\(\int \frac {1}{(a+\frac {b}{x^3})^{3/2} x} \, dx\) [2039]

Optimal result

Integrand size = 15, antiderivative size = 46 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x} \, dx=-\frac {2}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3 a^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 46, 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, 53, 65, 214} \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {2}{3 a \sqrt {a+\frac {b}{x^3}}} \]

[In]

Int[1/((a + b/x^3)^(3/2)*x),x]

[Out]

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

Rule 53

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] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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}{3} \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,\frac {1}{x^3}\right )\right ) \\ & = -\frac {2}{3 a \sqrt {a+\frac {b}{x^3}}}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x^3}\right )}{3 a} \\ & = -\frac {2}{3 a \sqrt {a+\frac {b}{x^3}}}-\frac {2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x^3}}\right )}{3 a b} \\ & = -\frac {2}{3 a \sqrt {a+\frac {b}{x^3}}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3 a^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.61 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x} \, dx=-\frac {2 \left (\sqrt {a} x^{3/2}-\sqrt {b+a x^3} \log \left (\sqrt {a} x^{3/2}+\sqrt {b+a x^3}\right )\right )}{3 a^{3/2} \sqrt {a+\frac {b}{x^3}} x^{3/2}} \]

[In]

Integrate[1/((a + b/x^3)^(3/2)*x),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(34)=68\).

Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.52

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (34) = 68\).

Time = 0.38 (sec) , antiderivative size = 186, normalized size of antiderivative = 4.04 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x} \, dx=\left [-\frac {4 \, a x^{3} \sqrt {\frac {a x^{3} + b}{x^{3}}} - {\left (a x^{3} + b\right )} \sqrt {a} \log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \, {\left (2 \, a x^{6} + b x^{3}\right )} \sqrt {a} \sqrt {\frac {a x^{3} + b}{x^{3}}}\right )}{6 \, {\left (a^{3} x^{3} + a^{2} b\right )}}, -\frac {2 \, a x^{3} \sqrt {\frac {a x^{3} + b}{x^{3}}} + {\left (a x^{3} + b\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a} x^{3} \sqrt {\frac {a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right )}{3 \, {\left (a^{3} x^{3} + a^{2} b\right )}}\right ] \]

[In]

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

[Out]

[-1/6*(4*a*x^3*sqrt((a*x^3 + b)/x^3) - (a*x^3 + b)*sqrt(a)*log(-8*a^2*x^6 - 8*a*b*x^3 - b^2 - 4*(2*a*x^6 + b*x
^3)*sqrt(a)*sqrt((a*x^3 + b)/x^3)))/(a^3*x^3 + a^2*b), -1/3*(2*a*x^3*sqrt((a*x^3 + b)/x^3) + (a*x^3 + b)*sqrt(
-a)*arctan(2*sqrt(-a)*x^3*sqrt((a*x^3 + b)/x^3)/(2*a*x^3 + b)))/(a^3*x^3 + a^2*b)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (39) = 78\).

Time = 0.99 (sec) , antiderivative size = 187, normalized size of antiderivative = 4.07 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x} \, dx=- \frac {2 a^{3} x^{3} \sqrt {1 + \frac {b}{a x^{3}}}}{3 a^{\frac {9}{2}} x^{3} + 3 a^{\frac {7}{2}} b} - \frac {a^{3} x^{3} \log {\left (\frac {b}{a x^{3}} \right )}}{3 a^{\frac {9}{2}} x^{3} + 3 a^{\frac {7}{2}} b} + \frac {2 a^{3} x^{3} \log {\left (\sqrt {1 + \frac {b}{a x^{3}}} + 1 \right )}}{3 a^{\frac {9}{2}} x^{3} + 3 a^{\frac {7}{2}} b} - \frac {a^{2} b \log {\left (\frac {b}{a x^{3}} \right )}}{3 a^{\frac {9}{2}} x^{3} + 3 a^{\frac {7}{2}} b} + \frac {2 a^{2} b \log {\left (\sqrt {1 + \frac {b}{a x^{3}}} + 1 \right )}}{3 a^{\frac {9}{2}} x^{3} + 3 a^{\frac {7}{2}} b} \]

[In]

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

[Out]

-2*a**3*x**3*sqrt(1 + b/(a*x**3))/(3*a**(9/2)*x**3 + 3*a**(7/2)*b) - a**3*x**3*log(b/(a*x**3))/(3*a**(9/2)*x**
3 + 3*a**(7/2)*b) + 2*a**3*x**3*log(sqrt(1 + b/(a*x**3)) + 1)/(3*a**(9/2)*x**3 + 3*a**(7/2)*b) - a**2*b*log(b/
(a*x**3))/(3*a**(9/2)*x**3 + 3*a**(7/2)*b) + 2*a**2*b*log(sqrt(1 + b/(a*x**3)) + 1)/(3*a**(9/2)*x**3 + 3*a**(7
/2)*b)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x} \, dx=-\frac {\log \left (\frac {\sqrt {a + \frac {b}{x^{3}}} - \sqrt {a}}{\sqrt {a + \frac {b}{x^{3}}} + \sqrt {a}}\right )}{3 \, a^{\frac {3}{2}}} - \frac {2}{3 \, \sqrt {a + \frac {b}{x^{3}}} a} \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value

Mupad [B] (verification not implemented)

Time = 5.78 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (a+\frac {b}{x^3}\right )^{3/2} x} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x^3}}}{\sqrt {a}}\right )}{3\,a^{3/2}}-\frac {2}{3\,a\,\sqrt {a+\frac {b}{x^3}}} \]

[In]

int(1/(x*(a + b/x^3)^(3/2)),x)

[Out]

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