3.11.23 \(\int \frac {x^2}{\sqrt {2+2 a-2 (1+a)+c x^4}} \, dx\) [1023]

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

[Out]

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

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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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Maple [A]
time = 0.03, size = 12, normalized size = 0.92

method	result	size
default	\(\frac {x^{3}}{\sqrt {c \,x^{4}}}\)	\(12\)
risch	\(\frac {x^{3}}{\sqrt {c \,x^{4}}}\)	\(12\)
trager	\(\frac {\left (-1+x \right ) \sqrt {c \,x^{4}}}{c \,x^{2}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, 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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Fricas [A]
time = 0.33, 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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Sympy [A]
time = 0.17, size = 10, normalized size = 0.77 \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)

[Out]

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

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Giac [A]
time = 5.20, 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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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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