3.72 \(\int \frac{1}{\left (a x+b x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=273 \[ \frac{3 \sqrt [4]{b} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{7/4} \sqrt{a x+b x^3}}-\frac{3 \sqrt [4]{b} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{7/4} \sqrt{a x+b x^3}}-\frac{3 \sqrt{a x+b x^3}}{a^2 x}+\frac{3 \sqrt{b} x \left (a+b x^2\right )}{a^2 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}+\frac{1}{a \sqrt{a x+b x^3}} \]

[Out]

1/(a*Sqrt[a*x + b*x^3]) + (3*Sqrt[b]*x*(a + b*x^2))/(a^2*(Sqrt[a] + Sqrt[b]*x)*S
qrt[a*x + b*x^3]) - (3*Sqrt[a*x + b*x^3])/(a^2*x) - (3*b^(1/4)*Sqrt[x]*(Sqrt[a]
+ Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/
4)*Sqrt[x])/a^(1/4)], 1/2])/(a^(7/4)*Sqrt[a*x + b*x^3]) + (3*b^(1/4)*Sqrt[x]*(Sq
rt[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])/(2*a^(7/4)*Sqrt[a*x + b*x^3])

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Rubi [A]  time = 0.4734, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{3 \sqrt [4]{b} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{7/4} \sqrt{a x+b x^3}}-\frac{3 \sqrt [4]{b} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{a^{7/4} \sqrt{a x+b x^3}}-\frac{3 \sqrt{a x+b x^3}}{a^2 x}+\frac{3 \sqrt{b} x \left (a+b x^2\right )}{a^2 \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{a x+b x^3}}+\frac{1}{a \sqrt{a x+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^3)^(-3/2),x]

[Out]

1/(a*Sqrt[a*x + b*x^3]) + (3*Sqrt[b]*x*(a + b*x^2))/(a^2*(Sqrt[a] + Sqrt[b]*x)*S
qrt[a*x + b*x^3]) - (3*Sqrt[a*x + b*x^3])/(a^2*x) - (3*b^(1/4)*Sqrt[x]*(Sqrt[a]
+ Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticE[2*ArcTan[(b^(1/
4)*Sqrt[x])/a^(1/4)], 1/2])/(a^(7/4)*Sqrt[a*x + b*x^3]) + (3*b^(1/4)*Sqrt[x]*(Sq
rt[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])/(2*a^(7/4)*Sqrt[a*x + b*x^3])

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Rubi in Sympy [A]  time = 49.6762, size = 255, normalized size = 0.93 \[ \frac{1}{a \sqrt{a x + b x^{3}}} + \frac{3 \sqrt{b} \sqrt{a x + b x^{3}}}{a^{2} \left (\sqrt{a} + \sqrt{b} x\right )} - \frac{3 \sqrt{a x + b x^{3}}}{a^{2} x} - \frac{3 \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{a^{\frac{7}{4}} \sqrt{x} \left (a + b x^{2}\right )} + \frac{3 \sqrt [4]{b} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{7}{4}} \sqrt{x} \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(a*sqrt(a*x + b*x**3)) + 3*sqrt(b)*sqrt(a*x + b*x**3)/(a**2*(sqrt(a) + sqrt(b)
*x)) - 3*sqrt(a*x + b*x**3)/(a**2*x) - 3*b**(1/4)*sqrt((a + b*x**2)/(sqrt(a) + s
qrt(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)/(a**(7/4)*sqrt(x)*(a + b*x**2)) + 3*b**(1/4)*sqrt((a +
b*x**2)/(sqrt(a) + sqrt(b)*x)**2)*(sqrt(a) + sqrt(b)*x)*sqrt(a*x + b*x**3)*elli
ptic_f(2*atan(b**(1/4)*sqrt(x)/a**(1/4)), 1/2)/(2*a**(7/4)*sqrt(x)*(a + b*x**2))

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Mathematica [C]  time = 0.179313, size = 174, normalized size = 0.64 \[ \frac{-\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \left (2 a+3 b x^2\right )-3 \sqrt{a} \sqrt{b} x \sqrt{\frac{b x^2}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )+3 \sqrt{a} \sqrt{b} x \sqrt{\frac{b x^2}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}}\right )\right |-1\right )}{a^2 \sqrt{\frac{i \sqrt{b} x}{\sqrt{a}}} \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x + b*x^3)^(-3/2),x]

[Out]

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

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Maple [A]  time = 0.029, size = 206, normalized size = 0.8 \[ -2\,{\frac{b{x}^{2}+a}{{a}^{2}\sqrt{x \left ( b{x}^{2}+a \right ) }}}-{\frac{b{x}^{2}}{{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{2}+{\frac{a}{b}} \right ) xb}}}}+{\frac{3}{2\,{a}^{2}}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}} \left ( -2\,{\frac{\sqrt{-ab}}{b}{\it EllipticE} \left ( \sqrt{{\frac{b}{\sqrt{-ab}} \left ( x+{\frac{\sqrt{-ab}}{b}} \right ) }},1/2\,\sqrt{2} \right ) }+{\frac{1}{b}\sqrt{-ab}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ) } \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^3+a*x)^(3/2),x)

[Out]

-2*(b*x^2+a)/a^2/(x*(b*x^2+a))^(1/2)-x^2*b/a^2/((x^2+a/b)*x*b)^(1/2)+3/2/a^2*(-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)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a*x)^(-3/2),x, algorithm="maxima")

[Out]

integrate((b*x^3 + a*x)^(-3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a*x)^(-3/2),x, algorithm="fricas")

[Out]

integral((b*x^3 + a*x)^(-3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a*x + b*x**3)**(-3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a*x)^(-3/2),x, algorithm="giac")

[Out]

integrate((b*x^3 + a*x)^(-3/2), x)