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

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

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1, 15, 8} \begin {gather*} \frac {x^3}{\sqrt {c x^4}} \end {gather*}

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 8

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

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]

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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} \frac {x^3}{\sqrt {c x^4}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.02, size = 16, normalized size = 1.23 \begin {gather*} \frac {\sqrt {c x^4}}{c x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.97, size = 14, normalized size = 1.08 \begin {gather*} \frac {\sqrt {c x^{4}}}{c x} \end {gather*}

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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giac [A]  time = 0.17, size = 5, normalized size = 0.38 \begin {gather*} \frac {x}{\sqrt {c}} \end {gather*}

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)

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maple [A]  time = 0.00, size = 12, normalized size = 0.92 \begin {gather*} \frac {x^{3}}{\sqrt {c \,x^{4}}} \end {gather*}

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 = 0.98, size = 11, normalized size = 0.85 \begin {gather*} \frac {x^{3}}{\sqrt {c x^{4}}} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.08 \begin {gather*} \int \frac {x^2}{\sqrt {c\,x^4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2/(c*x^4)^(1/2), x)

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sympy [A]  time = 0.48, size = 14, normalized size = 1.08 \begin {gather*} \frac {x^{3}}{\sqrt {c} \sqrt {x^{4}}} \end {gather*}

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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