Integrand size = 19, antiderivative size = 329 \[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=-\frac {3 \sqrt {b x^{2/3}+a x}}{10 b x^{11/3}}+\frac {19 a \sqrt {b x^{2/3}+a x}}{60 b^2 x^{10/3}}-\frac {323 a^2 \sqrt {b x^{2/3}+a x}}{960 b^3 x^3}+\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{896 b^4 x^{8/3}}-\frac {4199 a^4 \sqrt {b x^{2/3}+a x}}{10752 b^5 x^{7/3}}+\frac {46189 a^5 \sqrt {b x^{2/3}+a x}}{107520 b^6 x^2}-\frac {138567 a^6 \sqrt {b x^{2/3}+a x}}{286720 b^7 x^{5/3}}+\frac {46189 a^7 \sqrt {b x^{2/3}+a x}}{81920 b^8 x^{4/3}}-\frac {46189 a^8 \sqrt {b x^{2/3}+a x}}{65536 b^9 x}+\frac {138567 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^{10} x^{2/3}}-\frac {138567 a^{10} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{131072 b^{21/2}} \] Output:
-3/10*(b*x^(2/3)+a*x)^(1/2)/b/x^(11/3)+19/60*a*(b*x^(2/3)+a*x)^(1/2)/b^2/x ^(10/3)-323/960*a^2*(b*x^(2/3)+a*x)^(1/2)/b^3/x^3+323/896*a^3*(b*x^(2/3)+a *x)^(1/2)/b^4/x^(8/3)-4199/10752*a^4*(b*x^(2/3)+a*x)^(1/2)/b^5/x^(7/3)+461 89/107520*a^5*(b*x^(2/3)+a*x)^(1/2)/b^6/x^2-138567/286720*a^6*(b*x^(2/3)+a *x)^(1/2)/b^7/x^(5/3)+46189/81920*a^7*(b*x^(2/3)+a*x)^(1/2)/b^8/x^(4/3)-46 189/65536*a^8*(b*x^(2/3)+a*x)^(1/2)/b^9/x+138567/131072*a^9*(b*x^(2/3)+a*x )^(1/2)/b^10/x^(2/3)-138567/131072*a^10*arctanh(b^(1/2)*x^(1/3)/(b*x^(2/3) +a*x)^(1/2))/b^(21/2)
Time = 0.27 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=\frac {\sqrt {b x^{2/3}+a x} \left (-4128768 b^9+4358144 a b^8 \sqrt [3]{x}-4630528 a^2 b^7 x^{2/3}+4961280 a^3 b^6 x-5374720 a^4 b^5 x^{4/3}+5912192 a^5 b^4 x^{5/3}-6651216 a^6 b^3 x^2+7759752 a^7 b^2 x^{7/3}-9699690 a^8 b x^{8/3}+14549535 a^9 x^3\right )}{13762560 b^{10} x^{11/3}}-\frac {138567 a^{10} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{131072 b^{21/2}} \] Input:
Integrate[1/(x^4*Sqrt[b*x^(2/3) + a*x]),x]
Output:
(Sqrt[b*x^(2/3) + a*x]*(-4128768*b^9 + 4358144*a*b^8*x^(1/3) - 4630528*a^2 *b^7*x^(2/3) + 4961280*a^3*b^6*x - 5374720*a^4*b^5*x^(4/3) + 5912192*a^5*b ^4*x^(5/3) - 6651216*a^6*b^3*x^2 + 7759752*a^7*b^2*x^(7/3) - 9699690*a^8*b *x^(8/3) + 14549535*a^9*x^3))/(13762560*b^10*x^(11/3)) - (138567*a^10*ArcT anh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(131072*b^(21/2))
Time = 1.16 (sec) , antiderivative size = 377, 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 = {1931, 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.
| \(\displaystyle \int \frac {1}{x^4 \sqrt {a x+b x^{2/3}}} \, dx\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {19 a \int \frac {1}{x^{11/3} \sqrt {x^{2/3} b+a x}}dx}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {19 a \left (-\frac {17 a \int \frac {1}{x^{10/3} \sqrt {x^{2/3} b+a x}}dx}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \int \frac {1}{x^3 \sqrt {x^{2/3} b+a x}}dx}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \int \frac {1}{x^{8/3} \sqrt {x^{2/3} b+a x}}dx}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \int \frac {1}{x^{7/3} \sqrt {x^{2/3} b+a x}}dx}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \int \frac {1}{x^2 \sqrt {x^{2/3} b+a x}}dx}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \int \frac {1}{x^{5/3} \sqrt {x^{2/3} b+a x}}dx}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \int \frac {1}{x^{4/3} \sqrt {x^{2/3} b+a x}}dx}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \int \frac {1}{x \sqrt {x^{2/3} b+a x}}dx}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle -\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (-\frac {a \int \frac {1}{x^{2/3} \sqrt {x^{2/3} b+a x}}dx}{2 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\) |
| \(\Big \downarrow \) 1935 |
| \(\displaystyle -\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \int \frac {1}{1-\frac {b x^{2/3}}{x^{2/3} b+a x}}d\frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}}{b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\) |
Input:
Int[1/(x^4*Sqrt[b*x^(2/3) + a*x]),x]
Output:
(-3*Sqrt[b*x^(2/3) + a*x])/(10*b*x^(11/3)) - (19*a*(-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*Sq rt[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)))/(20*b)
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*(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 = 0.33 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(-\frac {\sqrt {x^{\frac {1}{3}} a +b}\, \left (4128768 \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {21}{2}}-4358144 b^{\frac {19}{2}} \sqrt {x^{\frac {1}{3}} a +b}\, a \,x^{\frac {1}{3}}+4630528 b^{\frac {17}{2}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{2} x^{\frac {2}{3}}-4961280 b^{\frac {15}{2}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{3} x +5374720 b^{\frac {13}{2}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{4} x^{\frac {4}{3}}-5912192 b^{\frac {11}{2}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{5} x^{\frac {5}{3}}+6651216 b^{\frac {9}{2}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{6} x^{2}-7759752 b^{\frac {7}{2}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{7} x^{\frac {7}{3}}+9699690 b^{\frac {5}{2}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{8} x^{\frac {8}{3}}-14549535 b^{\frac {3}{2}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{9} x^{3}+14549535 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right ) a^{10} b \,x^{\frac {10}{3}}\right )}{13762560 x^{3} \sqrt {b \,x^{\frac {2}{3}}+a x}\, b^{\frac {23}{2}}}\) | \(243\) |
| default | \(-\frac {\sqrt {x^{\frac {1}{3}} a +b}\, \left (9699690 \sqrt {x^{\frac {1}{3}} a +b}\, x^{\frac {17}{3}} b^{\frac {5}{2}} a^{8}-7759752 \sqrt {x^{\frac {1}{3}} a +b}\, x^{\frac {16}{3}} b^{\frac {7}{2}} a^{7}-5912192 \sqrt {x^{\frac {1}{3}} a +b}\, x^{\frac {14}{3}} b^{\frac {11}{2}} a^{5}+5374720 \sqrt {x^{\frac {1}{3}} a +b}\, x^{\frac {13}{3}} b^{\frac {13}{2}} a^{4}+4630528 \sqrt {x^{\frac {1}{3}} a +b}\, x^{\frac {11}{3}} b^{\frac {17}{2}} a^{2}-4358144 \sqrt {x^{\frac {1}{3}} a +b}\, x^{\frac {10}{3}} b^{\frac {19}{2}} a +14549535 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right ) x^{\frac {19}{3}} a^{10} b +4128768 \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {21}{2}} x^{3}-4961280 \sqrt {x^{\frac {1}{3}} a +b}\, x^{4} b^{\frac {15}{2}} a^{3}+6651216 \sqrt {x^{\frac {1}{3}} a +b}\, x^{5} b^{\frac {9}{2}} a^{6}-14549535 \sqrt {x^{\frac {1}{3}} a +b}\, x^{6} b^{\frac {3}{2}} a^{9}\right )}{13762560 x^{6} \sqrt {b \,x^{\frac {2}{3}}+a x}\, b^{\frac {23}{2}}}\) | \(248\) |
Input:
int(1/x^4/(b*x^(2/3)+a*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/13762560*(x^(1/3)*a+b)^(1/2)*(4128768*(x^(1/3)*a+b)^(1/2)*b^(21/2)-4358 144*b^(19/2)*(x^(1/3)*a+b)^(1/2)*a*x^(1/3)+4630528*b^(17/2)*(x^(1/3)*a+b)^ (1/2)*a^2*x^(2/3)-4961280*b^(15/2)*(x^(1/3)*a+b)^(1/2)*a^3*x+5374720*b^(13 /2)*(x^(1/3)*a+b)^(1/2)*a^4*x^(4/3)-5912192*b^(11/2)*(x^(1/3)*a+b)^(1/2)*a ^5*x^(5/3)+6651216*b^(9/2)*(x^(1/3)*a+b)^(1/2)*a^6*x^2-7759752*b^(7/2)*(x^ (1/3)*a+b)^(1/2)*a^7*x^(7/3)+9699690*b^(5/2)*(x^(1/3)*a+b)^(1/2)*a^8*x^(8/ 3)-14549535*b^(3/2)*(x^(1/3)*a+b)^(1/2)*a^9*x^3+14549535*arctanh((x^(1/3)* a+b)^(1/2)/b^(1/2))*a^10*b*x^(10/3))/x^3/(b*x^(2/3)+a*x)^(1/2)/b^(23/2)
Timed out. \[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=\text {Timed out} \] Input:
integrate(1/x^4/(b*x^(2/3)+a*x)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=\int \frac {1}{x^{4} \sqrt {a x + b x^{\frac {2}{3}}}}\, dx \] Input:
integrate(1/x**4/(b*x**(2/3)+a*x)**(1/2),x)
Output:
Integral(1/(x**4*sqrt(a*x + b*x**(2/3))), x)
\[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=\int { \frac {1}{\sqrt {a x + b x^{\frac {2}{3}}} x^{4}} \,d x } \] Input:
integrate(1/x^4/(b*x^(2/3)+a*x)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(a*x + b*x^(2/3))*x^4), x)
Time = 0.33 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=\frac {\frac {14549535 \, a^{11} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{10}} + \frac {14549535 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} a^{11} - 140645505 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{11} b + 609140532 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{11} b^{2} - 1554721740 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{11} b^{3} + 2585198330 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{11} b^{4} - 2918514950 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{11} b^{5} + 2255541300 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{11} b^{6} - 1168982220 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{11} b^{7} + 382331775 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{11} b^{8} - 68025825 \, \sqrt {a x^{\frac {1}{3}} + b} a^{11} b^{9}}{a^{10} b^{10} x^{\frac {10}{3}}}}{13762560 \, a} \] Input:
integrate(1/x^4/(b*x^(2/3)+a*x)^(1/2),x, algorithm="giac")
Output:
1/13762560*(14549535*a^11*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b ^10) + (14549535*(a*x^(1/3) + b)^(19/2)*a^11 - 140645505*(a*x^(1/3) + b)^( 17/2)*a^11*b + 609140532*(a*x^(1/3) + b)^(15/2)*a^11*b^2 - 1554721740*(a*x ^(1/3) + b)^(13/2)*a^11*b^3 + 2585198330*(a*x^(1/3) + b)^(11/2)*a^11*b^4 - 2918514950*(a*x^(1/3) + b)^(9/2)*a^11*b^5 + 2255541300*(a*x^(1/3) + b)^(7 /2)*a^11*b^6 - 1168982220*(a*x^(1/3) + b)^(5/2)*a^11*b^7 + 382331775*(a*x^ (1/3) + b)^(3/2)*a^11*b^8 - 68025825*sqrt(a*x^(1/3) + b)*a^11*b^9)/(a^10*b ^10*x^(10/3)))/a
Timed out. \[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=\int \frac {1}{x^4\,\sqrt {a\,x+b\,x^{2/3}}} \,d x \] Input:
int(1/(x^4*(a*x + b*x^(2/3))^(1/2)),x)
Output:
int(1/(x^4*(a*x + b*x^(2/3))^(1/2)), x)
Time = 0.20 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.71 \[ \int \frac {1}{x^4 \sqrt {b x^{2/3}+a x}} \, dx=\frac {-19399380 x^{\frac {8}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{8} b^{2}+11824384 x^{\frac {5}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{5} b^{5}-9261056 x^{\frac {2}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{2} b^{8}+15519504 x^{\frac {7}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{7} b^{3}-10749440 x^{\frac {4}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{4} b^{6}+8716288 x^{\frac {1}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a \,b^{9}+29099070 \sqrt {x^{\frac {1}{3}} a +b}\, a^{9} b \,x^{3}-13302432 \sqrt {x^{\frac {1}{3}} a +b}\, a^{6} b^{4} x^{2}+9922560 \sqrt {x^{\frac {1}{3}} a +b}\, a^{3} b^{7} x -8257536 \sqrt {x^{\frac {1}{3}} a +b}\, b^{10}+14549535 x^{\frac {10}{3}} \sqrt {b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}-\sqrt {b}\right ) a^{10}-14549535 x^{\frac {10}{3}} \sqrt {b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}+\sqrt {b}\right ) a^{10}}{27525120 x^{\frac {10}{3}} b^{11}} \] Input:
int(1/x^4/(b*x^(2/3)+a*x)^(1/2),x)
Output:
( - 19399380*x**(2/3)*sqrt(x**(1/3)*a + b)*a**8*b**2*x**2 + 11824384*x**(2 /3)*sqrt(x**(1/3)*a + b)*a**5*b**5*x - 9261056*x**(2/3)*sqrt(x**(1/3)*a + b)*a**2*b**8 + 15519504*x**(1/3)*sqrt(x**(1/3)*a + b)*a**7*b**3*x**2 - 107 49440*x**(1/3)*sqrt(x**(1/3)*a + b)*a**4*b**6*x + 8716288*x**(1/3)*sqrt(x* *(1/3)*a + b)*a*b**9 + 29099070*sqrt(x**(1/3)*a + b)*a**9*b*x**3 - 1330243 2*sqrt(x**(1/3)*a + b)*a**6*b**4*x**2 + 9922560*sqrt(x**(1/3)*a + b)*a**3* b**7*x - 8257536*sqrt(x**(1/3)*a + b)*b**10 + 14549535*x**(1/3)*sqrt(b)*lo g(sqrt(x**(1/3)*a + b) - sqrt(b))*a**10*x**3 - 14549535*x**(1/3)*sqrt(b)*l og(sqrt(x**(1/3)*a + b) + sqrt(b))*a**10*x**3)/(27525120*x**(1/3)*b**11*x* *3)