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

Optimal. Leaf size=146 \[ \frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 e \sqrt{b x+c x^2}}{d \sqrt{d+e x} (c d-b e)} \]

[Out]

(-2*e*Sqrt[b*x + c*x^2])/(d*(c*d - b*e)*Sqrt[d + e*x]) + (2*Sqrt[-b]*Sqrt[c]*Sqr
t[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]
], (b*e)/(c*d)])/(d*(c*d - b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 0.310685, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{d \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 e \sqrt{b x+c x^2}}{d \sqrt{d+e x} (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e*Sqrt[b*x + c*x^2])/(d*(c*d - b*e)*Sqrt[d + e*x]) + (2*Sqrt[-b]*Sqrt[c]*Sqr
t[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]
], (b*e)/(c*d)])/(d*(c*d - b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2])

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Rubi in Sympy [A]  time = 40.9853, size = 124, normalized size = 0.85 \[ - \frac{2 \sqrt{c} \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{d \sqrt{1 + \frac{e x}{d}} \left (b e - c d\right ) \sqrt{b x + c x^{2}}} + \frac{2 e \sqrt{b x + c x^{2}}}{d \sqrt{d + e x} \left (b e - c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(c)*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(d + e*x)*elliptic_e(asin(sqrt(c
)*sqrt(x)/sqrt(-b)), b*e/(c*d))/(d*sqrt(1 + e*x/d)*(b*e - c*d)*sqrt(b*x + c*x**2
)) + 2*e*sqrt(b*x + c*x**2)/(d*sqrt(d + e*x)*(b*e - c*d))

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Mathematica [A]  time = 0.275626, size = 127, normalized size = 0.87 \[ \frac{2 \sqrt{x (b+c x)} \left (e \sqrt{x} \sqrt{-\frac{d}{e}} \sqrt{\frac{d}{e x}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{d}{e}}}{\sqrt{x}}\right )|\frac{b e}{c d}\right )+d \sqrt{\frac{b}{c x}+1}\right )}{d x \sqrt{\frac{b}{c x}+1} \sqrt{d+e x} (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x*(b + c*x)]*(d*Sqrt[1 + b/(c*x)] + Sqrt[-(d/e)]*e*Sqrt[1 + d/(e*x)]*Sqr
t[x]*EllipticE[ArcSin[Sqrt[-(d/e)]/Sqrt[x]], (b*e)/(c*d)]))/(d*(c*d - b*e)*Sqrt[
1 + b/(c*x)]*x*Sqrt[d + e*x])

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Maple [A]  time = 0.036, size = 216, normalized size = 1.5 \[ 2\,{\frac{\sqrt{x \left ( cx+b \right ) }\sqrt{ex+d}}{ \left ( be-cd \right ) cdx \left ( ce{x}^{2}+bex+cdx+bd \right ) } \left ({\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ){b}^{2}e\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}-{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) bcd\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}+{x}^{2}{c}^{2}e+xbce \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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