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

Optimal. Leaf size=12 \[ \frac {x^5}{5 \sqrt {a}} \]

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2, 12, 30} \begin {gather*} \frac {x^5}{5 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^5/(5*Sqrt[a])

Rule 2

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {x^4}{\sqrt {a+(2+2 c-2 (1+c)) x^4}} \, dx &=\int \frac {x^4}{\sqrt {a}} \, dx\\ &=\frac {\int x^4 \, dx}{\sqrt {a}}\\ &=\frac {x^5}{5 \sqrt {a}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \frac {x^5}{5 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^5/(5*Sqrt[a])

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4}{\sqrt {a+(2+2 c-2 (1+c)) x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.84, size = 8, normalized size = 0.67 \begin {gather*} \frac {x^{5}}{5 \, \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5/sqrt(a)

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giac [A]  time = 0.15, size = 8, normalized size = 0.67 \begin {gather*} \frac {x^{5}}{5 \, \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5/sqrt(a)

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maple [A]  time = 0.00, size = 9, normalized size = 0.75 \begin {gather*} \frac {x^{5}}{5 \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/a^(1/2),x)

[Out]

1/5*x^5/a^(1/2)

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maxima [A]  time = 1.01, size = 8, normalized size = 0.67 \begin {gather*} \frac {x^{5}}{5 \, \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5/sqrt(a)

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mupad [B]  time = 0.02, size = 8, normalized size = 0.67 \begin {gather*} \frac {x^5}{5\,\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/a^(1/2),x)

[Out]

x^5/(5*a^(1/2))

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sympy [A]  time = 0.07, size = 8, normalized size = 0.67 \begin {gather*} \frac {x^{5}}{5 \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**5/(5*sqrt(a))

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