Integrand size = 19, antiderivative size = 324 \[ \int \frac {1}{x^3 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\frac {6}{b x^{8/3} \sqrt {b x^{2/3}+a x}}-\frac {19 \sqrt {b x^{2/3}+a x}}{3 b^2 x^{10/3}}+\frac {323 a \sqrt {b x^{2/3}+a x}}{48 b^3 x^3}-\frac {1615 a^2 \sqrt {b x^{2/3}+a x}}{224 b^4 x^{8/3}}+\frac {20995 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^5 x^{7/3}}-\frac {46189 a^4 \sqrt {b x^{2/3}+a x}}{5376 b^6 x^2}+\frac {138567 a^5 \sqrt {b x^{2/3}+a x}}{14336 b^7 x^{5/3}}-\frac {46189 a^6 \sqrt {b x^{2/3}+a x}}{4096 b^8 x^{4/3}}+\frac {230945 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^9 x}-\frac {692835 a^8 \sqrt {b x^{2/3}+a x}}{32768 b^{10} x^{2/3}}+\frac {692835 a^9 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{32768 b^{21/2}} \]
692835/32768*a^9*arctanh(x^(1/3)*b^(1/2)/(b*x^(2/3)+a*x)^(1/2))/b^(21/2)+6 /b/x^(8/3)/(b*x^(2/3)+a*x)^(1/2)-19/3*(b*x^(2/3)+a*x)^(1/2)/b^2/x^(10/3)+3 23/48*a*(b*x^(2/3)+a*x)^(1/2)/b^3/x^3-1615/224*a^2*(b*x^(2/3)+a*x)^(1/2)/b ^4/x^(8/3)+20995/2688*a^3*(b*x^(2/3)+a*x)^(1/2)/b^5/x^(7/3)-46189/5376*a^4 *(b*x^(2/3)+a*x)^(1/2)/b^6/x^2+138567/14336*a^5*(b*x^(2/3)+a*x)^(1/2)/b^7/ x^(5/3)-46189/4096*a^6*(b*x^(2/3)+a*x)^(1/2)/b^8/x^(4/3)+230945/16384*a^7* (b*x^(2/3)+a*x)^(1/2)/b^9/x-692835/32768*a^8*(b*x^(2/3)+a*x)^(1/2)/b^10/x^ (2/3)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.15 \[ \int \frac {1}{x^3 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=-\frac {6 a^9 \sqrt [3]{x} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},10,\frac {1}{2},1+\frac {a \sqrt [3]{x}}{b}\right )}{b^{10} \sqrt {b x^{2/3}+a x}} \]
(-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])
Time = 0.69 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.15, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {1929, 1931, 1931, 1931, 1931, 1931, 1931, 1931, 1931, 1931, 1935, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x^3 \left (a x+b x^{2/3}\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1929 |
| \(\displaystyle \frac {19 \int \frac {1}{x^{11/3} \sqrt {x^{2/3} b+a x}}dx}{b}+\frac {6}{b x^{8/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {19 \left (-\frac {17 a \int \frac {1}{x^{10/3} \sqrt {x^{2/3} b+a x}}dx}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{b}+\frac {6}{b x^{8/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {19 \left (-\frac {17 a \left (-\frac {15 a \int \frac {1}{x^3 \sqrt {x^{2/3} b+a x}}dx}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{b}+\frac {6}{b x^{8/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {19 \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \int \frac {1}{x^{8/3} \sqrt {x^{2/3} b+a x}}dx}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{b}+\frac {6}{b x^{8/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {19 \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \int \frac {1}{x^{7/3} \sqrt {x^{2/3} b+a x}}dx}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{b}+\frac {6}{b x^{8/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {19 \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \int \frac {1}{x^2 \sqrt {x^{2/3} b+a x}}dx}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{b}+\frac {6}{b x^{8/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {19 \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \int \frac {1}{x^{5/3} \sqrt {x^{2/3} b+a x}}dx}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{b}+\frac {6}{b x^{8/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {19 \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \int \frac {1}{x^{4/3} \sqrt {x^{2/3} b+a x}}dx}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{b}+\frac {6}{b x^{8/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {19 \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \int \frac {1}{x \sqrt {x^{2/3} b+a x}}dx}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{b}+\frac {6}{b x^{8/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {19 \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (-\frac {a \int \frac {1}{x^{2/3} \sqrt {x^{2/3} b+a x}}dx}{2 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{b}+\frac {6}{b x^{8/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 1935 |
| \(\displaystyle \frac {19 \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \int \frac {1}{1-\frac {b x^{2/3}}{x^{2/3} b+a x}}d\frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}}{b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{b}+\frac {6}{b x^{8/3} \sqrt {a x+b x^{2/3}}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {19 \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{b}+\frac {6}{b x^{8/3} \sqrt {a x+b x^{2/3}}}\) |
6/(b*x^(8/3)*Sqrt[b*x^(2/3) + a*x]) + (19*(-1/3*Sqrt[b*x^(2/3) + a*x]/(b*x ^(10/3)) - (17*a*((-3*Sqrt[b*x^(2/3) + a*x])/(8*b*x^3) - (15*a*((-3*Sqrt[b *x^(2/3) + a*x])/(7*b*x^(8/3)) - (13*a*(-1/2*Sqrt[b*x^(2/3) + a*x]/(b*x^(7 /3)) - (11*a*((-3*Sqrt[b*x^(2/3) + a*x])/(5*b*x^2) - (9*a*((-3*Sqrt[b*x^(2 /3) + a*x])/(4*b*x^(5/3)) - (7*a*(-(Sqrt[b*x^(2/3) + a*x]/(b*x^(4/3))) - ( 5*a*((-3*Sqrt[b*x^(2/3) + a*x])/(2*b*x) - (3*a*((-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)))/(4*b)))/(6*b)))/(8*b)))/(10*b)))/(12*b)))/(14*b)))/(16*b)))/(18*b))) /b
3.3.1.3.1 Defintions of rubi rules used
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] && (Gt Q[a, 0] || LtQ[b, 0])
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] + Simp[c^j*((m + n*p + n - j + 1)/(a*(n - j)*(p + 1))) In t[(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]
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] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1))) I nt[(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]
Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp [-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]
Time = 1.81 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.49
| method | result | size |
| derivativedivides | \(\frac {\left (b +a \,x^{\frac {1}{3}}\right ) \left (-229376 b^{\frac {19}{2}}+272384 b^{\frac {17}{2}} a \,x^{\frac {1}{3}}-330752 b^{\frac {15}{2}} a^{2} x^{\frac {2}{3}}+413440 b^{\frac {13}{2}} a^{3} x -537472 b^{\frac {11}{2}} a^{4} x^{\frac {4}{3}}+739024 b^{\frac {9}{2}} a^{5} x^{\frac {5}{3}}-1108536 b^{\frac {7}{2}} a^{6} x^{2}+1939938 b^{\frac {5}{2}} a^{7} x^{\frac {7}{3}}-4849845 b^{\frac {3}{2}} a^{8} x^{\frac {8}{3}}-14549535 a^{9} x^{3} \sqrt {b}+14549535 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{9} x^{3}\right )}{688128 x^{2} \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} b^{\frac {21}{2}}}\) | \(159\) |
| default | \(-\frac {\left (b +a \,x^{\frac {1}{3}}\right ) \left (-272384 b^{\frac {17}{2}} a \,x^{\frac {1}{3}}+229376 b^{\frac {19}{2}}+330752 b^{\frac {15}{2}} a^{2} x^{\frac {2}{3}}-413440 b^{\frac {13}{2}} a^{3} x +537472 b^{\frac {11}{2}} a^{4} x^{\frac {4}{3}}-739024 b^{\frac {9}{2}} a^{5} x^{\frac {5}{3}}+1108536 b^{\frac {7}{2}} a^{6} x^{2}-1939938 b^{\frac {5}{2}} a^{7} x^{\frac {7}{3}}+4849845 b^{\frac {3}{2}} a^{8} x^{\frac {8}{3}}+14549535 a^{9} x^{3} \sqrt {b}-14549535 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) \sqrt {b +a \,x^{\frac {1}{3}}}\, a^{9} x^{3}\right )}{688128 x^{2} \left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} b^{\frac {21}{2}}}\) | \(159\) |
1/688128*(b+a*x^(1/3))*(-229376*b^(19/2)+272384*b^(17/2)*a*x^(1/3)-330752* b^(15/2)*a^2*x^(2/3)+413440*b^(13/2)*a^3*x-537472*b^(11/2)*a^4*x^(4/3)+739 024*b^(9/2)*a^5*x^(5/3)-1108536*b^(7/2)*a^6*x^2+1939938*b^(5/2)*a^7*x^(7/3 )-4849845*b^(3/2)*a^8*x^(8/3)-14549535*a^9*x^3*b^(1/2)+14549535*arctanh((b +a*x^(1/3))^(1/2)/b^(1/2))*(b+a*x^(1/3))^(1/2)*a^9*x^3)/x^2/(b*x^(2/3)+a*x )^(3/2)/b^(21/2)
Timed out. \[ \int \frac {1}{x^3 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{x^3 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{x^3 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
Time = 0.36 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x^3 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=-\frac {692835 \, a^{9} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{32768 \, \sqrt {-b} b^{10}} - \frac {6 \, a^{9}}{\sqrt {a x^{\frac {1}{3}} + b} b^{10}} - \frac {10420767 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{9} - 88937058 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{9} b + 334408914 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{9} b^{2} - 724860666 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{9} b^{3} + 993296384 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{9} b^{4} - 884769030 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{9} b^{5} + 503730990 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{9} b^{6} - 169799070 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{9} b^{7} + 26738145 \, \sqrt {a x^{\frac {1}{3}} + b} a^{9} b^{8}}{688128 \, a^{9} b^{10} x^{3}} \]
-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 - 884769030*(a*x^(1/3) + b)^(7/2)*a^9*b^5 + 5037 30990*(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)
Timed out. \[ \int \frac {1}{x^3 \left (b x^{2/3}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2}} \,d x \]