3.8.91 \(\int \frac {x^3}{\sqrt {2+2 a-2 (1+a)+c x^4}} \, dx\)

Optimal. Leaf size=16 \[ \frac {x^4}{2 \sqrt {c x^4}} \]

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Rubi [A]  time = 0.00, antiderivative size = 16, 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, 30} \begin {gather*} \frac {x^4}{2 \sqrt {c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[2 + 2*a - 2*(1 + a) + c*x^4],x]

[Out]

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

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {x^4}{2 \sqrt {c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[2 + 2*a - 2*(1 + a) + c*x^4],x]

[Out]

x^4/(2*Sqrt[c*x^4])

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IntegrateAlgebraic [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {\sqrt {c x^4}}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^3/Sqrt[2 + 2*a - 2*(1 + a) + c*x^4],x]

[Out]

Sqrt[c*x^4]/(2*c)

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fricas [A]  time = 0.83, size = 12, normalized size = 0.75 \begin {gather*} \frac {\sqrt {c x^{4}}}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 8, normalized size = 0.50 \begin {gather*} \frac {x^{2}}{2 \, \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4)^(1/2),x, algorithm="giac")

[Out]

1/2*x^2/sqrt(c)

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maple [A]  time = 0.00, size = 13, normalized size = 0.81 \begin {gather*} \frac {x^{4}}{2 \sqrt {c \,x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.04, size = 12, normalized size = 0.75 \begin {gather*} \frac {\sqrt {c x^{4}}}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c*x^4)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.50, size = 10, normalized size = 0.62 \begin {gather*} \frac {\sqrt {x^4}}{2\,\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.53, size = 15, normalized size = 0.94 \begin {gather*} \frac {x^{4}}{2 \sqrt {c} \sqrt {x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(c*x**4)**(1/2),x)

[Out]

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

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