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

Optimal. Leaf size=298 \[ \frac{x \sqrt{d+e x}}{a \sqrt{a+c x^2}}+\frac{d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

[Out]

(x*Sqrt[d + e*x])/(a*Sqrt[a + c*x^2]) - (Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*Ellip
ticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*
d - a*e)])/(Sqrt[-a]*Sqrt[c]*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*
Sqrt[a + c*x^2]) + (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)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/
(Sqrt[-a]*Sqrt[c]*d - a*e)])/(Sqrt[-a]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.721086, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{x \sqrt{d+e x}}{a \sqrt{a+c x^2}}+\frac{d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{\sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[d + e*x])/(a*Sqrt[a + c*x^2]) - (Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*Ellip
ticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*
d - a*e)])/(Sqrt[-a]*Sqrt[c]*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*
Sqrt[a + c*x^2]) + (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)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/
(Sqrt[-a]*Sqrt[c]*d - a*e)])/(Sqrt[-a]*Sqrt[c]*Sqrt[d + e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 127.801, size = 277, normalized size = 0.93 \[ \frac{d \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{\sqrt{c} \sqrt{- a} \sqrt{a + c x^{2}} \sqrt{d + e x}} - \frac{\sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{\sqrt{c} \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} + \frac{x \sqrt{d + e x}}{a \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*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*s
qrt(-a)))/(sqrt(c)*sqrt(-a)*sqrt(a + c*x**2)*sqrt(d + e*x)) - 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(-a)*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e -
sqrt(c)*d*sqrt(-a)))*sqrt(a + c*x**2)) + x*sqrt(d + e*x)/(a*sqrt(a + c*x**2))

_______________________________________________________________________________________

Mathematica [C]  time = 5.84045, size = 408, normalized size = 1.37 \[ \frac{\sqrt{d+e x} \left (-\frac{e \left (a+c x^2\right )}{c (d+e x)}+\frac{\sqrt{a} \sqrt{d+e x} \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{c} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}-\frac{i \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{e}+x\right )}{a \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.056, size = 649, normalized size = 2.2 \[{\frac{1}{ce \left ( ce{x}^{3}+cd{x}^{2}+aex+ad \right ) a}\sqrt{ex+d}\sqrt{c{x}^{2}+a} \left ({\it EllipticE} \left ( \sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{1 \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) a{e}^{2}\sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}+{\it EllipticE} \left ( \sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{1 \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) c{d}^{2}\sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}-{\it EllipticF} \left ( \sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{1 \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) a{e}^{2}\sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}-{\it EllipticF} \left ( \sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}},\sqrt{-{1 \left ( \sqrt{-ac}e-cd \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}} \right ) de\sqrt{-ac}\sqrt{-{ \left ( ex+d \right ) c \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}\sqrt{{e \left ( -cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e+cd \right ) ^{-1}}}\sqrt{{e \left ( cx+\sqrt{-ac} \right ) \left ( \sqrt{-ac}e-cd \right ) ^{-1}}}+c{e}^{2}{x}^{2}+cdex \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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(e*x + d)/(c*x^2 + a)^(3/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError