3.2656 \(\int \frac{\sqrt{e+f x}}{(a+b x)^{3/2} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=184 \[ \frac{2 \sqrt{f} \sqrt{c+d x} \sqrt{a f-b e} \sqrt{\frac{b (e+f x)}{b e-a f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{a+b x}}{\sqrt{a f-b e}}\right )|\frac{d (b e-a f)}{(b c-a d) f}\right )}{b \sqrt{e+f x} (b c-a d) \sqrt{\frac{b (c+d x)}{b c-a d}}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x}}{\sqrt{a+b x} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.53936, antiderivative size = 184, 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 \[ \frac{2 \sqrt{f} \sqrt{c+d x} \sqrt{a f-b e} \sqrt{\frac{b (e+f x)}{b e-a f}} E\left (\sin ^{-1}\left (\frac{\sqrt{f} \sqrt{a+b x}}{\sqrt{a f-b e}}\right )|\frac{d (b e-a f)}{(b c-a d) f}\right )}{b \sqrt{e+f x} (b c-a d) \sqrt{\frac{b (c+d x)}{b c-a d}}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x}}{\sqrt{a+b x} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 59.0382, size = 150, normalized size = 0.82 \[ \frac{2 \sqrt{c + d x} \sqrt{e + f x}}{\sqrt{a + b x} \left (a d - b c\right )} - \frac{2 \sqrt{\frac{f \left (a + b x\right )}{a f - b e}} \sqrt{c + d x} \sqrt{- a f + b e} E\left (\operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{e + f x}}{\sqrt{- a f + b e}} \right )}\middle | \frac{d \left (a f - b e\right )}{b \left (c f - d e\right )}\right )}{\sqrt{b} \sqrt{\frac{f \left (c + d x\right )}{c f - d e}} \sqrt{a + b x} \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c + d*x)*sqrt(e + f*x)/(sqrt(a + b*x)*(a*d - b*c)) - 2*sqrt(f*(a + b*x)/(
a*f - b*e))*sqrt(c + d*x)*sqrt(-a*f + b*e)*elliptic_e(asin(sqrt(b)*sqrt(e + f*x)
/sqrt(-a*f + b*e)), d*(a*f - b*e)/(b*(c*f - d*e)))/(sqrt(b)*sqrt(f*(c + d*x)/(c*
f - d*e))*sqrt(a + b*x)*(a*d - b*c))

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Mathematica [A]  time = 1.29133, size = 126, normalized size = 0.68 \[ -\frac{2 \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{d (a+b x)}} E\left (\sin ^{-1}\left (\frac{\sqrt{a-\frac{b c}{d}}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )}{b \sqrt{c+d x} \sqrt{a-\frac{b c}{d}} \sqrt{\frac{b (e+f x)}{f (a+b x)}}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[e + f*x]*EllipticE[ArcSin[Sqrt[a - (b
*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])/(b*Sqrt[a - (b*c)/d]*Sq
rt[c + d*x]*Sqrt[(b*(e + f*x))/(f*(a + b*x))])

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Maple [B]  time = 0.082, size = 1022, normalized size = 5.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x+e)^(1/2)/(b*x+a)^(3/2)/(d*x+c)^(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*b*c
*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*d^2*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^2*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))*b^2*c*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))*a^2*d^2*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*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*d^2*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*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)-x^2*b^2*d^2*f-x*b^2*
c*d*f-x*b^2*d^2*e-b^2*c*d*e)*(d*x+c)^(1/2)*(b*x+a)^(1/2)*(f*x+e)^(1/2)/d/b^2/(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. \[ \int \frac{\sqrt{f x + e}}{{\left (b x + a\right )}^{\frac{3}{2}} \sqrt{d x + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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