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

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Rubi [A]  time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 1.00 \[ \frac {x^2 \log (x)}{\sqrt {c x^4}} \]

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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fricas [A]  time = 0.69, size = 16, normalized size = 1.07 \[ \frac {\sqrt {c x^{4}} \log \relax (x)}{c x^{2}} \]

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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giac [A]  time = 0.15, size = 7, normalized size = 0.47 \[ \frac {\log \left ({\left | x \right |}\right )}{\sqrt {c}} \]

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)

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maple [A]  time = 0.00, size = 14, normalized size = 0.93 \[ \frac {x^{2} \ln \relax (x )}{\sqrt {c \,x^{4}}} \]

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 = 0.98, size = 13, normalized size = 0.87 \[ \frac {x^{2} \log \relax (x)}{\sqrt {c x^{4}}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {x}{\sqrt {c\,x^4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(c*x^4)^(1/2),x)

[Out]

int(x/(c*x^4)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {c x^{4}}}\, dx \]

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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