Integrand size = 19, antiderivative size = 291 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=-\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {143 a^6 \sqrt {b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac {143 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^6 x}-\frac {429 a^8 \sqrt {b x^{2/3}+a x}}{32768 b^7 x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {429 a^9 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{32768 b^{15/2}} \] Output:
-1/16*a*(b*x^(2/3)+a*x)^(1/2)/x^3-1/224*a^2*(b*x^(2/3)+a*x)^(1/2)/b/x^(8/3 )+13/2688*a^3*(b*x^(2/3)+a*x)^(1/2)/b^2/x^(7/3)-143/26880*a^4*(b*x^(2/3)+a *x)^(1/2)/b^3/x^2+429/71680*a^5*(b*x^(2/3)+a*x)^(1/2)/b^4/x^(5/3)-143/2048 0*a^6*(b*x^(2/3)+a*x)^(1/2)/b^5/x^(4/3)+143/16384*a^7*(b*x^(2/3)+a*x)^(1/2 )/b^6/x-429/32768*a^8*(b*x^(2/3)+a*x)^(1/2)/b^7/x^(2/3)-1/3*(b*x^(2/3)+a*x )^(3/2)/x^4+429/32768*a^9*arctanh(b^(1/2)*x^(1/3)/(b*x^(2/3)+a*x)^(1/2))/b ^(15/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.21 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\frac {6 a^9 \left (b+a \sqrt [3]{x}\right )^2 \sqrt {b x^{2/3}+a x} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},10,\frac {7}{2},1+\frac {a \sqrt [3]{x}}{b}\right )}{5 b^{10} \sqrt [3]{x}} \] Input:
Integrate[(b*x^(2/3) + a*x)^(3/2)/x^5,x]
Output:
(6*a^9*(b + a*x^(1/3))^2*Sqrt[b*x^(2/3) + a*x]*Hypergeometric2F1[5/2, 10, 7/2, 1 + (a*x^(1/3))/b])/(5*b^10*x^(1/3))
Time = 1.03 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.12, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {1926, 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.
| \(\displaystyle \int \frac {\left (a x+b x^{2/3}\right )^{3/2}}{x^5} \, dx\) |
| \(\Big \downarrow \) 1926 |
| \(\displaystyle \frac {1}{6} a \int \frac {\sqrt {x^{2/3} b+a x}}{x^4}dx-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}\) |
| \(\Big \downarrow \) 1926 |
| \(\displaystyle \frac {1}{6} a \left (\frac {1}{16} a \int \frac {1}{x^3 \sqrt {x^{2/3} b+a x}}dx-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{6} a \left (\frac {1}{16} a \left (-\frac {13 a \int \frac {1}{x^{8/3} \sqrt {x^{2/3} b+a x}}dx}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{6} a \left (\frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \int \frac {1}{x^{7/3} \sqrt {x^{2/3} b+a x}}dx}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{6} a \left (\frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \int \frac {1}{x^2 \sqrt {x^{2/3} b+a x}}dx}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{6} a \left (\frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \int \frac {1}{x^{5/3} \sqrt {x^{2/3} b+a x}}dx}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{6} a \left (\frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \int \frac {1}{x^{4/3} \sqrt {x^{2/3} b+a x}}dx}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{6} a \left (\frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \int \frac {1}{x \sqrt {x^{2/3} b+a x}}dx}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}\) |
| \(\Big \downarrow \) 1931 |
| \(\displaystyle \frac {1}{6} a \left (\frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (-\frac {a \int \frac {1}{x^{2/3} \sqrt {x^{2/3} b+a x}}dx}{2 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}\) |
| \(\Big \downarrow \) 1935 |
| \(\displaystyle \frac {1}{6} a \left (\frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \int \frac {1}{1-\frac {b x^{2/3}}{x^{2/3} b+a x}}d\frac {\sqrt [3]{x}}{\sqrt {x^{2/3} b+a x}}}{b}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{6} a \left (\frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}\) |
Input:
Int[(b*x^(2/3) + a*x)^(3/2)/x^5,x]
Output:
-1/3*(b*x^(2/3) + a*x)^(3/2)/x^4 + (a*((-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*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)))/1 6))/6
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 = 0.36 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.62
| method | result | size |
| derivativedivides | \(-\frac {\left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (45045 \left (x^{\frac {1}{3}} a +b \right )^{\frac {17}{2}} b^{\frac {15}{2}}-390390 \left (x^{\frac {1}{3}} a +b \right )^{\frac {15}{2}} b^{\frac {17}{2}}+1495494 \left (x^{\frac {1}{3}} a +b \right )^{\frac {13}{2}} b^{\frac {19}{2}}-3317886 \left (x^{\frac {1}{3}} a +b \right )^{\frac {11}{2}} b^{\frac {21}{2}}+4685824 \left (x^{\frac {1}{3}} a +b \right )^{\frac {9}{2}} b^{\frac {23}{2}}-4349826 \left (x^{\frac {1}{3}} a +b \right )^{\frac {7}{2}} b^{\frac {25}{2}}+2633274 \left (x^{\frac {1}{3}} a +b \right )^{\frac {5}{2}} b^{\frac {27}{2}}-45045 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right ) b^{7} a^{9} x^{3}+390390 \left (x^{\frac {1}{3}} a +b \right )^{\frac {3}{2}} b^{\frac {29}{2}}-45045 \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {31}{2}}\right )}{3440640 x^{4} \left (x^{\frac {1}{3}} a +b \right )^{\frac {3}{2}} b^{\frac {29}{2}}}\) | \(181\) |
| default | \(-\frac {\left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (45045 \left (x^{\frac {1}{3}} a +b \right )^{\frac {17}{2}} b^{\frac {15}{2}}-390390 \left (x^{\frac {1}{3}} a +b \right )^{\frac {15}{2}} b^{\frac {17}{2}}+1495494 \left (x^{\frac {1}{3}} a +b \right )^{\frac {13}{2}} b^{\frac {19}{2}}-3317886 \left (x^{\frac {1}{3}} a +b \right )^{\frac {11}{2}} b^{\frac {21}{2}}+4685824 \left (x^{\frac {1}{3}} a +b \right )^{\frac {9}{2}} b^{\frac {23}{2}}-4349826 \left (x^{\frac {1}{3}} a +b \right )^{\frac {7}{2}} b^{\frac {25}{2}}+2633274 \left (x^{\frac {1}{3}} a +b \right )^{\frac {5}{2}} b^{\frac {27}{2}}-45045 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{\frac {1}{3}} a +b}}{\sqrt {b}}\right ) b^{7} a^{9} x^{3}+390390 \left (x^{\frac {1}{3}} a +b \right )^{\frac {3}{2}} b^{\frac {29}{2}}-45045 \sqrt {x^{\frac {1}{3}} a +b}\, b^{\frac {31}{2}}\right )}{3440640 x^{4} \left (x^{\frac {1}{3}} a +b \right )^{\frac {3}{2}} b^{\frac {29}{2}}}\) | \(181\) |
Input:
int((b*x^(2/3)+a*x)^(3/2)/x^5,x,method=_RETURNVERBOSE)
Output:
-1/3440640*(b*x^(2/3)+a*x)^(3/2)*(45045*(x^(1/3)*a+b)^(17/2)*b^(15/2)-3903 90*(x^(1/3)*a+b)^(15/2)*b^(17/2)+1495494*(x^(1/3)*a+b)^(13/2)*b^(19/2)-331 7886*(x^(1/3)*a+b)^(11/2)*b^(21/2)+4685824*(x^(1/3)*a+b)^(9/2)*b^(23/2)-43 49826*(x^(1/3)*a+b)^(7/2)*b^(25/2)+2633274*(x^(1/3)*a+b)^(5/2)*b^(27/2)-45 045*arctanh((x^(1/3)*a+b)^(1/2)/b^(1/2))*b^7*a^9*x^3+390390*(x^(1/3)*a+b)^ (3/2)*b^(29/2)-45045*(x^(1/3)*a+b)^(1/2)*b^(31/2))/x^4/(x^(1/3)*a+b)^(3/2) /b^(29/2)
Timed out. \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\text {Timed out} \] Input:
integrate((b*x^(2/3)+a*x)^(3/2)/x^5,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\int \frac {\left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}{x^{5}}\, dx \] Input:
integrate((b*x**(2/3)+a*x)**(3/2)/x**5,x)
Output:
Integral((a*x + b*x**(2/3))**(3/2)/x**5, x)
\[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\int { \frac {{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}}}{x^{5}} \,d x } \] Input:
integrate((b*x^(2/3)+a*x)^(3/2)/x^5,x, algorithm="maxima")
Output:
integrate((a*x + b*x^(2/3))^(3/2)/x^5, x)
Time = 0.34 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.56 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=-\frac {1}{3440640} \, a^{9} {\left (\frac {45045 \, \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{7}} + \frac {45045 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} - 390390 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b + 1495494 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{2} - 3317886 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{3} + 4685824 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{4} - 4349826 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{5} + 2633274 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{6} + 390390 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{7} - 45045 \, \sqrt {a x^{\frac {1}{3}} + b} b^{8}}{a^{9} b^{7} x^{3}}\right )} \] Input:
integrate((b*x^(2/3)+a*x)^(3/2)/x^5,x, algorithm="giac")
Output:
-1/3440640*a^9*(45045*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7) + (45045*(a*x^(1/3) + b)^(17/2) - 390390*(a*x^(1/3) + b)^(15/2)*b + 149549 4*(a*x^(1/3) + b)^(13/2)*b^2 - 3317886*(a*x^(1/3) + b)^(11/2)*b^3 + 468582 4*(a*x^(1/3) + b)^(9/2)*b^4 - 4349826*(a*x^(1/3) + b)^(7/2)*b^5 + 2633274* (a*x^(1/3) + b)^(5/2)*b^6 + 390390*(a*x^(1/3) + b)^(3/2)*b^7 - 45045*sqrt( a*x^(1/3) + b)*b^8)/(a^9*b^7*x^3))
Timed out. \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\int \frac {{\left (a\,x+b\,x^{2/3}\right )}^{3/2}}{x^5} \,d x \] Input:
int((a*x + b*x^(2/3))^(3/2)/x^5,x)
Output:
int((a*x + b*x^(2/3))^(3/2)/x^5, x)
Time = 0.22 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.75 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\frac {-48048 x^{\frac {8}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{6} b^{3}+33280 x^{\frac {5}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{3} b^{6}-2293760 x^{\frac {2}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, b^{9}-90090 x^{\frac {10}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{8} b +41184 x^{\frac {7}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{5} b^{4}-30720 x^{\frac {4}{3}} \sqrt {x^{\frac {1}{3}} a +b}\, a^{2} b^{7}+60060 \sqrt {x^{\frac {1}{3}} a +b}\, a^{7} b^{2} x^{3}-36608 \sqrt {x^{\frac {1}{3}} a +b}\, a^{4} b^{5} x^{2}-2723840 \sqrt {x^{\frac {1}{3}} a +b}\, a \,b^{8} x -45045 x^{\frac {11}{3}} \sqrt {b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}-\sqrt {b}\right ) a^{9}+45045 x^{\frac {11}{3}} \sqrt {b}\, \mathrm {log}\left (\sqrt {x^{\frac {1}{3}} a +b}+\sqrt {b}\right ) a^{9}}{6881280 x^{\frac {11}{3}} b^{8}} \] Input:
int((b*x^(2/3)+a*x)^(3/2)/x^5,x)
Output:
( - 48048*x**(2/3)*sqrt(x**(1/3)*a + b)*a**6*b**3*x**2 + 33280*x**(2/3)*sq rt(x**(1/3)*a + b)*a**3*b**6*x - 2293760*x**(2/3)*sqrt(x**(1/3)*a + b)*b** 9 - 90090*x**(1/3)*sqrt(x**(1/3)*a + b)*a**8*b*x**3 + 41184*x**(1/3)*sqrt( x**(1/3)*a + b)*a**5*b**4*x**2 - 30720*x**(1/3)*sqrt(x**(1/3)*a + b)*a**2* b**7*x + 60060*sqrt(x**(1/3)*a + b)*a**7*b**2*x**3 - 36608*sqrt(x**(1/3)*a + b)*a**4*b**5*x**2 - 2723840*sqrt(x**(1/3)*a + b)*a*b**8*x - 45045*x**(2 /3)*sqrt(b)*log(sqrt(x**(1/3)*a + b) - sqrt(b))*a**9*x**3 + 45045*x**(2/3) *sqrt(b)*log(sqrt(x**(1/3)*a + b) + sqrt(b))*a**9*x**3)/(6881280*x**(2/3)* b**8*x**3)