\(\int \sqrt {a+\frac {b}{x^4}} x^3 \, dx\) [2058]

Optimal result

Integrand size = 15, antiderivative size = 47 \[ \int \sqrt {a+\frac {b}{x^4}} x^3 \, dx=\frac {1}{4} \sqrt {a+\frac {b}{x^4}} x^4+\frac {b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4 \sqrt {a}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 43, 65, 214} \[ \int \sqrt {a+\frac {b}{x^4}} x^3 \, dx=\frac {b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4 \sqrt {a}}+\frac {1}{4} x^4 \sqrt {a+\frac {b}{x^4}} \]

[In]

Int[Sqrt[a + b/x^4]*x^3,x]

[Out]

(Sqrt[a + b/x^4]*x^4)/4 + (b*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/(4*Sqrt[a])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 272

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

Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{4} \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2} \, dx,x,\frac {1}{x^4}\right )\right ) \\ & = \frac {1}{4} \sqrt {a+\frac {b}{x^4}} x^4-\frac {1}{8} b \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^4}\right ) \\ & = \frac {1}{4} \sqrt {a+\frac {b}{x^4}} x^4-\frac {1}{4} \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^4}}\right ) \\ & = \frac {1}{4} \sqrt {a+\frac {b}{x^4}} x^4+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4 \sqrt {a}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.32 \[ \int \sqrt {a+\frac {b}{x^4}} x^3 \, dx=\frac {1}{4} \sqrt {a+\frac {b}{x^4}} x^2 \left (x^2+\frac {b \log \left (\sqrt {a} x^2+\sqrt {b+a x^4}\right )}{\sqrt {a} \sqrt {b+a x^4}}\right ) \]

[In]

Integrate[Sqrt[a + b/x^4]*x^3,x]

[Out]

(Sqrt[a + b/x^4]*x^2*(x^2 + (b*Log[Sqrt[a]*x^2 + Sqrt[b + a*x^4]])/(Sqrt[a]*Sqrt[b + a*x^4])))/4

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.45

method	result	size
default	\(\frac {\sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2} \left (x^{2} \sqrt {a \,x^{4}+b}\, \sqrt {a}+b \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right )\right )}{4 \sqrt {a \,x^{4}+b}\, \sqrt {a}}\)	\(68\)
risch	\(\frac {x^{4} \sqrt {\frac {a \,x^{4}+b}{x^{4}}}}{4}+\frac {b \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right ) \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}{4 \sqrt {a}\, \sqrt {a \,x^{4}+b}}\)	\(69\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.70 \[ \int \sqrt {a+\frac {b}{x^4}} x^3 \, dx=\left [\frac {2 \, a x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} + \sqrt {a} b \log \left (-2 \, a x^{4} - 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right )}{8 \, a}, \frac {a x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - \sqrt {-a} b \arctan \left (\frac {\sqrt {-a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right )}{4 \, a}\right ] \]

[In]

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

[Out]

[1/8*(2*a*x^4*sqrt((a*x^4 + b)/x^4) + sqrt(a)*b*log(-2*a*x^4 - 2*sqrt(a)*x^4*sqrt((a*x^4 + b)/x^4) - b))/a, 1/
4*(a*x^4*sqrt((a*x^4 + b)/x^4) - sqrt(-a)*b*arctan(sqrt(-a)*x^4*sqrt((a*x^4 + b)/x^4)/(a*x^4 + b)))/a]

Sympy [A] (verification not implemented)

Time = 1.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int \sqrt {a+\frac {b}{x^4}} x^3 \, dx=\frac {\sqrt {b} x^{2} \sqrt {\frac {a x^{4}}{b} + 1}}{4} + \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a} x^{2}}{\sqrt {b}} \right )}}{4 \sqrt {a}} \]

[In]

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

[Out]

sqrt(b)*x**2*sqrt(a*x**4/b + 1)/4 + b*asinh(sqrt(a)*x**2/sqrt(b))/(4*sqrt(a))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.13 \[ \int \sqrt {a+\frac {b}{x^4}} x^3 \, dx=\frac {1}{4} \, \sqrt {a + \frac {b}{x^{4}}} x^{4} - \frac {b \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right )}{8 \, \sqrt {a}} \]

[In]

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

[Out]

1/4*sqrt(a + b/x^4)*x^4 - 1/8*b*log((sqrt(a + b/x^4) - sqrt(a))/(sqrt(a + b/x^4) + sqrt(a)))/sqrt(a)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.87 \[ \int \sqrt {a+\frac {b}{x^4}} x^3 \, dx=\frac {1}{4} \, \sqrt {a x^{4} + b} x^{2} - \frac {b \log \left ({\left | -\sqrt {a} x^{2} + \sqrt {a x^{4} + b} \right |}\right )}{4 \, \sqrt {a}} \]

[In]

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

[Out]

1/4*sqrt(a*x^4 + b)*x^2 - 1/4*b*log(abs(-sqrt(a)*x^2 + sqrt(a*x^4 + b)))/sqrt(a)

Mupad [B] (verification not implemented)

Time = 6.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \sqrt {a+\frac {b}{x^4}} x^3 \, dx=\frac {x^4\,\sqrt {a+\frac {b}{x^4}}}{4}+\frac {b\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4\,\sqrt {a}} \]

[In]

int(x^3*(a + b/x^4)^(1/2),x)

[Out]

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