3.30 \(\int \frac{1}{\left (c+d x^3\right )^{4/3}} \, dx\)

Optimal. Leaf size=16 \[ \frac{x}{c \sqrt [3]{c+d x^3}} \]

[Out]

x/(c*(c + d*x^3)^(1/3))

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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]

[Out]

x/(c*(c + d*x^3)^(1/3))

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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]

x/(c*(c + d*x**3)**(1/3))

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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]

[Out]

((I/40)*Sqrt[Pi/3]*(((-1)^(2/3)*c^(1/3))/d^(1/3) + x)*((c^(1/3) + (-1)^(2/3)*d^(
1/3)*x)/((1 + (-1)^(1/3))*c^(1/3)))^(4/3)*(1 + (d^(1/3)*x)/c^(1/3))*Gamma[1/3]*(
48*(4*c + 2*(2 - I*Sqrt[3])*c^(2/3)*d^(1/3)*x + 2*(3 + I*Sqrt[3])*c^(1/3)*d^(2/3
)*x^2 + 3*(1 + I*Sqrt[3])*d*x^3)*Hypergeometric2F1[1, 4/3, 8/3, (6*((1 + I*Sqrt[
3])*c^(1/3) + (1 - I*Sqrt[3])*d^(1/3)*x))/((3*I + Sqrt[3])*((3*I + Sqrt[3])*c^(1
/3) - 2*Sqrt[3]*d^(1/3)*x))] - (12*I)*(c^(1/3) + d^(1/3)*x)*((-3*I + 7*Sqrt[3])*
c^(2/3) + 2*(-9*I + 2*Sqrt[3])*c^(1/3)*d^(1/3)*x - 9*(I + Sqrt[3])*d^(2/3)*x^2)*
Hypergeometric2F1[2, 7/3, 11/3, (6*((1 + I*Sqrt[3])*c^(1/3) + (1 - I*Sqrt[3])*d^
(1/3)*x))/((3*I + Sqrt[3])*((3*I + Sqrt[3])*c^(1/3) - 2*Sqrt[3]*d^(1/3)*x))] - (
36*I)*(c^(1/3) + d^(1/3)*x)^2*((-I + Sqrt[3])*c^(1/3) - (I + Sqrt[3])*d^(1/3)*x)
*HypergeometricPFQ[{2, 2, 7/3}, {1, 11/3}, (6*((1 + I*Sqrt[3])*c^(1/3) + (1 - I*
Sqrt[3])*d^(1/3)*x))/((3*I + Sqrt[3])*((3*I + Sqrt[3])*c^(1/3) - 2*Sqrt[3]*d^(1/
3)*x))]))/(2^(1/3)*(3*I + Sqrt[3])*c^(2/3)*((3*I + Sqrt[3])*c^(1/3) - 2*Sqrt[3]*
d^(1/3)*x)*(c + d*x^3)^(4/3)*(1 + (I*((-1)^(2/3)*c^(1/3) + d^(1/3)*x))/(Sqrt[3]*
c^(1/3)))^(4/3)*Gamma[2/3]*Gamma[7/6])

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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)

[Out]

x/c/(d*x^3+c)^(1/3)

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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")

[Out]

x/((d*x^3 + c)^(1/3)*c)

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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]

(d*x^3 + c)^(2/3)*x/(c*d*x^3 + c^2)

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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)

[Out]

x*gamma(1/3)/(3*c**(4/3)*(1 + d*x**3/c)**(1/3)*gamma(4/3))

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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]

integrate((d*x^3 + c)^(-4/3), x)