3.11.4 \(\int \frac {x^3}{\sqrt {a+(2+2 b-2 (1+b)) x^2+c x^4}} \, dx\) [1004]

Optimal. Leaf size=18 \[ \frac {\sqrt {a+c x^4}}{2 c} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {4, 267} \begin {gather*} \frac {\sqrt {a+c x^4}}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[a + c*x^4]/(2*c)

Rule 4

Int[(u_.)*((a_.) + (c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*(a + c*x^(2*n))^p, x] /; Fre
eQ[{a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[b, 0]

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a+c x^4}}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[a + c*x^4]/(2*c)

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Maple [A]
time = 0.14, size = 15, normalized size = 0.83

method	result	size
gosper	\(\frac {\sqrt {c \,x^{4}+a}}{2 c}\)	\(15\)
derivativedivides	\(\frac {\sqrt {c \,x^{4}+a}}{2 c}\)	\(15\)
default	\(\frac {\sqrt {c \,x^{4}+a}}{2 c}\)	\(15\)
trager	\(\frac {\sqrt {c \,x^{4}+a}}{2 c}\)	\(15\)
risch	\(\frac {\sqrt {c \,x^{4}+a}}{2 c}\)	\(15\)
elliptic	\(\frac {\sqrt {c \,x^{4}+a}}{2 c}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 14, normalized size = 0.78 \begin {gather*} \frac {\sqrt {c x^{4} + a}}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(c*x^4 + a)/c

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Fricas [A]
time = 0.35, size = 14, normalized size = 0.78 \begin {gather*} \frac {\sqrt {c x^{4} + a}}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(c*x^4 + a)/c

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Sympy [A]
time = 0.20, size = 22, normalized size = 1.22 \begin {gather*} \begin {cases} \frac {\sqrt {a + c x^{4}}}{2 c} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4 \sqrt {a}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(a + c*x**4)/(2*c), Ne(c, 0)), (x**4/(4*sqrt(a)), True))

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Giac [A]
time = 3.95, size = 14, normalized size = 0.78 \begin {gather*} \frac {\sqrt {c x^{4} + a}}{2 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(c*x^4 + a)/c

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Mupad [B]
time = 4.66, size = 14, normalized size = 0.78 \begin {gather*} \frac {\sqrt {c\,x^4+a}}{2\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + c*x^4)^(1/2),x)

[Out]

(a + c*x^4)^(1/2)/(2*c)

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