3.1023 \(\int \frac{x^2}{\sqrt{2+2 a-2 (1+a)+c x^4}} \, dx\)

Optimal. Leaf size=13 \[ \frac{x^3}{\sqrt{c x^4}} \]

[Out]

x^3/Sqrt[c*x^4]

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Rubi [A]  time = 0.0011957, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {1, 15, 8} \[ \frac{x^3}{\sqrt{c x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^3/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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{x^2}{\sqrt{2+2 a-2 (1+a)+c x^4}} \, dx &=\int \frac{x^2}{\sqrt{c x^4}} \, dx\\ &=\frac{x^2 \int 1 \, dx}{\sqrt{c x^4}}\\ &=\frac{x^3}{\sqrt{c x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0011859, size = 13, normalized size = 1. \[ \frac{x^3}{\sqrt{c x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^3/Sqrt[c*x^4]

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Maple [A]  time = 0.042, size = 12, normalized size = 0.9 \begin{align*}{{x}^{3}{\frac{1}{\sqrt{c{x}^{4}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3/(c*x^4)^(1/2)

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Maxima [A]  time = 0.960865, size = 15, normalized size = 1.15 \begin{align*} \frac{x^{3}}{\sqrt{c x^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3/sqrt(c*x^4)

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Fricas [A]  time = 1.21399, size = 26, normalized size = 2. \begin{align*} \frac{\sqrt{c x^{4}}}{c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.420185, size = 14, normalized size = 1.08 \begin{align*} \frac{x^{3}}{\sqrt{c} \sqrt{x^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(c*x**4)**(1/2),x)

[Out]

x**3/(sqrt(c)*sqrt(x**4))

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Giac [A]  time = 1.13143, size = 7, normalized size = 0.54 \begin{align*} \frac{x}{\sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/sqrt(c)