3.693 \(\int \frac{\left (a+b x^2\right )^{4/3}}{x} \, dx\)

Optimal. Leaf size=117 \[ \frac{3}{4} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )-\frac{1}{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{1}{2} a^{4/3} \log (x)+\frac{3}{2} a \sqrt [3]{a+b x^2}+\frac{3}{8} \left (a+b x^2\right )^{4/3} \]

[Out]

(3*a*(a + b*x^2)^(1/3))/2 + (3*(a + b*x^2)^(4/3))/8 - (Sqrt[3]*a^(4/3)*ArcTan[(a
^(1/3) + 2*(a + b*x^2)^(1/3))/(Sqrt[3]*a^(1/3))])/2 - (a^(4/3)*Log[x])/2 + (3*a^
(4/3)*Log[a^(1/3) - (a + b*x^2)^(1/3)])/4

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Rubi [A]  time = 0.224914, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{3}{4} a^{4/3} \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )-\frac{1}{2} \sqrt{3} a^{4/3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )-\frac{1}{2} a^{4/3} \log (x)+\frac{3}{2} a \sqrt [3]{a+b x^2}+\frac{3}{8} \left (a+b x^2\right )^{4/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(4/3)/x,x]

[Out]

(3*a*(a + b*x^2)^(1/3))/2 + (3*(a + b*x^2)^(4/3))/8 - (Sqrt[3]*a^(4/3)*ArcTan[(a
^(1/3) + 2*(a + b*x^2)^(1/3))/(Sqrt[3]*a^(1/3))])/2 - (a^(4/3)*Log[x])/2 + (3*a^
(4/3)*Log[a^(1/3) - (a + b*x^2)^(1/3)])/4

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Rubi in Sympy [A]  time = 13.4781, size = 109, normalized size = 0.93 \[ - \frac{a^{\frac{4}{3}} \log{\left (x^{2} \right )}}{4} + \frac{3 a^{\frac{4}{3}} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}} \right )}}{4} - \frac{\sqrt{3} a^{\frac{4}{3}} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{2}}}{3}\right )}{\sqrt [3]{a}} \right )}}{2} + \frac{3 a \sqrt [3]{a + b x^{2}}}{2} + \frac{3 \left (a + b x^{2}\right )^{\frac{4}{3}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(4/3)*log(x**2)/4 + 3*a**(4/3)*log(a**(1/3) - (a + b*x**2)**(1/3))/4 - sqrt(
3)*a**(4/3)*atan(sqrt(3)*(a**(1/3)/3 + 2*(a + b*x**2)**(1/3)/3)/a**(1/3))/2 + 3*
a*(a + b*x**2)**(1/3)/2 + 3*(a + b*x**2)**(4/3)/8

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Mathematica [C]  time = 0.0513, size = 76, normalized size = 0.65 \[ \frac{3 \left (5 a^2+6 a b x^2+b^2 x^4\right )-6 a^2 \left (\frac{a}{b x^2}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x^2}\right )}{8 \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(4/3)/x,x]

[Out]

(3*(5*a^2 + 6*a*b*x^2 + b^2*x^4) - 6*a^2*(1 + a/(b*x^2))^(2/3)*Hypergeometric2F1
[2/3, 2/3, 5/3, -(a/(b*x^2))])/(8*(a + b*x^2)^(2/3))

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(4/3)/x,x)

[Out]

int((b*x^2+a)^(4/3)/x,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.215985, size = 143, normalized size = 1.22 \[ -\frac{1}{2} \, \sqrt{3} a^{\frac{4}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right ) - \frac{1}{4} \, a^{\frac{4}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) + \frac{1}{2} \, a^{\frac{4}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) + \frac{3}{8} \,{\left (b x^{2} + 5 \, a\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)/x,x, algorithm="fricas")

[Out]

-1/2*sqrt(3)*a^(4/3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3))
- 1/4*a^(4/3)*log((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3)) + 1/
2*a^(4/3)*log((b*x^2 + a)^(1/3) - a^(1/3)) + 3/8*(b*x^2 + 5*a)*(b*x^2 + a)^(1/3)

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Sympy [A]  time = 5.39158, size = 49, normalized size = 0.42 \[ - \frac{b^{\frac{4}{3}} x^{\frac{8}{3}} \Gamma \left (- \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, - \frac{4}{3} \\ - \frac{1}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 \Gamma \left (- \frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**(4/3)*x**(8/3)*gamma(-4/3)*hyper((-4/3, -4/3), (-1/3,), a*exp_polar(I*pi)/(b
*x**2))/(2*gamma(-1/3))

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GIAC/XCAS [A]  time = 0.596838, size = 149, normalized size = 1.27 \[ -\frac{1}{2} \, \sqrt{3} a^{\frac{4}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right ) - \frac{1}{4} \, a^{\frac{4}{3}}{\rm ln}\left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right ) + \frac{1}{2} \, a^{\frac{4}{3}}{\rm ln}\left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right ) + \frac{3}{8} \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} + \frac{3}{2} \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)/x,x, algorithm="giac")

[Out]

-1/2*sqrt(3)*a^(4/3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3))
- 1/4*a^(4/3)*ln((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3)) + 1/2
*a^(4/3)*ln(abs((b*x^2 + a)^(1/3) - a^(1/3))) + 3/8*(b*x^2 + a)^(4/3) + 3/2*(b*x
^2 + a)^(1/3)*a