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

Optimal. Leaf size=78 \[ -6 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )+\frac{6 b \sqrt{a x+b x^{2/3}}}{\sqrt [3]{x}}+\frac{2 \left (a x+b x^{2/3}\right )^{3/2}}{x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.136998, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2021, 2029, 206} \[ -6 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )+\frac{6 b \sqrt{a x+b x^{2/3}}}{\sqrt [3]{x}}+\frac{2 \left (a x+b x^{2/3}\right )^{3/2}}{x} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^(2/3) + a*x)^(3/2)/x^2,x]

[Out]

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

Rule 2021

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

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{\left (b x^{2/3}+a x\right )^{3/2}}{x^2} \, dx &=\frac{2 \left (b x^{2/3}+a x\right )^{3/2}}{x}+b \int \frac{\sqrt{b x^{2/3}+a x}}{x^{4/3}} \, dx\\ &=\frac{6 b \sqrt{b x^{2/3}+a x}}{\sqrt [3]{x}}+\frac{2 \left (b x^{2/3}+a x\right )^{3/2}}{x}+b^2 \int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx\\ &=\frac{6 b \sqrt{b x^{2/3}+a x}}{\sqrt [3]{x}}+\frac{2 \left (b x^{2/3}+a x\right )^{3/2}}{x}-\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )\\ &=\frac{6 b \sqrt{b x^{2/3}+a x}}{\sqrt [3]{x}}+\frac{2 \left (b x^{2/3}+a x\right )^{3/2}}{x}-6 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0689394, size = 88, normalized size = 1.13 \[ \frac{2 \sqrt{a x+b x^{2/3}} \left (\sqrt{a \sqrt [3]{x}+b} \left (a \sqrt [3]{x}+4 b\right )-3 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a \sqrt [3]{x}+b}}{\sqrt{b}}\right )\right )}{\sqrt [3]{x} \sqrt{a \sqrt [3]{x}+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x^(2/3) + a*x]*(Sqrt[b + a*x^(1/3)]*(4*b + a*x^(1/3)) - 3*b^(3/2)*ArcTanh[Sqrt[b + a*x^(1/3)]/Sqrt[b
]]))/(Sqrt[b + a*x^(1/3)]*x^(1/3))

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 69, normalized size = 0.9 \begin{align*} -2\,{\frac{ \left ( b{x}^{2/3}+ax \right ) ^{3/2}}{x \left ( b+a\sqrt [3]{x} \right ) ^{3/2}} \left ( 3\,{b}^{3/2}{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ) - \left ( b+a\sqrt [3]{x} \right ) ^{3/2}-3\,b\sqrt{b+a\sqrt [3]{x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.21538, size = 112, normalized size = 1.44 \begin{align*} \frac{6 \, b^{2} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + 2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} + 6 \, \sqrt{a x^{\frac{1}{3}} + b} b - \frac{2 \,{\left (3 \, b^{2} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 4 \, \sqrt{-b} b^{\frac{3}{2}}\right )}}{\sqrt{-b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*b^2*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/sqrt(-b) + 2*(a*x^(1/3) + b)^(3/2) + 6*sqrt(a*x^(1/3) + b)*b - 2*(3
*b^2*arctan(sqrt(b)/sqrt(-b)) + 4*sqrt(-b)*b^(3/2))/sqrt(-b)