3.683 \(\int \frac{\left (a+b x^2\right )^{2/3}}{x^5} \, dx\)

Optimal. Leaf size=135 \[ -\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{4/3}}-\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt{3} a^{4/3}}+\frac{b^2 \log (x)}{18 a^{4/3}}-\frac{b \left (a+b x^2\right )^{2/3}}{6 a x^2}-\frac{\left (a+b x^2\right )^{2/3}}{4 x^4} \]

[Out]

-(a + b*x^2)^(2/3)/(4*x^4) - (b*(a + b*x^2)^(2/3))/(6*a*x^2) - (b^2*ArcTan[(a^(1
/3) + 2*(a + b*x^2)^(1/3))/(Sqrt[3]*a^(1/3))])/(6*Sqrt[3]*a^(4/3)) + (b^2*Log[x]
)/(18*a^(4/3)) - (b^2*Log[a^(1/3) - (a + b*x^2)^(1/3)])/(12*a^(4/3))

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Rubi [A]  time = 0.219052, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ -\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{4/3}}-\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt{3} a^{4/3}}+\frac{b^2 \log (x)}{18 a^{4/3}}-\frac{b \left (a+b x^2\right )^{2/3}}{6 a x^2}-\frac{\left (a+b x^2\right )^{2/3}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(2/3)/x^5,x]

[Out]

-(a + b*x^2)^(2/3)/(4*x^4) - (b*(a + b*x^2)^(2/3))/(6*a*x^2) - (b^2*ArcTan[(a^(1
/3) + 2*(a + b*x^2)^(1/3))/(Sqrt[3]*a^(1/3))])/(6*Sqrt[3]*a^(4/3)) + (b^2*Log[x]
)/(18*a^(4/3)) - (b^2*Log[a^(1/3) - (a + b*x^2)^(1/3)])/(12*a^(4/3))

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Rubi in Sympy [A]  time = 15.4821, size = 122, normalized size = 0.9 \[ - \frac{\left (a + b x^{2}\right )^{\frac{2}{3}}}{4 x^{4}} - \frac{b \left (a + b x^{2}\right )^{\frac{2}{3}}}{6 a x^{2}} + \frac{b^{2} \log{\left (x^{2} \right )}}{36 a^{\frac{4}{3}}} - \frac{b^{2} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}} \right )}}{12 a^{\frac{4}{3}}} - \frac{\sqrt{3} b^{2} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{2}}}{3}\right )}{\sqrt [3]{a}} \right )}}{18 a^{\frac{4}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(2/3)/x**5,x)

[Out]

-(a + b*x**2)**(2/3)/(4*x**4) - b*(a + b*x**2)**(2/3)/(6*a*x**2) + b**2*log(x**2
)/(36*a**(4/3)) - b**2*log(a**(1/3) - (a + b*x**2)**(1/3))/(12*a**(4/3)) - sqrt(
3)*b**2*atan(sqrt(3)*(a**(1/3)/3 + 2*(a + b*x**2)**(1/3)/3)/a**(1/3))/(18*a**(4/
3))

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Mathematica [C]  time = 0.0455761, size = 83, normalized size = 0.61 \[ \frac{-3 a^2+2 b^2 x^4 \sqrt [3]{\frac{a}{b x^2}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};-\frac{a}{b x^2}\right )-5 a b x^2-2 b^2 x^4}{12 a x^4 \sqrt [3]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(2/3)/x^5,x]

[Out]

(-3*a^2 - 5*a*b*x^2 - 2*b^2*x^4 + 2*b^2*(1 + a/(b*x^2))^(1/3)*x^4*Hypergeometric
2F1[1/3, 1/3, 4/3, -(a/(b*x^2))])/(12*a*x^4*(a + b*x^2)^(1/3))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(2/3)/x^5,x)

[Out]

int((b*x^2+a)^(2/3)/x^5,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)^(2/3)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220498, size = 219, normalized size = 1.62 \[ -\frac{\sqrt{3}{\left (\sqrt{3} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (-a\right )^{\frac{1}{3}} -{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} - a\right ) - 2 \, \sqrt{3} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} - a\right ) - 6 \, b^{2} x^{4} \arctan \left (\frac{2 \, \sqrt{3}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (-a\right )^{\frac{2}{3}} + \sqrt{3} a}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (2 \, b x^{2} + 3 \, a\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (-a\right )^{\frac{1}{3}}\right )}}{108 \, \left (-a\right )^{\frac{1}{3}} a x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(2/3)/x^5,x, algorithm="fricas")

[Out]

-1/108*sqrt(3)*(sqrt(3)*b^2*x^4*log((b*x^2 + a)^(2/3)*(-a)^(1/3) - (b*x^2 + a)^(
1/3)*(-a)^(2/3) - a) - 2*sqrt(3)*b^2*x^4*log((b*x^2 + a)^(1/3)*(-a)^(2/3) - a) -
6*b^2*x^4*arctan(1/3*(2*sqrt(3)*(b*x^2 + a)^(1/3)*(-a)^(2/3) + sqrt(3)*a)/a) +
3*sqrt(3)*(2*b*x^2 + 3*a)*(b*x^2 + a)^(2/3)*(-a)^(1/3))/((-a)^(1/3)*a*x^4)

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Sympy [A]  time = 5.76814, size = 42, normalized size = 0.31 \[ - \frac{b^{\frac{2}{3}} \Gamma \left (\frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{4}{3} \\ \frac{7}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{\frac{8}{3}} \Gamma \left (\frac{7}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(2/3)/x**5,x)

[Out]

-b**(2/3)*gamma(4/3)*hyper((-2/3, 4/3), (7/3,), a*exp_polar(I*pi)/(b*x**2))/(2*x
**(8/3)*gamma(7/3))

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GIAC/XCAS [A]  time = 0.611818, size = 170, normalized size = 1.26 \[ -\frac{1}{36} \, b^{2}{\left (\frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{4}{3}}} - \frac{{\rm ln}\left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{4}{3}}} + \frac{2 \,{\rm ln}\left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{4}{3}}} + \frac{3 \,{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} +{\left (b x^{2} + a\right )}^{\frac{2}{3}} a\right )}}{a b^{2} x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(2/3)/x^5,x, algorithm="giac")

[Out]

-1/36*b^2*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3))
/a^(4/3) - ln((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3))/a^(4/3) +
2*ln(abs((b*x^2 + a)^(1/3) - a^(1/3)))/a^(4/3) + 3*(2*(b*x^2 + a)^(5/3) + (b*x^
2 + a)^(2/3)*a)/(a*b^2*x^4))