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

Optimal. Leaf size=324 \[ -\frac{692835 a^8 \sqrt{a x+b x^{2/3}}}{32768 b^{10} x^{2/3}}+\frac{230945 a^7 \sqrt{a x+b x^{2/3}}}{16384 b^9 x}-\frac{46189 a^6 \sqrt{a x+b x^{2/3}}}{4096 b^8 x^{4/3}}+\frac{138567 a^5 \sqrt{a x+b x^{2/3}}}{14336 b^7 x^{5/3}}-\frac{46189 a^4 \sqrt{a x+b x^{2/3}}}{5376 b^6 x^2}+\frac{20995 a^3 \sqrt{a x+b x^{2/3}}}{2688 b^5 x^{7/3}}-\frac{1615 a^2 \sqrt{a x+b x^{2/3}}}{224 b^4 x^{8/3}}+\frac{692835 a^9 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{32768 b^{21/2}}+\frac{323 a \sqrt{a x+b x^{2/3}}}{48 b^3 x^3}-\frac{19 \sqrt{a x+b x^{2/3}}}{3 b^2 x^{10/3}}+\frac{6}{b x^{8/3} \sqrt{a x+b x^{2/3}}} \]

[Out]

6/(b*x^(8/3)*Sqrt[b*x^(2/3) + a*x]) - (19*Sqrt[b*x^(2/3) + a*x])/(3*b^2*x^(10/3)) + (323*a*Sqrt[b*x^(2/3) + a*
x])/(48*b^3*x^3) - (1615*a^2*Sqrt[b*x^(2/3) + a*x])/(224*b^4*x^(8/3)) + (20995*a^3*Sqrt[b*x^(2/3) + a*x])/(268
8*b^5*x^(7/3)) - (46189*a^4*Sqrt[b*x^(2/3) + a*x])/(5376*b^6*x^2) + (138567*a^5*Sqrt[b*x^(2/3) + a*x])/(14336*
b^7*x^(5/3)) - (46189*a^6*Sqrt[b*x^(2/3) + a*x])/(4096*b^8*x^(4/3)) + (230945*a^7*Sqrt[b*x^(2/3) + a*x])/(1638
4*b^9*x) - (692835*a^8*Sqrt[b*x^(2/3) + a*x])/(32768*b^10*x^(2/3)) + (692835*a^9*ArcTanh[(Sqrt[b]*x^(1/3))/Sqr
t[b*x^(2/3) + a*x]])/(32768*b^(21/2))

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Rubi [A]  time = 0.597076, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2023, 2025, 2029, 206} \[ -\frac{692835 a^8 \sqrt{a x+b x^{2/3}}}{32768 b^{10} x^{2/3}}+\frac{230945 a^7 \sqrt{a x+b x^{2/3}}}{16384 b^9 x}-\frac{46189 a^6 \sqrt{a x+b x^{2/3}}}{4096 b^8 x^{4/3}}+\frac{138567 a^5 \sqrt{a x+b x^{2/3}}}{14336 b^7 x^{5/3}}-\frac{46189 a^4 \sqrt{a x+b x^{2/3}}}{5376 b^6 x^2}+\frac{20995 a^3 \sqrt{a x+b x^{2/3}}}{2688 b^5 x^{7/3}}-\frac{1615 a^2 \sqrt{a x+b x^{2/3}}}{224 b^4 x^{8/3}}+\frac{692835 a^9 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{a x+b x^{2/3}}}\right )}{32768 b^{21/2}}+\frac{323 a \sqrt{a x+b x^{2/3}}}{48 b^3 x^3}-\frac{19 \sqrt{a x+b x^{2/3}}}{3 b^2 x^{10/3}}+\frac{6}{b x^{8/3} \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

6/(b*x^(8/3)*Sqrt[b*x^(2/3) + a*x]) - (19*Sqrt[b*x^(2/3) + a*x])/(3*b^2*x^(10/3)) + (323*a*Sqrt[b*x^(2/3) + a*
x])/(48*b^3*x^3) - (1615*a^2*Sqrt[b*x^(2/3) + a*x])/(224*b^4*x^(8/3)) + (20995*a^3*Sqrt[b*x^(2/3) + a*x])/(268
8*b^5*x^(7/3)) - (46189*a^4*Sqrt[b*x^(2/3) + a*x])/(5376*b^6*x^2) + (138567*a^5*Sqrt[b*x^(2/3) + a*x])/(14336*
b^7*x^(5/3)) - (46189*a^6*Sqrt[b*x^(2/3) + a*x])/(4096*b^8*x^(4/3)) + (230945*a^7*Sqrt[b*x^(2/3) + a*x])/(1638
4*b^9*x) - (692835*a^8*Sqrt[b*x^(2/3) + a*x])/(32768*b^10*x^(2/3)) + (692835*a^9*ArcTanh[(Sqrt[b]*x^(1/3))/Sqr
t[b*x^(2/3) + a*x]])/(32768*b^(21/2))

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^3 \left (b x^{2/3}+a x\right )^{3/2}} \, dx &=\frac{6}{b x^{8/3} \sqrt{b x^{2/3}+a x}}+\frac{19 \int \frac{1}{x^{11/3} \sqrt{b x^{2/3}+a x}} \, dx}{b}\\ &=\frac{6}{b x^{8/3} \sqrt{b x^{2/3}+a x}}-\frac{19 \sqrt{b x^{2/3}+a x}}{3 b^2 x^{10/3}}-\frac{(323 a) \int \frac{1}{x^{10/3} \sqrt{b x^{2/3}+a x}} \, dx}{18 b^2}\\ &=\frac{6}{b x^{8/3} \sqrt{b x^{2/3}+a x}}-\frac{19 \sqrt{b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac{323 a \sqrt{b x^{2/3}+a x}}{48 b^3 x^3}+\frac{\left (1615 a^2\right ) \int \frac{1}{x^3 \sqrt{b x^{2/3}+a x}} \, dx}{96 b^3}\\ &=\frac{6}{b x^{8/3} \sqrt{b x^{2/3}+a x}}-\frac{19 \sqrt{b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac{323 a \sqrt{b x^{2/3}+a x}}{48 b^3 x^3}-\frac{1615 a^2 \sqrt{b x^{2/3}+a x}}{224 b^4 x^{8/3}}-\frac{\left (20995 a^3\right ) \int \frac{1}{x^{8/3} \sqrt{b x^{2/3}+a x}} \, dx}{1344 b^4}\\ &=\frac{6}{b x^{8/3} \sqrt{b x^{2/3}+a x}}-\frac{19 \sqrt{b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac{323 a \sqrt{b x^{2/3}+a x}}{48 b^3 x^3}-\frac{1615 a^2 \sqrt{b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac{20995 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^5 x^{7/3}}+\frac{\left (230945 a^4\right ) \int \frac{1}{x^{7/3} \sqrt{b x^{2/3}+a x}} \, dx}{16128 b^5}\\ &=\frac{6}{b x^{8/3} \sqrt{b x^{2/3}+a x}}-\frac{19 \sqrt{b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac{323 a \sqrt{b x^{2/3}+a x}}{48 b^3 x^3}-\frac{1615 a^2 \sqrt{b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac{20995 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac{46189 a^4 \sqrt{b x^{2/3}+a x}}{5376 b^6 x^2}-\frac{\left (46189 a^5\right ) \int \frac{1}{x^2 \sqrt{b x^{2/3}+a x}} \, dx}{3584 b^6}\\ &=\frac{6}{b x^{8/3} \sqrt{b x^{2/3}+a x}}-\frac{19 \sqrt{b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac{323 a \sqrt{b x^{2/3}+a x}}{48 b^3 x^3}-\frac{1615 a^2 \sqrt{b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac{20995 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac{46189 a^4 \sqrt{b x^{2/3}+a x}}{5376 b^6 x^2}+\frac{138567 a^5 \sqrt{b x^{2/3}+a x}}{14336 b^7 x^{5/3}}+\frac{\left (46189 a^6\right ) \int \frac{1}{x^{5/3} \sqrt{b x^{2/3}+a x}} \, dx}{4096 b^7}\\ &=\frac{6}{b x^{8/3} \sqrt{b x^{2/3}+a x}}-\frac{19 \sqrt{b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac{323 a \sqrt{b x^{2/3}+a x}}{48 b^3 x^3}-\frac{1615 a^2 \sqrt{b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac{20995 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac{46189 a^4 \sqrt{b x^{2/3}+a x}}{5376 b^6 x^2}+\frac{138567 a^5 \sqrt{b x^{2/3}+a x}}{14336 b^7 x^{5/3}}-\frac{46189 a^6 \sqrt{b x^{2/3}+a x}}{4096 b^8 x^{4/3}}-\frac{\left (230945 a^7\right ) \int \frac{1}{x^{4/3} \sqrt{b x^{2/3}+a x}} \, dx}{24576 b^8}\\ &=\frac{6}{b x^{8/3} \sqrt{b x^{2/3}+a x}}-\frac{19 \sqrt{b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac{323 a \sqrt{b x^{2/3}+a x}}{48 b^3 x^3}-\frac{1615 a^2 \sqrt{b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac{20995 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac{46189 a^4 \sqrt{b x^{2/3}+a x}}{5376 b^6 x^2}+\frac{138567 a^5 \sqrt{b x^{2/3}+a x}}{14336 b^7 x^{5/3}}-\frac{46189 a^6 \sqrt{b x^{2/3}+a x}}{4096 b^8 x^{4/3}}+\frac{230945 a^7 \sqrt{b x^{2/3}+a x}}{16384 b^9 x}+\frac{\left (230945 a^8\right ) \int \frac{1}{x \sqrt{b x^{2/3}+a x}} \, dx}{32768 b^9}\\ &=\frac{6}{b x^{8/3} \sqrt{b x^{2/3}+a x}}-\frac{19 \sqrt{b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac{323 a \sqrt{b x^{2/3}+a x}}{48 b^3 x^3}-\frac{1615 a^2 \sqrt{b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac{20995 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac{46189 a^4 \sqrt{b x^{2/3}+a x}}{5376 b^6 x^2}+\frac{138567 a^5 \sqrt{b x^{2/3}+a x}}{14336 b^7 x^{5/3}}-\frac{46189 a^6 \sqrt{b x^{2/3}+a x}}{4096 b^8 x^{4/3}}+\frac{230945 a^7 \sqrt{b x^{2/3}+a x}}{16384 b^9 x}-\frac{692835 a^8 \sqrt{b x^{2/3}+a x}}{32768 b^{10} x^{2/3}}-\frac{\left (230945 a^9\right ) \int \frac{1}{x^{2/3} \sqrt{b x^{2/3}+a x}} \, dx}{65536 b^{10}}\\ &=\frac{6}{b x^{8/3} \sqrt{b x^{2/3}+a x}}-\frac{19 \sqrt{b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac{323 a \sqrt{b x^{2/3}+a x}}{48 b^3 x^3}-\frac{1615 a^2 \sqrt{b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac{20995 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac{46189 a^4 \sqrt{b x^{2/3}+a x}}{5376 b^6 x^2}+\frac{138567 a^5 \sqrt{b x^{2/3}+a x}}{14336 b^7 x^{5/3}}-\frac{46189 a^6 \sqrt{b x^{2/3}+a x}}{4096 b^8 x^{4/3}}+\frac{230945 a^7 \sqrt{b x^{2/3}+a x}}{16384 b^9 x}-\frac{692835 a^8 \sqrt{b x^{2/3}+a x}}{32768 b^{10} x^{2/3}}+\frac{\left (692835 a^9\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 )}{32768 b^{10}}\\ &=\frac{6}{b x^{8/3} \sqrt{b x^{2/3}+a x}}-\frac{19 \sqrt{b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac{323 a \sqrt{b x^{2/3}+a x}}{48 b^3 x^3}-\frac{1615 a^2 \sqrt{b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac{20995 a^3 \sqrt{b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac{46189 a^4 \sqrt{b x^{2/3}+a x}}{5376 b^6 x^2}+\frac{138567 a^5 \sqrt{b x^{2/3}+a x}}{14336 b^7 x^{5/3}}-\frac{46189 a^6 \sqrt{b x^{2/3}+a x}}{4096 b^8 x^{4/3}}+\frac{230945 a^7 \sqrt{b x^{2/3}+a x}}{16384 b^9 x}-\frac{692835 a^8 \sqrt{b x^{2/3}+a x}}{32768 b^{10} x^{2/3}}+\frac{692835 a^9 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt [3]{x}}{\sqrt{b x^{2/3}+a x}}\right )}{32768 b^{21/2}}\\ \end{align*}

Mathematica [C]  time = 0.0566937, size = 48, normalized size = 0.15 \[ -\frac{6 a^9 \sqrt [3]{x} \, _2F_1\left (-\frac{1}{2},10;\frac{1}{2};\frac{\sqrt [3]{x} a}{b}+1\right )}{b^{10} \sqrt{a x+b x^{2/3}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*a^9*x^(1/3)*Hypergeometric2F1[-1/2, 10, 1/2, 1 + (a*x^(1/3))/b])/(b^10*Sqrt[b*x^(2/3) + a*x])

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Maple [A]  time = 0.018, size = 159, normalized size = 0.5 \begin{align*}{\frac{1}{688128\,{x}^{2}} \left ( b+a\sqrt [3]{x} \right ) \left ( -537472\,{b}^{11/2}{x}^{4/3}{a}^{4}+739024\,{b}^{9/2}{x}^{5/3}{a}^{5}-1108536\,{b}^{7/2}{x}^{2}{a}^{6}+1939938\,{b}^{5/2}{x}^{7/3}{a}^{7}-4849845\,{b}^{3/2}{x}^{8/3}{a}^{8}-14549535\,{x}^{3}{a}^{9}\sqrt{b}+272384\,{b}^{17/2}\sqrt [3]{x}a-330752\,{b}^{15/2}{x}^{2/3}{a}^{2}+413440\,{b}^{13/2}x{a}^{3}-229376\,{b}^{19/2}[phone]\,\sqrt{b+a\sqrt [3]{x}}{\it Artanh} \left ({\frac{\sqrt{b+a\sqrt [3]{x}}}{\sqrt{b}}} \right ){x}^{3}{a}^{9} \right ) \left ( b{x}^{{\frac{2}{3}}}+ax \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{21}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/688128*(b+a*x^(1/3))*(-537472*b^(11/2)*x^(4/3)*a^4+739024*b^(9/2)*x^(5/3)*a^5-1108536*b^(7/2)*x^2*a^6+193993
8*b^(5/2)*x^(7/3)*a^7-4849845*b^(3/2)*x^(8/3)*a^8-14549535*x^3*a^9*b^(1/2)+272384*b^(17/2)*x^(1/3)*a-330752*b^
(15/2)*x^(2/3)*a^2+413440*b^(13/2)*x*a^3-229376*b^(19/2)[phone]*(b+a*x^(1/3))^(1/2)*arctanh((b+a*x^(1/3))^(1
/2)/b^(1/2))*x^3*a^9)/x^2/(b*x^(2/3)+a*x)^(3/2)/b^(21/2)

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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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Giac [A]  time = 1.36586, size = 279, normalized size = 0.86 \begin{align*} -\frac{692835 \, a^{9} \arctan \left (\frac{\sqrt{a x^{\frac{1}{3}} + b}}{\sqrt{-b}}\right )}{32768 \, \sqrt{-b} b^{10}} - \frac{6 \, a^{9}}{\sqrt{a x^{\frac{1}{3}} + b} b^{10}} - \frac{10420767 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{17}{2}} a^{9} - 88937058 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{15}{2}} a^{9} b + 334408914 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{13}{2}} a^{9} b^{2} - 724860666 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{11}{2}} a^{9} b^{3} + 993296384 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{9}{2}} a^{9} b^{4} - 884769030 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{7}{2}} a^{9} b^{5} + 503730990 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{5}{2}} a^{9} b^{6} - 169799070 \,{\left (a x^{\frac{1}{3}} + b\right )}^{\frac{3}{2}} a^{9} b^{7} + 26738145 \, \sqrt{a x^{\frac{1}{3}} + b} a^{9} b^{8}}{688128 \, a^{9} b^{10} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-692835/32768*a^9*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^10) - 6*a^9/(sqrt(a*x^(1/3) + b)*b^10) - 1/
688128*(10420767*(a*x^(1/3) + b)^(17/2)*a^9 - 88937058*(a*x^(1/3) + b)^(15/2)*a^9*b + 334408914*(a*x^(1/3) + b
)^(13/2)*a^9*b^2 - 724860666*(a*x^(1/3) + b)^(11/2)*a^9*b^3 + 993296384*(a*x^(1/3) + b)^(9/2)*a^9*b^4 - 884769
030*(a*x^(1/3) + b)^(7/2)*a^9*b^5 + 503730990*(a*x^(1/3) + b)^(5/2)*a^9*b^6 - 169799070*(a*x^(1/3) + b)^(3/2)*
a^9*b^7 + 26738145*sqrt(a*x^(1/3) + b)*a^9*b^8)/(a^9*b^10*x^3)