Integrand size = 19, antiderivative size = 354 \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^5} \, dx=-\frac {3 \sqrt {b x^{2/3}+a x}}{11 x^4}-\frac {3 a \sqrt {b x^{2/3}+a x}}{220 b x^{11/3}}+\frac {19 a^2 \sqrt {b x^{2/3}+a x}}{1320 b^2 x^{10/3}}-\frac {323 a^3 \sqrt {b x^{2/3}+a x}}{21120 b^3 x^3}+\frac {323 a^4 \sqrt {b x^{2/3}+a x}}{19712 b^4 x^{8/3}}-\frac {4199 a^5 \sqrt {b x^{2/3}+a x}}{236544 b^5 x^{7/3}}+\frac {4199 a^6 \sqrt {b x^{2/3}+a x}}{215040 b^6 x^2}-\frac {12597 a^7 \sqrt {b x^{2/3}+a x}}{573440 b^7 x^{5/3}}+\frac {4199 a^8 \sqrt {b x^{2/3}+a x}}{163840 b^8 x^{4/3}}-\frac {4199 a^9 \sqrt {b x^{2/3}+a x}}{131072 b^9 x}+\frac {12597 a^{10} \sqrt {b x^{2/3}+a x}}{262144 b^{10} x^{2/3}}-\frac {12597 a^{11} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{262144 b^{21/2}} \]
-12597/262144*a^11*arctanh(x^(1/3)*b^(1/2)/(b*x^(2/3)+a*x)^(1/2))/b^(21/2) -3/11*(b*x^(2/3)+a*x)^(1/2)/x^4-3/220*a*(b*x^(2/3)+a*x)^(1/2)/b/x^(11/3)+1 9/1320*a^2*(b*x^(2/3)+a*x)^(1/2)/b^2/x^(10/3)-323/21120*a^3*(b*x^(2/3)+a*x )^(1/2)/b^3/x^3+323/19712*a^4*(b*x^(2/3)+a*x)^(1/2)/b^4/x^(8/3)-4199/23654 4*a^5*(b*x^(2/3)+a*x)^(1/2)/b^5/x^(7/3)+4199/215040*a^6*(b*x^(2/3)+a*x)^(1 /2)/b^6/x^2-12597/573440*a^7*(b*x^(2/3)+a*x)^(1/2)/b^7/x^(5/3)+4199/163840 *a^8*(b*x^(2/3)+a*x)^(1/2)/b^8/x^(4/3)-4199/131072*a^9*(b*x^(2/3)+a*x)^(1/ 2)/b^9/x+12597/262144*a^10*(b*x^(2/3)+a*x)^(1/2)/b^10/x^(2/3)
Time = 0.51 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^5} \, dx=\frac {\sqrt {b x^{2/3}+a x} \left (-82575360 b^{10}-4128768 a b^9 \sqrt [3]{x}+4358144 a^2 b^8 x^{2/3}-4630528 a^3 b^7 x+4961280 a^4 b^6 x^{4/3}-5374720 a^5 b^5 x^{5/3}+5912192 a^6 b^4 x^2-6651216 a^7 b^3 x^{7/3}+7759752 a^8 b^2 x^{8/3}-9699690 a^9 b x^3+14549535 a^{10} x^{10/3}\right )}{302776320 b^{10} x^4}-\frac {12597 a^{11} \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{262144 b^{21/2}} \]
(Sqrt[b*x^(2/3) + a*x]*(-82575360*b^10 - 4128768*a*b^9*x^(1/3) + 4358144*a ^2*b^8*x^(2/3) - 4630528*a^3*b^7*x + 4961280*a^4*b^6*x^(4/3) - 5374720*a^5 *b^5*x^(5/3) + 5912192*a^6*b^4*x^2 - 6651216*a^7*b^3*x^(7/3) + 7759752*a^8 *b^2*x^(8/3) - 9699690*a^9*b*x^3 + 14549535*a^10*x^(10/3)))/(302776320*b^1 0*x^4) - (12597*a^11*ArcTanh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(26 2144*b^(21/2))
Time = 0.79 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {1926, 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.
| \(\displaystyle \int \frac {\sqrt {a x+b x^{2/3}}}{x^5} \, dx\) |
| \(\Big \downarrow \) 1926 |
| \(\displaystyle \frac {1}{22} a \int \frac {1}{x^4 \sqrt {x^{2/3} b+a x}}dx-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{22} a \left (-\frac {19 a \int \frac {1}{x^{11/3} \sqrt {x^{2/3} b+a x}}dx}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \int \frac {1}{x^{10/3} \sqrt {x^{2/3} b+a x}}dx}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \int \frac {1}{x^3 \sqrt {x^{2/3} b+a x}}dx}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \int \frac {1}{x^{8/3} \sqrt {x^{2/3} b+a x}}dx}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \int \frac {1}{x^{7/3} \sqrt {x^{2/3} b+a x}}dx}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \int \frac {1}{x^2 \sqrt {x^{2/3} b+a x}}dx}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \int \frac {1}{x^{5/3} \sqrt {x^{2/3} b+a x}}dx}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \int \frac {1}{x^{4/3} \sqrt {x^{2/3} b+a x}}dx}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \int \frac {1}{x \sqrt {x^{2/3} b+a x}}dx}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (-\frac {a \int \frac {1}{x^{2/3} \sqrt {x^{2/3} b+a x}}dx}{2 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\) |
| \(\Big \downarrow \) 1935 |
| \(\displaystyle \frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \int \frac {1}{1-\frac {b x^{2/3}}{x^{2/3} b+a x}}d\frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}}{b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{22} a \left (-\frac {19 a \left (-\frac {17 a \left (-\frac {15 a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )}{16 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{8 b x^3}\right )}{18 b}-\frac {\sqrt {a x+b x^{2/3}}}{3 b x^{10/3}}\right )}{20 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{10 b x^{11/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{11 x^4}\) |
(-3*Sqrt[b*x^(2/3) + a*x])/(11*x^4) + (a*((-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*S qrt[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*Arc Tanh[(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)))/22
3.2.75.3.1 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*x)^(m + 1)*((a*x^j + b*x^n)^p/(c*(m + j*p + 1))), x] - Simp[b*p *((n - j)/(c^n*(m + j*p + 1))) Int[(c*x)^(m + n)*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && !IntegerQ[p] && LtQ[0, j, n] && (Integer sQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && LtQ[m + j*p + 1, 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 = 2.10 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.59
| method | result | size |
| derivativedivides | \(-\frac {\sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (-14549535 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {21}{2}} b^{\frac {21}{2}}+155195040 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {19}{2}} b^{\frac {23}{2}}-749786037 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {17}{2}} b^{\frac {25}{2}}+2163862272 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {15}{2}} b^{\frac {27}{2}}-4139920070 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {13}{2}} b^{\frac {29}{2}}+5503713280 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {11}{2}} b^{\frac {31}{2}}-5174056250 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}} b^{\frac {33}{2}}+3424523520 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}} b^{\frac {35}{2}}-1551313995 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}} b^{\frac {37}{2}}+14549535 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{10} a^{11} x^{\frac {11}{3}}+450357600 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {39}{2}}+14549535 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {41}{2}}\right )}{302776320 x^{4} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {41}{2}}}\) | \(209\) |
| default | \(-\frac {\sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (-14549535 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {21}{2}} b^{\frac {21}{2}}+155195040 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {19}{2}} b^{\frac {23}{2}}-749786037 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {17}{2}} b^{\frac {25}{2}}+2163862272 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {15}{2}} b^{\frac {27}{2}}-4139920070 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {13}{2}} b^{\frac {29}{2}}+5503713280 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {11}{2}} b^{\frac {31}{2}}-5174056250 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}} b^{\frac {33}{2}}+3424523520 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}} b^{\frac {35}{2}}-1551313995 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}} b^{\frac {37}{2}}+14549535 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{10} a^{11} x^{\frac {11}{3}}+450357600 \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {39}{2}}+14549535 \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {41}{2}}\right )}{302776320 x^{4} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {41}{2}}}\) | \(209\) |
-1/302776320*(b*x^(2/3)+a*x)^(1/2)*(-14549535*(b+a*x^(1/3))^(21/2)*b^(21/2 )+155195040*(b+a*x^(1/3))^(19/2)*b^(23/2)-749786037*(b+a*x^(1/3))^(17/2)*b ^(25/2)+2163862272*(b+a*x^(1/3))^(15/2)*b^(27/2)-4139920070*(b+a*x^(1/3))^ (13/2)*b^(29/2)+5503713280*(b+a*x^(1/3))^(11/2)*b^(31/2)-5174056250*(b+a*x ^(1/3))^(9/2)*b^(33/2)+3424523520*(b+a*x^(1/3))^(7/2)*b^(35/2)-1551313995* (b+a*x^(1/3))^(5/2)*b^(37/2)+14549535*arctanh((b+a*x^(1/3))^(1/2)/b^(1/2)) *b^10*a^11*x^(11/3)+450357600*(b+a*x^(1/3))^(3/2)*b^(39/2)+14549535*(b+a*x ^(1/3))^(1/2)*b^(41/2))/x^4/(b+a*x^(1/3))^(1/2)/b^(41/2)
Timed out. \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^5} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^5} \, dx=\int \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x^{5}}\, dx \]
\[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^5} \, dx=\int { \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x^{5}} \,d x } \]
Time = 0.38 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^5} \, dx=\frac {\frac {14549535 \, a^{12} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{10}} + \frac {14549535 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} a^{12} - 155195040 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} a^{12} b + 749786037 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{12} b^{2} - 2163862272 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{12} b^{3} + 4139920070 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{12} b^{4} - 5503713280 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{12} b^{5} + 5174056250 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{12} b^{6} - 3424523520 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{12} b^{7} + 1551313995 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{12} b^{8} - 450357600 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{12} b^{9} - 14549535 \, \sqrt {a x^{\frac {1}{3}} + b} a^{12} b^{10}}{a^{11} b^{10} x^{\frac {11}{3}}}}{302776320 \, a} \]
1/302776320*(14549535*a^12*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)* b^10) + (14549535*(a*x^(1/3) + b)^(21/2)*a^12 - 155195040*(a*x^(1/3) + b)^ (19/2)*a^12*b + 749786037*(a*x^(1/3) + b)^(17/2)*a^12*b^2 - 2163862272*(a* x^(1/3) + b)^(15/2)*a^12*b^3 + 4139920070*(a*x^(1/3) + b)^(13/2)*a^12*b^4 - 5503713280*(a*x^(1/3) + b)^(11/2)*a^12*b^5 + 5174056250*(a*x^(1/3) + b)^ (9/2)*a^12*b^6 - 3424523520*(a*x^(1/3) + b)^(7/2)*a^12*b^7 + 1551313995*(a *x^(1/3) + b)^(5/2)*a^12*b^8 - 450357600*(a*x^(1/3) + b)^(3/2)*a^12*b^9 - 14549535*sqrt(a*x^(1/3) + b)*a^12*b^10)/(a^11*b^10*x^(11/3)))/a
Timed out. \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^5} \, dx=\int \frac {\sqrt {a\,x+b\,x^{2/3}}}{x^5} \,d x \]