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

Optimal. Leaf size=60 \[ \frac{6 \sqrt [3]{x}}{b \sqrt{a x+b x^{2/3}}}-\frac{6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{b^{3/2}} \]

[Out]

(6*x^(1/3))/(b*Sqrt[b*x^(2/3) + a*x]) - (6*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/b^(3/2)

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Rubi [A]  time = 0.0560228, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2006, 2029, 206} \[ \frac{6 \sqrt [3]{x}}{b \sqrt{a x+b x^{2/3}}}-\frac{6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^(2/3) + a*x)^(-3/2),x]

[Out]

(6*x^(1/3))/(b*Sqrt[b*x^(2/3) + a*x]) - (6*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/b^(3/2)

Rule 2006

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

Rule 2029

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]

Rule 206

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

Rubi steps

\begin{align*} \int \frac{1}{\left (b x^{2/3}+a x\right )^{3/2}} \, dx &=\frac{6 \sqrt [3]{x}}{b \sqrt{b x^{2/3}+a x}}+\frac{\int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx}{b}\\ &=\frac{6 \sqrt [3]{x}}{b \sqrt{b x^{2/3}+a x}}-\frac{6 \operatorname{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{6 \sqrt [3]{x}}{b \sqrt{b x^{2/3}+a x}}-\frac{6 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{b^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.0259092, size = 45, normalized size = 0.75 \[ \frac{6 \sqrt [3]{x} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{b \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^(2/3) + a*x)^(-3/2),x]

[Out]

(6*x^(1/3)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (a*x^(1/3))/b])/(b*Sqrt[b*x^(2/3) + a*x])

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Maple [A]  time = 0.006, size = 55, normalized size = 0.9 \begin{align*} 6\,{\frac{x \left ( b+a\sqrt [3]{x} \right ) }{ \left ( b{x}^{2/3}+ax \right ) ^{3/2}{b}^{5/2}} \left ({b}^{3/2}-{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ) b\sqrt{b+a\sqrt [3]{x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a x + b x^{\frac{2}{3}}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a x + b x^{\frac{2}{3}}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16026, size = 96, normalized size = 1.6 \begin{align*} \frac{6 \, \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} - \frac{6 \,{\left (\sqrt{b} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + \sqrt{-b}\right )}}{\sqrt{-b} b^{\frac{3}{2}}} + \frac{6}{\sqrt{a x^{\frac{1}{3}} + b} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b) - 6*(sqrt(b)*arctan(sqrt(b)/sqrt(-b)) + sqrt(-b))/(sqrt(-b
)*b^(3/2)) + 6/(sqrt(a*x^(1/3) + b)*b)