3.1024 \(\int \frac{x}{\sqrt{2+2 a-2 (1+a)+c x^4}} \, dx\)

Optimal. Leaf size=15 \[ \frac{x^2 \log (x)}{\sqrt{c x^4}} \]

[Out]

(x^2*Log[x])/Sqrt[c*x^4]

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Rubi [A]  time = 0.00614463, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{x^2 \log (x)}{\sqrt{c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[2 + 2*a - 2*(1 + a) + c*x^4],x]

[Out]

(x^2*Log[x])/Sqrt[c*x^4]

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Rubi in Sympy [A]  time = 1.93994, size = 15, normalized size = 1. \[ \frac{\sqrt{c x^{4}} \log{\left (x \right )}}{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(c*x**4)**(1/2),x)

[Out]

sqrt(c*x**4)*log(x)/(c*x**2)

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Mathematica [A]  time = 0.00298992, size = 15, normalized size = 1. \[ \frac{x^2 \log (x)}{\sqrt{c x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[2 + 2*a - 2*(1 + a) + c*x^4],x]

[Out]

(x^2*Log[x])/Sqrt[c*x^4]

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Maple [A]  time = 0.004, size = 14, normalized size = 0.9 \[{{x}^{2}\ln \left ( x \right ){\frac{1}{\sqrt{c{x}^{4}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(c*x^4)^(1/2),x)

[Out]

x^2*ln(x)/(c*x^4)^(1/2)

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Maxima [A]  time = 0.713645, size = 18, normalized size = 1.2 \[ \frac{x^{2} \log \left (x\right )}{\sqrt{c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(c*x^4),x, algorithm="maxima")

[Out]

x^2*log(x)/sqrt(c*x^4)

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Fricas [A]  time = 0.270119, size = 22, normalized size = 1.47 \[ \frac{\sqrt{c x^{4}} \log \left (x\right )}{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(c*x^4),x, algorithm="fricas")

[Out]

sqrt(c*x^4)*log(x)/(c*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(c*x**4)**(1/2),x)

[Out]

Integral(x/sqrt(c*x**4), x)

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GIAC/XCAS [A]  time = 0.271099, size = 16, normalized size = 1.07 \[ \frac{{\rm ln}\left (x^{4}{\left | c \right |}\right )}{4 \, \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(c*x^4),x, algorithm="giac")

[Out]

1/4*ln(x^4*abs(c))/sqrt(c)