\(\int \frac {x^3}{\sqrt {a+\frac {b}{x^4}}} \, dx\) [2082]

Optimal result

Integrand size = 15, antiderivative size = 50 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^4}}} \, dx=\frac {\sqrt {a+\frac {b}{x^4}} x^4}{4 a}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4 a^{3/2}} \]

[Out]

-1/4*b*arctanh((a+b/x^4)^(1/2)/a^(1/2))/a^(3/2)+1/4*x^4*(a+b/x^4)^(1/2)/a

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 50, 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, 44, 65, 214} \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^4}}} \, dx=\frac {x^4 \sqrt {a+\frac {b}{x^4}}}{4 a}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4 a^{3/2}} \]

[In]

Int[x^3/Sqrt[a + b/x^4],x]

[Out]

(Sqrt[a + b/x^4]*x^4)/(4*a) - (b*ArcTanh[Sqrt[a + b/x^4]/Sqrt[a]])/(4*a^(3/2))

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[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 {1}{x^2 \sqrt {a+b x}} \, dx,x,\frac {1}{x^4}\right )\right ) \\ & = \frac {\sqrt {a+\frac {b}{x^4}} x^4}{4 a}+\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^4}\right )}{8 a} \\ & = \frac {\sqrt {a+\frac {b}{x^4}} x^4}{4 a}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^4}}\right )}{4 a} \\ & = \frac {\sqrt {a+\frac {b}{x^4}} x^4}{4 a}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4 a^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.52 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^4}}} \, dx=\frac {\sqrt {a} x^2 \left (b+a x^4\right )-b \sqrt {b+a x^4} \log \left (\sqrt {a} x^2+\sqrt {b+a x^4}\right )}{4 a^{3/2} \sqrt {a+\frac {b}{x^4}} x^2} \]

[In]

Integrate[x^3/Sqrt[a + b/x^4],x]

[Out]

(Sqrt[a]*x^2*(b + a*x^4) - b*Sqrt[b + a*x^4]*Log[Sqrt[a]*x^2 + Sqrt[b + a*x^4]])/(4*a^(3/2)*Sqrt[a + b/x^4]*x^
2)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.40

method	result	size
default	\(\frac {\sqrt {a \,x^{4}+b}\, \left (x^{2} \sqrt {a \,x^{4}+b}\, a^{\frac {3}{2}}-b \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right ) a \right )}{4 \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2} a^{\frac {5}{2}}}\)	\(70\)
risch	\(\frac {a \,x^{4}+b}{4 a \sqrt {\frac {a \,x^{4}+b}{x^{4}}}}-\frac {b \ln \left (x^{2} \sqrt {a}+\sqrt {a \,x^{4}+b}\right ) \sqrt {a \,x^{4}+b}}{4 a^{\frac {3}{2}} \sqrt {\frac {a \,x^{4}+b}{x^{4}}}\, x^{2}}\)	\(76\)

[In]

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

[Out]

1/4*(a*x^4+b)^(1/2)*(x^2*(a*x^4+b)^(1/2)*a^(3/2)-b*ln(x^2*a^(1/2)+(a*x^4+b)^(1/2))*a)/((a*x^4+b)/x^4)^(1/2)/x^
2/a^(5/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.52 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^4}}} \, dx=\left [\frac {2 \, a x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} + \sqrt {a} b \log \left (-2 \, a x^{4} + 2 \, \sqrt {a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} - b\right )}{8 \, a^{2}}, \frac {a x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}} + \sqrt {-a} b \arctan \left (\frac {\sqrt {-a} x^{4} \sqrt {\frac {a x^{4} + b}{x^{4}}}}{a x^{4} + b}\right )}{4 \, a^{2}}\right ] \]

[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^2,
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^2]

Sympy [A] (verification not implemented)

Time = 1.30 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.92 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^4}}} \, dx=\frac {\sqrt {b} x^{2} \sqrt {\frac {a x^{4}}{b} + 1}}{4 a} - \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a} x^{2}}{\sqrt {b}} \right )}}{4 a^{\frac {3}{2}}} \]

[In]

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

[Out]

sqrt(b)*x**2*sqrt(a*x**4/b + 1)/(4*a) - b*asinh(sqrt(a)*x**2/sqrt(b))/(4*a**(3/2))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.36 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^4}}} \, dx=\frac {\sqrt {a + \frac {b}{x^{4}}} b}{4 \, {\left ({\left (a + \frac {b}{x^{4}}\right )} a - a^{2}\right )}} + \frac {b \log \left (\frac {\sqrt {a + \frac {b}{x^{4}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{4}}} + \sqrt {a}}\right )}{8 \, a^{\frac {3}{2}}} \]

[In]

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

[Out]

1/4*sqrt(a + b/x^4)*b/((a + b/x^4)*a - a^2) + 1/8*b*log((sqrt(a + b/x^4) - sqrt(a))/(sqrt(a + b/x^4) + sqrt(a)
))/a^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^4}}} \, dx=\frac {\sqrt {a x^{4} + b} x^{2}}{4 \, a} + \frac {b \log \left ({\left | -\sqrt {a} x^{2} + \sqrt {a x^{4} + b} \right |}\right )}{4 \, a^{\frac {3}{2}}} \]

[In]

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

[Out]

1/4*sqrt(a*x^4 + b)*x^2/a + 1/4*b*log(abs(-sqrt(a)*x^2 + sqrt(a*x^4 + b)))/a^(3/2)

Mupad [B] (verification not implemented)

Time = 6.10 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.76 \[ \int \frac {x^3}{\sqrt {a+\frac {b}{x^4}}} \, dx=\frac {x^4\,\sqrt {a+\frac {b}{x^4}}}{4\,a}-\frac {b\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^4}}}{\sqrt {a}}\right )}{4\,a^{3/2}} \]

[In]

int(x^3/(a + b/x^4)^(1/2),x)

[Out]

(x^4*(a + b/x^4)^(1/2))/(4*a) - (b*atanh((a + b/x^4)^(1/2)/a^(1/2)))/(4*a^(3/2))