3.669 \(\int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

[Out]

(-2*Sqrt[-a]*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c
]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(Sqrt[c]*Sqrt[(Sq
rt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.233475, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]/Sqrt[a + c*x^2],x]

[Out]

(-2*Sqrt[-a]*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c
]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(Sqrt[c]*Sqrt[(Sq
rt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 37.896, size = 129, normalized size = 0.95 \[ - \frac{2 \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{\sqrt{c} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.711662, size = 294, normalized size = 2.16 \[ \frac{2 i \sqrt{d+e x} \left (\sqrt{c} d+i \sqrt{a} e\right ) \sqrt{\frac{e \left (\sqrt{a}+i \sqrt{c} x\right )}{\sqrt{a} e-i \sqrt{c} d}} \left (E\left (i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-i \sqrt{a} e}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )-F\left (i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{c} (d+e x)}{\sqrt{c} d-i \sqrt{a} e}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{\sqrt{c} e \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{e \left (\sqrt{c} x+i \sqrt{a}\right )}}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]/Sqrt[a + c*x^2],x]

[Out]

((2*I)*(Sqrt[c]*d + I*Sqrt[a]*e)*Sqrt[(e*(Sqrt[a] + I*Sqrt[c]*x))/((-I)*Sqrt[c]*
d + Sqrt[a]*e)]*Sqrt[d + e*x]*(EllipticE[I*ArcSinh[Sqrt[-((Sqrt[c]*(d + e*x))/(S
qrt[c]*d - I*Sqrt[a]*e))]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]
- EllipticF[I*ArcSinh[Sqrt[-((Sqrt[c]*(d + e*x))/(Sqrt[c]*d - I*Sqrt[a]*e))]],
(Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(Sqrt[c]*e*Sqrt[(Sqrt[c]*
(d + e*x))/(e*(I*Sqrt[a] + Sqrt[c]*x))]*Sqrt[a + c*x^2])

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Maple [B]  time = 0.043, size = 396, normalized size = 2.9 \[ 2\,{\frac{\sqrt{ex+d}\sqrt{c{x}^{2}+a} \left ( -\sqrt{-ac}e+cd \right ) }{e \left ( ce{x}^{3}+cd{x}^{2}+aex+ad \right ){c}^{2}}\sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e+cd}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e-cd}}} \left ( \sqrt{-ac}{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) e+d{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) c-{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) \sqrt{-ac}e-{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) cd \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)*(-(-a*c)^(1/2)*e+c*d)*(-(e*x+d)*c/((-a*c)^(1/2)*
e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1
/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*((-a*c)^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(
1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*e+d*Ellip
ticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2
)*e+c*d))^(1/2))*c-EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(
1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*(-a*c)^(1/2)*e-EllipticE((-(e*x+d)*c/((
-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c*
d)/e/(c*e*x^3+c*d*x^2+a*e*x+a*d)/c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{\sqrt{c x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d + e x}}{\sqrt{a + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{\sqrt{c x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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