3.670 \(\int \frac{\sqrt [3]{a+b x^2}}{x^3} \, dx\)

Optimal. Leaf size=107 \[ \frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{2/3}}-\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3} a^{2/3}}-\frac{b \log (x)}{6 a^{2/3}}-\frac{\sqrt [3]{a+b x^2}}{2 x^2} \]

[Out]

-(a + b*x^2)^(1/3)/(2*x^2) - (b*ArcTan[(a^(1/3) + 2*(a + b*x^2)^(1/3))/(Sqrt[3]*
a^(1/3))])/(2*Sqrt[3]*a^(2/3)) - (b*Log[x])/(6*a^(2/3)) + (b*Log[a^(1/3) - (a +
b*x^2)^(1/3)])/(4*a^(2/3))

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Rubi [A]  time = 0.172615, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{2/3}}-\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{2 \sqrt{3} a^{2/3}}-\frac{b \log (x)}{6 a^{2/3}}-\frac{\sqrt [3]{a+b x^2}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(1/3)/x^3,x]

[Out]

-(a + b*x^2)^(1/3)/(2*x^2) - (b*ArcTan[(a^(1/3) + 2*(a + b*x^2)^(1/3))/(Sqrt[3]*
a^(1/3))])/(2*Sqrt[3]*a^(2/3)) - (b*Log[x])/(6*a^(2/3)) + (b*Log[a^(1/3) - (a +
b*x^2)^(1/3)])/(4*a^(2/3))

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Rubi in Sympy [A]  time = 10.8191, size = 99, normalized size = 0.93 \[ - \frac{\sqrt [3]{a + b x^{2}}}{2 x^{2}} - \frac{b \log{\left (x^{2} \right )}}{12 a^{\frac{2}{3}}} + \frac{b \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}} \right )}}{4 a^{\frac{2}{3}}} - \frac{\sqrt{3} b \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{2}}}{3}\right )}{\sqrt [3]{a}} \right )}}{6 a^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x**2)**(1/3)/(2*x**2) - b*log(x**2)/(12*a**(2/3)) + b*log(a**(1/3) - (a
+ b*x**2)**(1/3))/(4*a**(2/3)) - sqrt(3)*b*atan(sqrt(3)*(a**(1/3)/3 + 2*(a + b*x
**2)**(1/3)/3)/a**(1/3))/(6*a**(2/3))

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Mathematica [C]  time = 0.0446799, size = 67, normalized size = 0.63 \[ \frac{-b x^2 \left (\frac{a}{b x^2}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x^2}\right )-2 \left (a+b x^2\right )}{4 x^2 \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}\sqrt [3]{b{x}^{2}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/3)/x^3,x)

[Out]

int((b*x^2+a)^(1/3)/x^3,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)^(1/3)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.216654, size = 193, normalized size = 1.8 \[ -\frac{\sqrt{3}{\left (\sqrt{3} b x^{2} \log \left (a^{2} +{\left (b x^{2} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{1}{3}} a +{\left (b x^{2} + a\right )}^{\frac{2}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} b x^{2} \log \left (-a +{\left (b x^{2} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{1}{3}}\right ) + 6 \, b x^{2} \arctan \left (\frac{\sqrt{3} a + 2 \, \sqrt{3}{\left (b x^{2} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{1}{3}}}{3 \, a}\right ) + 6 \, \sqrt{3}{\left (b x^{2} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{1}{3}}\right )}}{36 \,{\left (a^{2}\right )}^{\frac{1}{3}} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/36*sqrt(3)*(sqrt(3)*b*x^2*log(a^2 + (b*x^2 + a)^(1/3)*(a^2)^(1/3)*a + (b*x^2
+ a)^(2/3)*(a^2)^(2/3)) - 2*sqrt(3)*b*x^2*log(-a + (b*x^2 + a)^(1/3)*(a^2)^(1/3)
) + 6*b*x^2*arctan(1/3*(sqrt(3)*a + 2*sqrt(3)*(b*x^2 + a)^(1/3)*(a^2)^(1/3))/a)
+ 6*sqrt(3)*(b*x^2 + a)^(1/3)*(a^2)^(1/3))/((a^2)^(1/3)*x^2)

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Sympy [A]  time = 4.40067, size = 42, normalized size = 0.39 \[ - \frac{\sqrt [3]{b} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{\frac{4}{3}} \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**(1/3)*gamma(2/3)*hyper((-1/3, 2/3), (5/3,), a*exp_polar(I*pi)/(b*x**2))/(2*x
**(4/3)*gamma(5/3))

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GIAC/XCAS [A]  time = 0.581174, size = 143, normalized size = 1.34 \[ -\frac{1}{12} \, b{\left (\frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{2}{3}}} + \frac{{\rm ln}\left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{2}{3}}} - \frac{2 \,{\rm ln}\left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{2}{3}}} + \frac{6 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{b x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*b*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3))/a
^(2/3) + ln((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3))/a^(2/3) - 2
*ln(abs((b*x^2 + a)^(1/3) - a^(1/3)))/a^(2/3) + 6*(b*x^2 + a)^(1/3)/(b*x^2))