Integrand size = 19, antiderivative size = 266 \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^4} \, dx=-\frac {3 \sqrt {b x^{2/3}+a x}}{8 x^3}-\frac {3 a \sqrt {b x^{2/3}+a x}}{112 b x^{8/3}}+\frac {13 a^2 \sqrt {b x^{2/3}+a x}}{448 b^2 x^{7/3}}-\frac {143 a^3 \sqrt {b x^{2/3}+a x}}{4480 b^3 x^2}+\frac {1287 a^4 \sqrt {b x^{2/3}+a x}}{35840 b^4 x^{5/3}}-\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{10240 b^5 x^{4/3}}+\frac {429 a^6 \sqrt {b x^{2/3}+a x}}{8192 b^6 x}-\frac {1287 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^7 x^{2/3}}+\frac {1287 a^8 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{16384 b^{15/2}} \] Output:
-3/8*(b*x^(2/3)+a*x)^(1/2)/x^3-3/112*a*(b*x^(2/3)+a*x)^(1/2)/b/x^(8/3)+13/ 448*a^2*(b*x^(2/3)+a*x)^(1/2)/b^2/x^(7/3)-143/4480*a^3*(b*x^(2/3)+a*x)^(1/ 2)/b^3/x^2+1287/35840*a^4*(b*x^(2/3)+a*x)^(1/2)/b^4/x^(5/3)-429/10240*a^5* (b*x^(2/3)+a*x)^(1/2)/b^5/x^(4/3)+429/8192*a^6*(b*x^(2/3)+a*x)^(1/2)/b^6/x -1287/16384*a^7*(b*x^(2/3)+a*x)^(1/2)/b^7/x^(2/3)+1287/16384*a^8*arctanh(b ^(1/2)*x^(1/3)/(b*x^(2/3)+a*x)^(1/2))/b^(15/2)
Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^4} \, dx=\frac {\sqrt {b x^{2/3}+a x} \left (-215040 b^7-15360 a b^6 \sqrt [3]{x}+16640 a^2 b^5 x^{2/3}-18304 a^3 b^4 x+20592 a^4 b^3 x^{4/3}-24024 a^5 b^2 x^{5/3}+30030 a^6 b x^2-45045 a^7 x^{7/3}\right )}{573440 b^7 x^3}+\frac {1287 a^8 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{16384 b^{15/2}} \] Input:
Integrate[Sqrt[b*x^(2/3) + a*x]/x^4,x]
Output:
(Sqrt[b*x^(2/3) + a*x]*(-215040*b^7 - 15360*a*b^6*x^(1/3) + 16640*a^2*b^5* x^(2/3) - 18304*a^3*b^4*x + 20592*a^4*b^3*x^(4/3) - 24024*a^5*b^2*x^(5/3) + 30030*a^6*b*x^2 - 45045*a^7*x^(7/3)))/(573440*b^7*x^3) + (1287*a^8*ArcTa nh[(Sqrt[b]*x^(1/3))/Sqrt[b*x^(2/3) + a*x]])/(16384*b^(15/2))
Time = 0.95 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {1926, 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.
| \(\displaystyle \int \frac {\sqrt {a x+b x^{2/3}}}{x^4} \, dx\) |
| \(\Big \downarrow \) 1926 |
| \(\displaystyle \frac {1}{16} a \int \frac {1}{x^3 \sqrt {x^{2/3} b+a x}}dx-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{16} a \left (-\frac {13 a \int \frac {1}{x^{8/3} \sqrt {x^{2/3} b+a x}}dx}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \int \frac {1}{x^{7/3} \sqrt {x^{2/3} b+a x}}dx}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \int \frac {1}{x^2 \sqrt {x^{2/3} b+a x}}dx}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \int \frac {1}{x^{5/3} \sqrt {x^{2/3} b+a x}}dx}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \int \frac {1}{x^{4/3} \sqrt {x^{2/3} b+a x}}dx}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \int \frac {1}{x \sqrt {x^{2/3} b+a x}}dx}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (-\frac {a \int \frac {1}{x^{2/3} \sqrt {x^{2/3} b+a x}}dx}{2 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\) |
| \(\Big \downarrow \) 1935 |
| \(\displaystyle \frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \int \frac {1}{1-\frac {b x^{2/3}}{x^{2/3} b+a x}}d\frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}}{b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\) |
Input:
Int[Sqrt[b*x^(2/3) + a*x]/x^4,x]
Output:
(-3*Sqrt[b*x^(2/3) + a*x])/(8*x^3) + (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
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 = 0.38 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.63
| method | result | size |
| derivativedivides | \(-\frac {\sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (45045 \left (x^{\frac {1}{3}} a +b \right )^{\frac {15}{2}} b^{\frac {15}{2}}-345345 \left (x^{\frac {1}{3}} a +b \right )^{\frac {13}{2}} b^{\frac {17}{2}}+1150149 \left (x^{\frac {1}{3}} a +b \right )^{\frac {11}{2}} b^{\frac {19}{2}}-2167737 \left (x^{\frac {1}{3}} a +b \right )^{\frac {9}{2}} b^{\frac {21}{2}}+2518087 \left (x^{\frac {1}{3}} a +b \right )^{\frac {7}{2}} b^{\frac {23}{2}}-1831739 \left (x^{\frac {1}{3}} a +b \right )^{\frac {5}{2}} b^{\frac {25}{2}}+801535 \left (x^{\frac {1}{3}} a +b \right )^{\frac {3}{2}} b^{\frac {27}{2}}-45045 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right ) b^{7} a^{8} x^{\frac {8}{3}}+45045 \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {29}{2}}\right )}{573440 x^{3} \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {29}{2}}}\) | \(167\) |
| default | \(-\frac {\sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (45045 \left (x^{\frac {1}{3}} a +b \right )^{\frac {15}{2}} b^{\frac {15}{2}}-345345 \left (x^{\frac {1}{3}} a +b \right )^{\frac {13}{2}} b^{\frac {17}{2}}+1150149 \left (x^{\frac {1}{3}} a +b \right )^{\frac {11}{2}} b^{\frac {19}{2}}-2167737 \left (x^{\frac {1}{3}} a +b \right )^{\frac {9}{2}} b^{\frac {21}{2}}+2518087 \left (x^{\frac {1}{3}} a +b \right )^{\frac {7}{2}} b^{\frac {23}{2}}-1831739 \left (x^{\frac {1}{3}} a +b \right )^{\frac {5}{2}} b^{\frac {25}{2}}+801535 \left (x^{\frac {1}{3}} a +b \right )^{\frac {3}{2}} b^{\frac {27}{2}}-45045 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right ) b^{7} a^{8} x^{\frac {8}{3}}+45045 \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {29}{2}}\right )}{573440 x^{3} \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {29}{2}}}\) | \(167\) |
Input:
int((b*x^(2/3)+a*x)^(1/2)/x^4,x,method=_RETURNVERBOSE)
Output:
-1/573440*(b*x^(2/3)+a*x)^(1/2)*(45045*(x^(1/3)*a+b)^(15/2)*b^(15/2)-34534 5*(x^(1/3)*a+b)^(13/2)*b^(17/2)+1150149*(x^(1/3)*a+b)^(11/2)*b^(19/2)-2167 737*(x^(1/3)*a+b)^(9/2)*b^(21/2)+2518087*(x^(1/3)*a+b)^(7/2)*b^(23/2)-1831 739*(x^(1/3)*a+b)^(5/2)*b^(25/2)+801535*(x^(1/3)*a+b)^(3/2)*b^(27/2)-45045 *arctanh((x^(1/3)*a+b)^(1/2)/b^(1/2))*b^7*a^8*x^(8/3)+45045*(x^(1/3)*a+b)^ (1/2)*b^(29/2))/x^3/(x^(1/3)*a+b)^(1/2)/b^(29/2)
Timed out. \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^4} \, dx=\text {Timed out} \] Input:
integrate((b*x^(2/3)+a*x)^(1/2)/x^4,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^4} \, dx=\int \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x^{4}}\, dx \] Input:
integrate((b*x**(2/3)+a*x)**(1/2)/x**4,x)
Output:
Integral(sqrt(a*x + b*x**(2/3))/x**4, x)
\[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^4} \, dx=\int { \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x^{4}} \,d x } \] Input:
integrate((b*x^(2/3)+a*x)^(1/2)/x^4,x, algorithm="maxima")
Output:
integrate(sqrt(a*x + b*x^(2/3))/x^4, x)
Time = 0.33 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^4} \, dx=-\frac {\frac {45045 \, a^{9} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{7}} + \frac {45045 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{9} - 345345 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{9} b + 1150149 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{9} b^{2} - 2167737 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{9} b^{3} + 2518087 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{9} b^{4} - 1831739 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{9} b^{5} + 801535 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{9} b^{6} + 45045 \, \sqrt {a x^{\frac {1}{3}} + b} a^{9} b^{7}}{a^{8} b^{7} x^{\frac {8}{3}}}}{573440 \, a} \] Input:
integrate((b*x^(2/3)+a*x)^(1/2)/x^4,x, algorithm="giac")
Output:
-1/573440*(45045*a^9*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7) + (45045*(a*x^(1/3) + b)^(15/2)*a^9 - 345345*(a*x^(1/3) + b)^(13/2)*a^9*b + 1150149*(a*x^(1/3) + b)^(11/2)*a^9*b^2 - 2167737*(a*x^(1/3) + b)^(9/2)*a^ 9*b^3 + 2518087*(a*x^(1/3) + b)^(7/2)*a^9*b^4 - 1831739*(a*x^(1/3) + b)^(5 /2)*a^9*b^5 + 801535*(a*x^(1/3) + b)^(3/2)*a^9*b^6 + 45045*sqrt(a*x^(1/3) + b)*a^9*b^7)/(a^8*b^7*x^(8/3)))/a
Timed out. \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^4} \, dx=\int \frac {\sqrt {a\,x+b\,x^{2/3}}}{x^4} \,d x \] Input:
int((a*x + b*x^(2/3))^(1/2)/x^4,x)
Output:
int((a*x + b*x^(2/3))^(1/2)/x^4, x)
Time = 0.21 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^4} \, dx=\frac {-48048 x^{\frac {5}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{5} b^{3}+33280 x^{\frac {2}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{2} b^{6}-90090 x^{\frac {7}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{7} b +41184 x^{\frac {4}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{4} b^{4}-30720 x^{\frac {1}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a \,b^{7}+60060 \sqrt {x^{\frac {1}{3}} a +b}\, a^{6} b^{2} x^{2}-36608 \sqrt {x^{\frac {1}{3}} a +b}\, a^{3} b^{5} x -430080 \sqrt {x^{\frac {1}{3}} a +b}\, b^{8}-45045 x^{\frac {8}{3}} \sqrt {b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}-\sqrt {b}\right ) a^{8}+45045 x^{\frac {8}{3}} \sqrt {b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}+\sqrt {b}\right ) a^{8}}{1146880 x^{\frac {8}{3}} b^{8}} \] Input:
int((b*x^(2/3)+a*x)^(1/2)/x^4,x)
Output:
( - 48048*x**(2/3)*sqrt(x**(1/3)*a + b)*a**5*b**3*x + 33280*x**(2/3)*sqrt( x**(1/3)*a + b)*a**2*b**6 - 90090*x**(1/3)*sqrt(x**(1/3)*a + b)*a**7*b*x** 2 + 41184*x**(1/3)*sqrt(x**(1/3)*a + b)*a**4*b**4*x - 30720*x**(1/3)*sqrt( x**(1/3)*a + b)*a*b**7 + 60060*sqrt(x**(1/3)*a + b)*a**6*b**2*x**2 - 36608 *sqrt(x**(1/3)*a + b)*a**3*b**5*x - 430080*sqrt(x**(1/3)*a + b)*b**8 - 450 45*x**(2/3)*sqrt(b)*log(sqrt(x**(1/3)*a + b) - sqrt(b))*a**8*x**2 + 45045* x**(2/3)*sqrt(b)*log(sqrt(x**(1/3)*a + b) + sqrt(b))*a**8*x**2)/(1146880*x **(2/3)*b**8*x**2)