Optimal. Leaf size=16 \[ \frac{x}{c \sqrt [3]{c+d x^3}} \]
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Rubi [A] time = 0.00856531, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x}{c \sqrt [3]{c+d x^3}} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^3)^(-4/3),x]
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Rubi in Sympy [A] time = 1.25739, size = 12, normalized size = 0.75 \[ \frac{x}{c \sqrt [3]{c + d x^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(d*x**3+c)**(4/3),x)
[Out]
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Mathematica [C] time = 2.2023, size = 674, normalized size = 42.12 \[ \frac{i \sqrt{\frac{\pi }{3}} \Gamma \left (\frac{1}{3}\right ) \left (\frac{(-1)^{2/3} \sqrt [3]{c}}{\sqrt [3]{d}}+x\right ) \left (\frac{\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{c}}\right )^{4/3} \left (\frac{\sqrt [3]{d} x}{\sqrt [3]{c}}+1\right ) \left (-36 i \left (\left (\sqrt{3}-i\right ) \sqrt [3]{c}-\left (\sqrt{3}+i\right ) \sqrt [3]{d} x\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )^2 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{11}{3};\frac{6 \left (\left (1-i \sqrt{3}\right ) \sqrt [3]{d} x+\left (1+i \sqrt{3}\right ) \sqrt [3]{c}\right )}{\left (3 i+\sqrt{3}\right ) \left (\left (3 i+\sqrt{3}\right ) \sqrt [3]{c}-2 \sqrt{3} \sqrt [3]{d} x\right )}\right )-12 i \left (\left (7 \sqrt{3}-3 i\right ) c^{2/3}+2 \left (2 \sqrt{3}-9 i\right ) \sqrt [3]{c} \sqrt [3]{d} x-9 \left (\sqrt{3}+i\right ) d^{2/3} x^2\right ) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \, _2F_1\left (2,\frac{7}{3};\frac{11}{3};\frac{6 \left (\left (1-i \sqrt{3}\right ) \sqrt [3]{d} x+\left (1+i \sqrt{3}\right ) \sqrt [3]{c}\right )}{\left (3 i+\sqrt{3}\right ) \left (\left (3 i+\sqrt{3}\right ) \sqrt [3]{c}-2 \sqrt{3} \sqrt [3]{d} x\right )}\right )+48 \left (2 \left (2-i \sqrt{3}\right ) c^{2/3} \sqrt [3]{d} x+2 \left (3+i \sqrt{3}\right ) \sqrt [3]{c} d^{2/3} x^2+4 c+3 \left (1+i \sqrt{3}\right ) d x^3\right ) \, _2F_1\left (1,\frac{4}{3};\frac{8}{3};\frac{6 \left (\left (1-i \sqrt{3}\right ) \sqrt [3]{d} x+\left (1+i \sqrt{3}\right ) \sqrt [3]{c}\right )}{\left (3 i+\sqrt{3}\right ) \left (\left (3 i+\sqrt{3}\right ) \sqrt [3]{c}-2 \sqrt{3} \sqrt [3]{d} x\right )}\right )\right )}{40 \sqrt [3]{2} \left (\sqrt{3}+3 i\right ) c^{2/3} \Gamma \left (\frac{2}{3}\right ) \Gamma \left (\frac{7}{6}\right ) \left (-2 \sqrt{3} \sqrt [3]{d} x+\left (\sqrt{3}+3 i\right ) \sqrt [3]{c}\right ) \left (c+d x^3\right )^{4/3} \left (1+\frac{i \left ((-1)^{2/3} \sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt{3} \sqrt [3]{c}}\right )^{4/3}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(c + d*x^3)^(-4/3),x]
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Maple [A] time = 0.003, size = 15, normalized size = 0.9 \[{\frac{x}{c}{\frac{1}{\sqrt [3]{d{x}^{3}+c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(d*x^3+c)^(4/3),x)
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Maxima [A] time = 1.40159, size = 19, normalized size = 1.19 \[ \frac{x}{{\left (d x^{3} + c\right )}^{\frac{1}{3}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^(-4/3),x, algorithm="maxima")
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Fricas [A] time = 0.211339, size = 31, normalized size = 1.94 \[ \frac{{\left (d x^{3} + c\right )}^{\frac{2}{3}} x}{c d x^{3} + c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^(-4/3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.95163, size = 29, normalized size = 1.81 \[ \frac{x \Gamma \left (\frac{1}{3}\right )}{3 c^{\frac{4}{3}} \sqrt [3]{1 + \frac{d x^{3}}{c}} \Gamma \left (\frac{4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(d*x**3+c)**(4/3),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (d x^{3} + c\right )}^{\frac{4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^3 + c)^(-4/3),x, algorithm="giac")
[Out]