3.159 \(\int \frac{d x^3}{2+3 x^4} \, dx\)

Optimal. Leaf size=13 \[ \frac{1}{12} d \log \left (3 x^4+2\right ) \]

[Out]

(d*Log[2 + 3*x^4])/12

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Rubi [A]  time = 0.0108337, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{1}{12} d \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]  Int[(d*x^3)/(2 + 3*x^4),x]

[Out]

(d*Log[2 + 3*x^4])/12

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Rubi in Sympy [A]  time = 2.1779, size = 10, normalized size = 0.77 \[ \frac{d \log{\left (3 x^{4} + 2 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(d*x**3/(3*x**4+2),x)

[Out]

d*log(3*x**4 + 2)/12

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Mathematica [A]  time = 0.00477415, size = 13, normalized size = 1. \[ \frac{1}{12} d \log \left (3 x^4+2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d*x^3)/(2 + 3*x^4),x]

[Out]

(d*Log[2 + 3*x^4])/12

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Maple [A]  time = 0.002, size = 12, normalized size = 0.9 \[{\frac{d\ln \left ( 3\,{x}^{4}+2 \right ) }{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(d*x^3/(3*x^4+2),x)

[Out]

1/12*d*ln(3*x^4+2)

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Maxima [A]  time = 1.396, size = 15, normalized size = 1.15 \[ \frac{1}{12} \, d \log \left (3 \, x^{4} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d*x^3/(3*x^4 + 2),x, algorithm="maxima")

[Out]

1/12*d*log(3*x^4 + 2)

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Fricas [A]  time = 0.217436, size = 15, normalized size = 1.15 \[ \frac{1}{12} \, d \log \left (3 \, x^{4} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*d*log(3*x^4 + 2)

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Sympy [A]  time = 0.08263, size = 10, normalized size = 0.77 \[ \frac{d \log{\left (3 x^{4} + 2 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d*x**3/(3*x**4+2),x)

[Out]

d*log(3*x**4 + 2)/12

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GIAC/XCAS [A]  time = 0.216708, size = 15, normalized size = 1.15 \[ \frac{1}{12} \, d{\rm ln}\left (3 \, x^{4} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*d*ln(3*x^4 + 2)