*Sqrt[a
+ b*x])

_______________________________________________________________________________________

Maple [B]  time = 0.049, size = 1011, normalized size = 5. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c} \sqrt{f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c} \sqrt{f x + e}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \sqrt{e + f x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**(3/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c} \sqrt{f x + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*sqrt(d*x + c)*sqrt(f*x + e)),x, algorithm="giac")

[Out]

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