\(\int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx\) [190]

Optimal result

Integrand size = 19, antiderivative size = 61 \[ \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx=-\frac {3 \sqrt {b x^{2/3}+a x}}{b x^{2/3}}+\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{b^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2050, 2054, 212} \[ \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx=\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}} \]

[In]

Int[1/(x*Sqrt[b*x^(2/3) + a*x]),x]

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(b*x^(2/3)) + (3*a*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/b^(3/2)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2050

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[c^(j - 1)*(c*x)^(m - j +
1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p + 1))), x] - Dist[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))
), Int[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j,
n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]

Rule 2054

Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[-2/(n - j), Subst[Int[1/(1 - a*x^2
), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx=-\frac {3 \sqrt {b x^{2/3}+a x}}{b x^{2/3}}+\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{b^{3/2}} \]

[In]

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

[Out]

(-3*Sqrt[b*x^(2/3) + a*x])/(b*x^(2/3)) + (3*a*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/b^(3/2)

Maple [A] (verified)

Time = 1.81 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(-\frac {3 \sqrt {b +a \,x^{\frac {1}{3}}}\, \left (\sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {3}{2}}-\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b a \,x^{\frac {1}{3}}\right )}{\sqrt {b \,x^{\frac {2}{3}}+a x}\, b^{\frac {5}{2}}}\)	\(61\)
default	\(\frac {3 \sqrt {b +a \,x^{\frac {1}{3}}}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b a \,x^{\frac {1}{3}}-\sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {3}{2}}\right )}{\sqrt {b \,x^{\frac {2}{3}}+a x}\, b^{\frac {5}{2}}}\)	\(61\)

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx=\int \frac {1}{x \sqrt {a x + b x^{\frac {2}{3}}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx=\int { \frac {1}{\sqrt {a x + b x^{\frac {2}{3}}} x} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx=-\frac {3 \, {\left (\frac {a^{2} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} + \frac {\sqrt {a x^{\frac {1}{3}} + b} a}{b x^{\frac {1}{3}}}\right )}}{a} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx=\int \frac {1}{x\,\sqrt {a\,x+b\,x^{2/3}}} \,d x \]

[In]

int(1/(x*(a*x + b*x^(2/3))^(1/2)),x)

[Out]

int(1/(x*(a*x + b*x^(2/3))^(1/2)), x)