3.45 \(\int \frac{\sqrt{a x+b x^3}}{x^4} \, dx\)

Optimal. Leaf size=283 \[ \frac{2 b^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a x+b x^3}}-\frac{4 b^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a x+b x^3}}+\frac{4 b^{3/2} x \left (a+b x^2\right )}{5 a \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}-\frac{4 b \sqrt{a x+b x^3}}{5 a x}-\frac{2 \sqrt{a x+b x^3}}{5 x^3} \]

[Out]

(4*b^(3/2)*x*(a + b*x^2))/(5*a*(Sqrt[a] + Sqrt[b]*x)*Sqrt[a*x + b*x^3]) - (2*Sqr
t[a*x + b*x^3])/(5*x^3) - (4*b*Sqrt[a*x + b*x^3])/(5*a*x) - (4*b^(5/4)*Sqrt[x]*(
Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTa
n[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(5*a^(3/4)*Sqrt[a*x + b*x^3]) + (2*b^(5/4)*S
qrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF
[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(5*a^(3/4)*Sqrt[a*x + b*x^3])

_______________________________________________________________________________________

Rubi [A]  time = 0.509398, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412 \[ \frac{2 b^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a x+b x^3}}-\frac{4 b^{5/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 a^{3/4} \sqrt{a x+b x^3}}+\frac{4 b^{3/2} x \left (a+b x^2\right )}{5 a \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}-\frac{4 b \sqrt{a x+b x^3}}{5 a x}-\frac{2 \sqrt{a x+b x^3}}{5 x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x + b*x^3]/x^4,x]

[Out]

(4*b^(3/2)*x*(a + b*x^2))/(5*a*(Sqrt[a] + Sqrt[b]*x)*Sqrt[a*x + b*x^3]) - (2*Sqr
t[a*x + b*x^3])/(5*x^3) - (4*b*Sqrt[a*x + b*x^3])/(5*a*x) - (4*b^(5/4)*Sqrt[x]*(
Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTa
n[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(5*a^(3/4)*Sqrt[a*x + b*x^3]) + (2*b^(5/4)*S
qrt[x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF
[2*ArcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(5*a^(3/4)*Sqrt[a*x + b*x^3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 50.7563, size = 262, normalized size = 0.93 \[ - \frac{2 \sqrt{a x + b x^{3}}}{5 x^{3}} + \frac{4 b^{\frac{3}{2}} \sqrt{a x + b x^{3}}}{5 a \left (\sqrt{a} + \sqrt{b} x\right )} - \frac{4 b \sqrt{a x + b x^{3}}}{5 a x} - \frac{4 b^{\frac{5}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{3}{4}} \sqrt{x} \left (a + b x^{2}\right )} + \frac{2 b^{\frac{5}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 a^{\frac{3}{4}} \sqrt{x} \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a*x)**(1/2)/x**4,x)

[Out]

-2*sqrt(a*x + b*x**3)/(5*x**3) + 4*b**(3/2)*sqrt(a*x + b*x**3)/(5*a*(sqrt(a) + s
qrt(b)*x)) - 4*b*sqrt(a*x + b*x**3)/(5*a*x) - 4*b**(5/4)*sqrt((a + b*x**2)/(sqrt
(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*sqrt(a*x + b*x**3)*elliptic_e(2*atan(
b**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(5*a**(3/4)*sqrt(x)*(a + b*x**2)) + 2*b**(5/4)*
sqrt((a + b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*sqrt(a*x + b*x
**3)*elliptic_f(2*atan(b**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(5*a**(3/4)*sqrt(x)*(a +
b*x**2))

_______________________________________________________________________________________

Mathematica [C]  time = 0.347662, size = 192, normalized size = 0.68 \[ -\frac{2 \left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (a^2+3 a b x^2+2 b^2 x^4\right )+2 \sqrt{a} b^{3/2} x^3 \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )-2 \sqrt{a} b^{3/2} x^3 \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )\right )}{5 a x^2 \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a*x + b*x^3]/x^4,x]

[Out]

(-2*(Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*(a^2 + 3*a*b*x^2 + 2*b^2*x^4) - 2*Sqrt[a]*b^(3/
2)*x^3*Sqrt[1 + (b*x^2)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]], -1]
+ 2*Sqrt[a]*b^(3/2)*x^3*Sqrt[1 + (b*x^2)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b]
*x)/Sqrt[a]]], -1]))/(5*a*x^2*Sqrt[(I*Sqrt[b]*x)/Sqrt[a]]*Sqrt[x*(a + b*x^2)])

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 201, normalized size = 0.7 \[ -{\frac{2}{5\,{x}^{3}}\sqrt{b{x}^{3}+ax}}-{\frac{ \left ( 4\,b{x}^{2}+4\,a \right ) b}{5\,a}{\frac{1}{\sqrt{x \left ( b{x}^{2}+a \right ) }}}}+{\frac{2\,b}{5\,a}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}} \left ( -2\,{\frac{\sqrt{-ab}}{b}{\it EllipticE} \left ( \sqrt{{\frac{b}{\sqrt{-ab}} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }},1/2\,\sqrt{2} \right ) }+{\frac{1}{b}\sqrt{-ab}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a*x)^(1/2)/x^4,x)

[Out]

-2/5*(b*x^3+a*x)^(1/2)/x^3-4/5*(b*x^2+a)*b/a/(x*(b*x^2+a))^(1/2)+2/5/a*b*(-a*b)^
(1/2)*((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2)*(-2*(x-1/b*(-a*b)^(1/2))*b/(-a
*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)/(b*x^3+a*x)^(1/2)*(-2/b*(-a*b)^(1/2)*
EllipticE(((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))+1/b*(-a*b)^(1
/2)*EllipticF(((x+1/b*(-a*b)^(1/2))*b/(-a*b)^(1/2))^(1/2),1/2*2^(1/2)))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{3} + a x}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a*x)/x^4,x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^3 + a*x)/x^4, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{b x^{3} + a x}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a*x)/x^4,x, algorithm="fricas")

[Out]

integral(sqrt(b*x^3 + a*x)/x^4, x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (a + b x^{2}\right )}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a*x)**(1/2)/x**4,x)

[Out]

Integral(sqrt(x*(a + b*x**2))/x**4, x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{3} + a x}}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^3 + a*x)/x^4,x, algorithm="giac")

[Out]

integrate(sqrt(b*x^3 + a*x)/x^4, x)