\(\int \frac {x^3}{\sqrt {a+b x^4}} \, dx\) [813]

Optimal result

Integrand size = 15, antiderivative size = 18 \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {a+b x^4}}{2 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {267} \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {a+b x^4}}{2 b} \]

[In]

Int[x^3/Sqrt[a + b*x^4],x]

[Out]

Sqrt[a + b*x^4]/(2*b)

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*} \text {integral}& = \frac {\sqrt {a+b x^4}}{2 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {a+b x^4}}{2 b} \]

[In]

Integrate[x^3/Sqrt[a + b*x^4],x]

[Out]

Sqrt[a + b*x^4]/(2*b)

Maple [A] (verified)

Time = 4.30 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {b x^{4} + a}}{2 \, b} \]

[In]

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

[Out]

1/2*sqrt(b*x^4 + a)/b

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\begin {cases} \frac {\sqrt {a + b x^{4}}}{2 b} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 \sqrt {a}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {b x^{4} + a}}{2 \, b} \]

[In]

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

[Out]

1/2*sqrt(b*x^4 + a)/b

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {b x^{4} + a}}{2 \, b} \]

[In]

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

[Out]

1/2*sqrt(b*x^4 + a)/b

Mupad [B] (verification not implemented)

Time = 5.51 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\sqrt {a+b x^4}} \, dx=\frac {\sqrt {b\,x^4+a}}{2\,b} \]

[In]

int(x^3/(a + b*x^4)^(1/2),x)

[Out]

(a + b*x^4)^(1/2)/(2*b)