Optimal. Leaf size=425 \[ \frac{27\ 3^{3/4} b^2 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{110 a^3 c^{17/3} \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac{27 b \sqrt [3]{a+b x^2}}{55 a^2 c^3 (c x)^{5/3}}-\frac{3 \sqrt [3]{a+b x^2}}{11 a c (c x)^{11/3}} \]
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Rubi [A] time = 1.54505, antiderivative size = 425, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{27\ 3^{3/4} b^2 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{110 a^3 c^{17/3} \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac{27 b \sqrt [3]{a+b x^2}}{55 a^2 c^3 (c x)^{5/3}}-\frac{3 \sqrt [3]{a+b x^2}}{11 a c (c x)^{11/3}} \]
Antiderivative was successfully verified.
[In] Int[1/((c*x)^(14/3)*(a + b*x^2)^(2/3)),x]
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Rubi in Sympy [A] time = 37.2542, size = 413, normalized size = 0.97 \[ - \frac{3 \sqrt [3]{a + b x^{2}}}{11 a c \left (c x\right )^{\frac{11}{3}}} + \frac{27 \cdot 3^{\frac{3}{4}} b^{2} \sqrt [3]{c x} \sqrt{\frac{\frac{b^{\frac{2}{3}} \left (c x\right )^{\frac{4}{3}}}{\left (a + b x^{2}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} c^{\frac{2}{3}} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{4}{3}}}{\left (\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )^{2}}} \left (- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right ) F\left (\operatorname{acos}{\left (\frac{\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (-1 + \sqrt{3}\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}}{\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{110 a^{2} c^{\frac{17}{3}} \sqrt{\frac{a}{a + b x^{2}}} \sqrt{- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}}}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )}{\sqrt [3]{a + b x^{2}} \left (\frac{\sqrt [3]{b} \left (c x\right )^{\frac{2}{3}} \left (- \sqrt{3} - 1\right )}{\sqrt [3]{a + b x^{2}}} + c^{\frac{2}{3}}\right )^{2}}} \left (a + b x^{2}\right )^{\frac{2}{3}} \sqrt{- \frac{b x^{2}}{a + b x^{2}} + 1}} + \frac{27 b \sqrt [3]{a + b x^{2}}}{55 a^{2} c^{3} \left (c x\right )^{\frac{5}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x)**(14/3)/(b*x**2+a)**(2/3),x)
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Mathematica [C] time = 0.084455, size = 93, normalized size = 0.22 \[ \frac{3 \sqrt [3]{c x} \left (-5 a^2+27 b^2 x^4 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{6},\frac{2}{3};\frac{7}{6};-\frac{b x^2}{a}\right )+4 a b x^2+9 b^2 x^4\right )}{55 a^2 c^5 x^4 \left (a+b x^2\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c*x)^(14/3)*(a + b*x^2)^(2/3)),x]
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Maple [F] time = 0.043, size = 0, normalized size = 0. \[ \int{1 \left ( cx \right ) ^{-{\frac{14}{3}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x)^(14/3)/(b*x^2+a)^(2/3),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (c x\right )^{\frac{14}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(14/3)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (c x\right )^{\frac{2}{3}} c^{4} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(14/3)),x, algorithm="fricas")
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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)**(14/3)/(b*x**2+a)**(2/3),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (c x\right )^{\frac{14}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^(2/3)*(c*x)^(14/3)),x, algorithm="giac")
[Out]