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 = 0.0013116, 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, Rules used = {1, 15, 29} \[ \frac{x^2 \log (x)}{\sqrt{c x^4}} \]

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]

Rule 1

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*(b*x^n)^p, x] /; FreeQ[{a, b, n, p}, x] && EqQ[a
, 0]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{2+2 a-2 (1+a)+c x^4}} \, dx &=\int \frac{x}{\sqrt{c x^4}} \, dx\\ &=\frac{x^2 \int \frac{1}{x} \, dx}{\sqrt{c x^4}}\\ &=\frac{x^2 \log (x)}{\sqrt{c x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0015753, 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 = 0.046, size = 14, normalized size = 0.9 \begin{align*}{{x}^{2}\ln \left ( x \right ){\frac{1}{\sqrt{c{x}^{4}}}}} \end{align*}

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 = 0.951511, size = 18, normalized size = 1.2 \begin{align*} \frac{x^{2} \log \left (x\right )}{\sqrt{c x^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(c*x^4)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.29413, size = 38, normalized size = 2.53 \begin{align*} \frac{\sqrt{c x^{4}} \log \left (x\right )}{c x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(c*x^4)^(1/2),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. \begin{align*} \int \frac{x}{\sqrt{c x^{4}}}\, dx \end{align*}

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 [A]  time = 1.17966, size = 9, normalized size = 0.6 \begin{align*} \frac{\log \left ({\left | x \right |}\right )}{\sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(c*x^4)^(1/2),x, algorithm="giac")

[Out]

log(abs(x))/sqrt(c)