3.11.28 \(\int \frac {1}{x^3 \sqrt {2+2 a-2 (1+a)+c x^4}} \, dx\) [1028]

Optimal. Leaf size=16 \[ -\frac {1}{4 x^2 \sqrt {c x^4}} \]

[Out]

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

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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 {1}{4 x^2 \sqrt {c x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*1/(x^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 {1}{x^3 \sqrt {2+2 a-2 (1+a)+c x^4}} \, dx &=\int \frac {1}{x^3 \sqrt {c x^4}} \, dx\\ &=\frac {x^2 \int \frac {1}{x^5} \, dx}{\sqrt {c x^4}}\\ &=-\frac {1}{4 x^2 \sqrt {c x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.06 \begin {gather*} -\frac {c x^2}{4 \left (c x^4\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 13, normalized size = 0.81

method	result	size
gosper	\(-\frac {1}{4 x^{2} \sqrt {c \,x^{4}}}\)	\(13\)
default	\(-\frac {1}{4 x^{2} \sqrt {c \,x^{4}}}\)	\(13\)
risch	\(-\frac {1}{4 x^{2} \sqrt {c \,x^{4}}}\)	\(13\)
trager	\(\frac {\left (-1+x \right ) \left (x^{3}+x^{2}+x +1\right ) \sqrt {c \,x^{4}}}{4 c \,x^{6}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 12, normalized size = 0.75 \begin {gather*} -\frac {1}{4 \, \sqrt {c x^{4}} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.34, size = 15, normalized size = 0.94 \begin {gather*} -\frac {\sqrt {c x^{4}}}{4 \, c x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(c*x^4)/(c*x^6)

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Sympy [A]
time = 0.21, size = 15, normalized size = 0.94 \begin {gather*} - \frac {1}{4 x^{2} \sqrt {c x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 4.04, size = 8, normalized size = 0.50 \begin {gather*} -\frac {1}{4 \, \sqrt {c} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 4.27, size = 13, normalized size = 0.81 \begin {gather*} -\frac {1}{4\,\sqrt {c}\,x^2\,\sqrt {x^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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