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

Optimal. Leaf size=146 \[ -\frac{105 a^2 \sqrt{a x+b x^{2/3}}}{8 b^4 x^{2/3}}+\frac{105 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{8 b^{9/2}}+\frac{35 a \sqrt{a x+b x^{2/3}}}{4 b^3 x}-\frac{7 \sqrt{a x+b x^{2/3}}}{b^2 x^{4/3}}+\frac{6}{b x^{2/3} \sqrt{a x+b x^{2/3}}} \]

[Out]

6/(b*x^(2/3)*Sqrt[b*x^(2/3) + a*x]) - (7*Sqrt[b*x^(2/3) + a*x])/(b^2*x^(4/3)) + (35*a*Sqrt[b*x^(2/3) + a*x])/(
4*b^3*x) - (105*a^2*Sqrt[b*x^(2/3) + a*x])/(8*b^4*x^(2/3)) + (105*a^3*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3)
+ a*x]])/(8*b^(9/2))

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Rubi [A]  time = 0.240522, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2023, 2025, 2029, 206} \[ -\frac{105 a^2 \sqrt{a x+b x^{2/3}}}{8 b^4 x^{2/3}}+\frac{105 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{8 b^{9/2}}+\frac{35 a \sqrt{a x+b x^{2/3}}}{4 b^3 x}-\frac{7 \sqrt{a x+b x^{2/3}}}{b^2 x^{4/3}}+\frac{6}{b x^{2/3} \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(b*x^(2/3) + a*x)^(3/2)),x]

[Out]

6/(b*x^(2/3)*Sqrt[b*x^(2/3) + a*x]) - (7*Sqrt[b*x^(2/3) + a*x])/(b^2*x^(4/3)) + (35*a*Sqrt[b*x^(2/3) + a*x])/(
4*b^3*x) - (105*a^2*Sqrt[b*x^(2/3) + a*x])/(8*b^4*x^(2/3)) + (105*a^3*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3)
+ a*x]])/(8*b^(9/2))

Rule 2023

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*(n - j)*(p + 1)), x] + Dist[(c^j*(m + n*p + n - j + 1))/(a*(n - j)*(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]) && LtQ[p, -1]

Rule 2025

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 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}{x \left (b x^{2/3}+a x\right )^{3/2}} \, dx &=\frac{6}{b x^{2/3} \sqrt{b x^{2/3}+a x}}+\frac{7 \int \frac{1}{x^{5/3} \sqrt{b x^{2/3}+a x}} \, dx}{b}\\ &=\frac{6}{b x^{2/3} \sqrt{b x^{2/3}+a x}}-\frac{7 \sqrt{b x^{2/3}+a x}}{b^2 x^{4/3}}-\frac{(35 a) \int \frac{1}{x^{4/3} \sqrt{b x^{2/3}+a x}} \, dx}{6 b^2}\\ &=\frac{6}{b x^{2/3} \sqrt{b x^{2/3}+a x}}-\frac{7 \sqrt{b x^{2/3}+a x}}{b^2 x^{4/3}}+\frac{35 a \sqrt{b x^{2/3}+a x}}{4 b^3 x}+\frac{\left (35 a^2\right ) \int \frac{1}{x \sqrt{b x^{2/3}+a x}} \, dx}{8 b^3}\\ &=\frac{6}{b x^{2/3} \sqrt{b x^{2/3}+a x}}-\frac{7 \sqrt{b x^{2/3}+a x}}{b^2 x^{4/3}}+\frac{35 a \sqrt{b x^{2/3}+a x}}{4 b^3 x}-\frac{105 a^2 \sqrt{b x^{2/3}+a x}}{8 b^4 x^{2/3}}-\frac{\left (35 a^3\right ) \int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx}{16 b^4}\\ &=\frac{6}{b x^{2/3} \sqrt{b x^{2/3}+a x}}-\frac{7 \sqrt{b x^{2/3}+a x}}{b^2 x^{4/3}}+\frac{35 a \sqrt{b x^{2/3}+a x}}{4 b^3 x}-\frac{105 a^2 \sqrt{b x^{2/3}+a x}}{8 b^4 x^{2/3}}+\frac{\left (105 a^3\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 )}{8 b^4}\\ &=\frac{6}{b x^{2/3} \sqrt{b x^{2/3}+a x}}-\frac{7 \sqrt{b x^{2/3}+a x}}{b^2 x^{4/3}}+\frac{35 a \sqrt{b x^{2/3}+a x}}{4 b^3 x}-\frac{105 a^2 \sqrt{b x^{2/3}+a x}}{8 b^4 x^{2/3}}+\frac{105 a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{8 b^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0565704, size = 48, normalized size = 0.33 \[ -\frac{6 a^3 \sqrt [3]{x} \, _2F_1\left (-\frac{1}{2},4;\frac{1}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{b^4 \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*a^3*x^(1/3)*Hypergeometric2F1[-1/2, 4, 1/2, 1 + (a*x^(1/3))/b])/(b^4*Sqrt[b*x^(2/3) + a*x])

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Maple [A]  time = 0.013, size = 88, normalized size = 0.6 \begin{align*} -{\frac{1}{8} \left ( b+a\sqrt [3]{x} \right ) \left ( 105\,\sqrt{b}x{a}^{3}+35\,{x}^{2/3}{b}^{3/2}{a}^{2}-14\,\sqrt [3]{x}{b}^{5/2}a+8\,{b}^{7/2}-105\,{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ) \sqrt{b+a\sqrt [3]{x}}x{a}^{3} \right ) \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(b+a*x^(1/3))*(105*b^(1/2)*x*a^3+35*x^(2/3)*b^(3/2)*a^2-14*x^(1/3)*b^(5/2)*a+8*b^(7/2)-105*arctanh((b+a*x
^(1/3))^(1/2)/b^(1/2))*(b+a*x^(1/3))^(1/2)*x*a^3)/(b*x^(2/3)+a*x)^(3/2)/b^(9/2)

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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}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((a*x + b*x^(2/3))^(3/2)*x), 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/x/(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}{x \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/x/(b*x**(2/3)+a*x)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.23205, size = 142, normalized size = 0.97 \begin{align*} -\frac{105 \, a^{3} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{8 \, \sqrt{-b} b^{4}} - \frac{6 \, a^{3}}{\sqrt{a x^{\frac{1}{3}} + b} b^{4}} - \frac{57 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{3} - 136 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{3} b + 87 \, \sqrt{a x^{\frac{1}{3}} + b} a^{3} b^{2}}{8 \, a^{3} b^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-105/8*a^3*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^4) - 6*a^3/(sqrt(a*x^(1/3) + b)*b^4) - 1/8*(57*(a*
x^(1/3) + b)^(5/2)*a^3 - 136*(a*x^(1/3) + b)^(3/2)*a^3*b + 87*sqrt(a*x^(1/3) + b)*a^3*b^2)/(a^3*b^4*x)