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

Optimal. Leaf size=139 \[ -\frac{5 b^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 a^{9/4} \sqrt{a x+b x^3}}-\frac{5 \sqrt{a x+b x^3}}{3 a^2 x^2}+\frac{1}{a x \sqrt{a x+b x^3}} \]

[Out]

1/(a*x*Sqrt[a*x + b*x^3]) - (5*Sqrt[a*x + b*x^3])/(3*a^2*x^2) - (5*b^(3/4)*Sqrt[
x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*A
rcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(6*a^(9/4)*Sqrt[a*x + b*x^3])

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Rubi [A]  time = 0.267133, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ -\frac{5 b^{3/4} \sqrt{x} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{6 a^{9/4} \sqrt{a x+b x^3}}-\frac{5 \sqrt{a x+b x^3}}{3 a^2 x^2}+\frac{1}{a x \sqrt{a x+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

1/(a*x*Sqrt[a*x + b*x^3]) - (5*Sqrt[a*x + b*x^3])/(3*a^2*x^2) - (5*b^(3/4)*Sqrt[
x]*(Sqrt[a] + Sqrt[b]*x)*Sqrt[(a + b*x^2)/(Sqrt[a] + Sqrt[b]*x)^2]*EllipticF[2*A
rcTan[(b^(1/4)*Sqrt[x])/a^(1/4)], 1/2])/(6*a^(9/4)*Sqrt[a*x + b*x^3])

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Rubi in Sympy [A]  time = 26.0352, size = 133, normalized size = 0.96 \[ \frac{1}{a x \sqrt{a x + b x^{3}}} - \frac{5 \sqrt{a x + b x^{3}}}{3 a^{2} x^{2}} - \frac{5 b^{\frac{3}{4}} \sqrt{\frac{a + b x^{2}}{\left (\sqrt{a} + \sqrt{b} x\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x\right ) \sqrt{a x + b x^{3}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{6 a^{\frac{9}{4}} \sqrt{x} \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(a*x*sqrt(a*x + b*x**3)) - 5*sqrt(a*x + b*x**3)/(3*a**2*x**2) - 5*b**(3/4)*sqr
t((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)/(6*a**(9/4)*sqrt(x)*(a + b*
x**2))

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Mathematica [C]  time = 0.238528, size = 106, normalized size = 0.76 \[ \frac{-\frac{5 i b x^{5/2} \sqrt{\frac{a}{b x^2}+1} F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}{\sqrt{x}}\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{a}}{\sqrt{b}}}}-2 a-5 b x^2}{3 a^2 x \sqrt{x \left (a+b x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.032, size = 150, normalized size = 1.1 \[ -{\frac{bx}{{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{2}+{\frac{a}{b}} \right ) xb}}}}-{\frac{2}{3\,{a}^{2}{x}^{2}}\sqrt{b{x}^{3}+ax}}-{\frac{5}{6\,{a}^{2}}\sqrt{-ab}\sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}}\sqrt{-2\,{\frac{b}{\sqrt{-ab}} \left ( x-{\frac{\sqrt{-ab}}{b}} \right ) }}\sqrt{-{bx{\frac{1}{\sqrt{-ab}}}}}{\it EllipticF} \left ( \sqrt{{b \left ( x+{\frac{1}{b}\sqrt{-ab}} \right ){\frac{1}{\sqrt{-ab}}}}},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{b{x}^{3}+ax}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^3+a*x)^(3/2),x)

[Out]

-x*b/a^2/((x^2+a/b)*x*b)^(1/2)-2/3*(b*x^3+a*x)^(1/2)/a^2/x^2-5/6/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)*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. \[ \int \frac{1}{{\left (b x^{3} + a x\right )}^{\frac{3}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a*x)^(3/2)*x),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{4} + a x^{2}\right )} \sqrt{b x^{3} + a x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a*x)^(3/2)*x),x, algorithm="fricas")

[Out]

integral(1/((b*x^4 + a*x^2)*sqrt(b*x^3 + a*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(x*(a + b*x**2))**(3/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a*x)^(3/2)*x),x, algorithm="giac")

[Out]

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