3.782 \(\int \frac{1}{(c x)^{8/3} \left (a+b x^2\right )^{2/3}} \, dx\)

Optimal. Leaf size=394 \[ -\frac{3\ 3^{3/4} b \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{10 a^2 c^{11/3} \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}-\frac{3 \sqrt [3]{a+b x^2}}{5 a c (c x)^{5/3}} \]

[Out]

(-3*(a + b*x^2)^(1/3))/(5*a*c*(c*x)^(5/3)) - (3*3^(3/4)*b*(c*x)^(1/3)*(a + b*x^2
)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*Sqrt[(c^(4/3) + (b^(
2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^
(1/3))/(c^(2/3) - ((1 + Sqrt[3])*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2]*Elli
pticF[ArcCos[(c^(2/3) - ((1 - Sqrt[3])*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(
c^(2/3) - ((1 + Sqrt[3])*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))], (2 + Sqrt[3])
/4])/(10*a^2*c^(11/3)*Sqrt[-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3
))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + Sqrt[3])*b^(1/3)*(c*x
)^(2/3))/(a + b*x^2)^(1/3))^2))])

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Rubi [A]  time = 1.40602, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{3\ 3^{3/4} b \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{10 a^2 c^{11/3} \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}-\frac{3 \sqrt [3]{a+b x^2}}{5 a c (c x)^{5/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*x)^(8/3)*(a + b*x^2)^(2/3)),x]

[Out]

(-3*(a + b*x^2)^(1/3))/(5*a*c*(c*x)^(5/3)) - (3*3^(3/4)*b*(c*x)^(1/3)*(a + b*x^2
)^(1/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))*Sqrt[(c^(4/3) + (b^(
2/3)*(c*x)^(4/3))/(a + b*x^2)^(2/3) + (b^(1/3)*c^(2/3)*(c*x)^(2/3))/(a + b*x^2)^
(1/3))/(c^(2/3) - ((1 + Sqrt[3])*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))^2]*Elli
pticF[ArcCos[(c^(2/3) - ((1 - Sqrt[3])*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))/(
c^(2/3) - ((1 + Sqrt[3])*b^(1/3)*(c*x)^(2/3))/(a + b*x^2)^(1/3))], (2 + Sqrt[3])
/4])/(10*a^2*c^(11/3)*Sqrt[-((b^(1/3)*(c*x)^(2/3)*(c^(2/3) - (b^(1/3)*(c*x)^(2/3
))/(a + b*x^2)^(1/3)))/((a + b*x^2)^(1/3)*(c^(2/3) - ((1 + Sqrt[3])*b^(1/3)*(c*x
)^(2/3))/(a + b*x^2)^(1/3))^2))])

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Rubi in Sympy [A]  time = 28.4713, size = 382, normalized size = 0.97 \[ - \frac{3 \cdot 3^{\frac{3}{4}} b \sqrt [3]{c x} \sqrt{\frac{\frac{b^{\frac{2}{3}} \left (c x\right )^{\frac{4}{3}}}{\left (a + b x^{2}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} c^{\frac{2}{3}} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{4}{3}}}{\left (\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )^{2}}} \left (- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right ) F\left (\operatorname{acos}{\left (\frac{\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (-1 + \sqrt{3}\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}}{\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{10 a c^{\frac{11}{3}} \sqrt{\frac{a}{a + b x^{2}}} \sqrt{- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )}{\sqrt [3]{a + b x^{2}} \left (\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )^{2}}} \left (a + b x^{2}\right )^{\frac{2}{3}} \sqrt{- \frac{b x^{2}}{a + b x^{2}} + 1}} - \frac{3 \sqrt [3]{a + b x^{2}}}{5 a c \left (c x\right )^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c*x)**(8/3)/(b*x**2+a)**(2/3),x)

[Out]

-3*3**(3/4)*b*(c*x)**(1/3)*sqrt((b**(2/3)*(c*x)**(4/3)/(a + b*x**2)**(2/3) + b**
(1/3)*c**(2/3)*(c*x)**(2/3)/(a + b*x**2)**(1/3) + c**(4/3))/(b**(1/3)*(c*x)**(2/
3)*(-sqrt(3) - 1)/(a + b*x**2)**(1/3) + c**(2/3))**2)*(-b**(1/3)*(c*x)**(2/3)/(a
+ b*x**2)**(1/3) + c**(2/3))*elliptic_f(acos((b**(1/3)*(c*x)**(2/3)*(-1 + sqrt(
3))/(a + b*x**2)**(1/3) + c**(2/3))/(b**(1/3)*(c*x)**(2/3)*(-sqrt(3) - 1)/(a + b
*x**2)**(1/3) + c**(2/3))), sqrt(3)/4 + 1/2)/(10*a*c**(11/3)*sqrt(a/(a + b*x**2)
)*sqrt(-b**(1/3)*(c*x)**(2/3)*(-b**(1/3)*(c*x)**(2/3)/(a + b*x**2)**(1/3) + c**(
2/3))/((a + b*x**2)**(1/3)*(b**(1/3)*(c*x)**(2/3)*(-sqrt(3) - 1)/(a + b*x**2)**(
1/3) + c**(2/3))**2))*(a + b*x**2)**(2/3)*sqrt(-b*x**2/(a + b*x**2) + 1)) - 3*(a
+ b*x**2)**(1/3)/(5*a*c*(c*x)**(5/3))

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Mathematica [C]  time = 0.058139, size = 72, normalized size = 0.18 \[ -\frac{3 x \left (3 b x^2 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{6},\frac{2}{3};\frac{7}{6};-\frac{b x^2}{a}\right )+a+b x^2\right )}{5 a (c x)^{8/3} \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*x)^(8/3)*(a + b*x^2)^(2/3)),x]

[Out]

(-3*x*(a + b*x^2 + 3*b*x^2*(1 + (b*x^2)/a)^(2/3)*Hypergeometric2F1[1/6, 2/3, 7/6
, -((b*x^2)/a)]))/(5*a*(c*x)^(8/3)*(a + b*x^2)^(2/3))

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{8}{3}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x)^(8/3)/(b*x^2+a)^(2/3),x)

[Out]

int(1/(c*x)^(8/3)/(b*x^2+a)^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (c x\right )^{\frac{8}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(8/3)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(8/3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(8/3)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c*x)**(8/3)/(b*x**2+a)**(2/3),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (c x\right )^{\frac{8}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(8/3)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(8/3)), x)