3.695 \(\int \frac{\left (a+b x^2\right )^{4/3}}{x^5} \, dx\)

Optimal. Leaf size=132 \[ \frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{6 a^{2/3}}-\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3}}-\frac{b^2 \log (x)}{9 a^{2/3}}-\frac{b \sqrt [3]{a+b x^2}}{3 x^2}-\frac{\left (a+b x^2\right )^{4/3}}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.222703, antiderivative size = 132, 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 \[ \frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{6 a^{2/3}}-\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3}}-\frac{b^2 \log (x)}{9 a^{2/3}}-\frac{b \sqrt [3]{a+b x^2}}{3 x^2}-\frac{\left (a+b x^2\right )^{4/3}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(4/3)/x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 15.0453, size = 121, normalized size = 0.92 \[ - \frac{b \sqrt [3]{a + b x^{2}}}{3 x^{2}} - \frac{\left (a + b x^{2}\right )^{\frac{4}{3}}}{4 x^{4}} - \frac{b^{2} \log{\left (x^{2} \right )}}{18 a^{\frac{2}{3}}} + \frac{b^{2} \log{\left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}} \right )}}{6 a^{\frac{2}{3}}} - \frac{\sqrt{3} b^{2} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{\sqrt [3]{a}}{3} + \frac{2 \sqrt [3]{a + b x^{2}}}{3}\right )}{\sqrt [3]{a}} \right )}}{9 a^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(4/3)/x**5,x)

[Out]

-b*(a + b*x**2)**(1/3)/(3*x**2) - (a + b*x**2)**(4/3)/(4*x**4) - b**2*log(x**2)/
(18*a**(2/3)) + b**2*log(a**(1/3) - (a + b*x**2)**(1/3))/(6*a**(2/3)) - sqrt(3)*
b**2*atan(sqrt(3)*(a**(1/3)/3 + 2*(a + b*x**2)**(1/3)/3)/a**(1/3))/(9*a**(2/3))

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Mathematica [C]  time = 0.0502357, size = 80, normalized size = 0.61 \[ \frac{-3 a^2-2 b^2 x^4 \left (\frac{a}{b x^2}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{a}{b x^2}\right )-10 a b x^2-7 b^2 x^4}{12 x^4 \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(4/3)/x^5,x]

[Out]

(-3*a^2 - 10*a*b*x^2 - 7*b^2*x^4 - 2*b^2*(1 + a/(b*x^2))^(2/3)*x^4*Hypergeometri
c2F1[2/3, 2/3, 5/3, -(a/(b*x^2))])/(12*x^4*(a + b*x^2)^(2/3))

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(4/3)/x^5,x)

[Out]

int((b*x^2+a)^(4/3)/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)^(4/3)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217347, size = 216, normalized size = 1.64 \[ -\frac{\sqrt{3}{\left (2 \, \sqrt{3} b^{2} x^{4} \log \left (a^{2} +{\left (b x^{2} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{1}{3}} a +{\left (b x^{2} + a\right )}^{\frac{2}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right ) - 4 \, \sqrt{3} b^{2} x^{4} \log \left (-a +{\left (b x^{2} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{1}{3}}\right ) + 12 \, b^{2} x^{4} \arctan \left (\frac{\sqrt{3} a + 2 \, \sqrt{3}{\left (b x^{2} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{1}{3}}}{3 \, a}\right ) + 3 \, \sqrt{3}{\left (7 \, b x^{2} + 3 \, a\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{1}{3}}\right )}}{108 \,{\left (a^{2}\right )}^{\frac{1}{3}} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)/x^5,x, algorithm="fricas")

[Out]

-1/108*sqrt(3)*(2*sqrt(3)*b^2*x^4*log(a^2 + (b*x^2 + a)^(1/3)*(a^2)^(1/3)*a + (b
*x^2 + a)^(2/3)*(a^2)^(2/3)) - 4*sqrt(3)*b^2*x^4*log(-a + (b*x^2 + a)^(1/3)*(a^2
)^(1/3)) + 12*b^2*x^4*arctan(1/3*(sqrt(3)*a + 2*sqrt(3)*(b*x^2 + a)^(1/3)*(a^2)^
(1/3))/a) + 3*sqrt(3)*(7*b*x^2 + 3*a)*(b*x^2 + a)^(1/3)*(a^2)^(1/3))/((a^2)^(1/3
)*x^4)

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Sympy [A]  time = 6.96548, size = 42, normalized size = 0.32 \[ - \frac{b^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{\frac{4}{3}} \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(4/3)/x**5,x)

[Out]

-b**(4/3)*gamma(2/3)*hyper((-4/3, 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 = 0.593371, size = 167, normalized size = 1.27 \[ -\frac{1}{36} \, b^{2}{\left (\frac{4 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{2}{3}}} + \frac{2 \,{\rm ln}\left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{2}{3}}} - \frac{4 \,{\rm ln}\left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{2}{3}}} + \frac{3 \,{\left (7 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} - 4 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a\right )}}{b^{2} x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(4/3)/x^5,x, algorithm="giac")

[Out]

-1/36*b^2*(4*sqrt(3)*arctan(1/3*sqrt(3)*(2*(b*x^2 + a)^(1/3) + a^(1/3))/a^(1/3))
/a^(2/3) + 2*ln((b*x^2 + a)^(2/3) + (b*x^2 + a)^(1/3)*a^(1/3) + a^(2/3))/a^(2/3)
- 4*ln(abs((b*x^2 + a)^(1/3) - a^(1/3)))/a^(2/3) + 3*(7*(b*x^2 + a)^(4/3) - 4*(
b*x^2 + a)^(1/3)*a)/(b^2*x^4))