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

Optimal. Leaf size=305 \[ \frac{4 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{e^2 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{4 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{2 \sqrt{a+c x^2}}{e \sqrt{d+e x}} \]

[Out]

(-2*Sqrt[a + c*x^2])/(e*Sqrt[d + e*x]) - (4*Sqrt[-a]*Sqrt[c]*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)])/(e^2*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-
a]*e)]*Sqrt[a + c*x^2]) + (4*Sqrt[-a]*Sqrt[c]*d*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c
]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqr
t[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(e^2*Sqrt[d + e*x]*Sqrt[a
+ c*x^2])

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Rubi [A]  time = 0.665677, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{4 \sqrt{-a} \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{e^2 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{4 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{2 \sqrt{a+c x^2}}{e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a + c*x^2])/(e*Sqrt[d + e*x]) - (4*Sqrt[-a]*Sqrt[c]*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)])/(e^2*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-
a]*e)]*Sqrt[a + c*x^2]) + (4*Sqrt[-a]*Sqrt[c]*d*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c
]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqr
t[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(e^2*Sqrt[d + e*x]*Sqrt[a
+ c*x^2])

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Rubi in Sympy [A]  time = 109.264, size = 287, normalized size = 0.94 \[ \frac{4 \sqrt{c} d \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{e^{2} \sqrt{a + c x^{2}} \sqrt{d + e x}} - \frac{4 \sqrt{c} \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{e^{2} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} - \frac{2 \sqrt{a + c x^{2}}}{e \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(c)*d*sqrt(-a)*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a))
)*sqrt(1 + c*x**2/a)*elliptic_f(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*e
/(a*e - sqrt(c)*d*sqrt(-a)))/(e**2*sqrt(a + c*x**2)*sqrt(d + e*x)) - 4*sqrt(c)*s
qrt(-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)))/(e**2*sqrt(sqrt(c)*sqrt(-a)*(-d
- e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(a + c*x**2)) - 2*sqrt(a + c*x**2)/(e*sq
rt(d + e*x))

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Mathematica [C]  time = 2.3417, size = 419, normalized size = 1.37 \[ \frac{2 \left (-\frac{2 \sqrt{a} \sqrt{c} e (d+e x)^{3/2} \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}+\frac{2 \sqrt{c} (d+e x)^{3/2} \left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}+e^2 \left (a+c x^2\right )\right )}{e^3 \sqrt{a+c x^2} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(e^2*(a + c*x^2) + (2*Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)*Sqrt[(e*((I*Sqrt[a
])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d
+ e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]]
, (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-d - (I*Sqrt[a]*e)/
Sqrt[c]] - (2*Sqrt[a]*Sqrt[c]*e*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sq
rt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSi
nh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(S
qrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(e^3*Sqrt[d + e*x]*
Sqrt[a + c*x^2])

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Maple [B]  time = 0.064, size = 646, normalized size = 2.1 \[ -2\,{\frac{\sqrt{ex+d}\sqrt{c{x}^{2}+a}}{ \left ( ce{x}^{3}+cd{x}^{2}+aex+ad \right ){e}^{3}} \left ( 2\,{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) a{e}^{2}\sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e+cd}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e-cd}}}+2\,{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) c{d}^{2}\sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e+cd}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e-cd}}}-2\,{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) a{e}^{2}\sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e+cd}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e-cd}}}-2\,{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}},\sqrt{-{\frac{\sqrt{-ac}e-cd}{\sqrt{-ac}e+cd}}} \right ) de\sqrt{-ac}\sqrt{-{\frac{ \left ( ex+d \right ) c}{\sqrt{-ac}e-cd}}}\sqrt{{\frac{ \left ( -cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e+cd}}}\sqrt{{\frac{ \left ( cx+\sqrt{-ac} \right ) e}{\sqrt{-ac}e-cd}}}+c{e}^{2}{x}^{2}+a{e}^{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError