Integrand size = 19, antiderivative size = 291 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=-\frac {a \sqrt {b x^{2/3}+a x}}{16 x^3}-\frac {a^2 \sqrt {b x^{2/3}+a x}}{224 b x^{8/3}}+\frac {13 a^3 \sqrt {b x^{2/3}+a x}}{2688 b^2 x^{7/3}}-\frac {143 a^4 \sqrt {b x^{2/3}+a x}}{26880 b^3 x^2}+\frac {429 a^5 \sqrt {b x^{2/3}+a x}}{71680 b^4 x^{5/3}}-\frac {143 a^6 \sqrt {b x^{2/3}+a x}}{20480 b^5 x^{4/3}}+\frac {143 a^7 \sqrt {b x^{2/3}+a x}}{16384 b^6 x}-\frac {429 a^8 \sqrt {b x^{2/3}+a x}}{32768 b^7 x^{2/3}}-\frac {\left (b x^{2/3}+a x\right )^{3/2}}{3 x^4}+\frac {429 a^9 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{32768 b^{15/2}} \]
-1/3*(b*x^(2/3)+a*x)^(3/2)/x^4+429/32768*a^9*arctanh(x^(1/3)*b^(1/2)/(b*x^ (2/3)+a*x)^(1/2))/b^(15/2)-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)-143/20480*a^6*(b*x^(2/3)+a*x)^(1/2)/b^5/x^(4/3)+143/1638 4*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)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.21 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\frac {6 a^9 \left (b+a \sqrt [3]{x}\right )^2 \sqrt {b x^{2/3}+a x} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},10,\frac {7}{2},1+\frac {a \sqrt [3]{x}}{b}\right )}{5 b^{10} \sqrt [3]{x}} \]
(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 = 0.62 (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 |
| \(\displaystyle \frac {1}{6} a \left (\frac {1}{16} a \left (-\frac {13 a \left (-\frac {11 a \left (-\frac {9 a \left (-\frac {7 a \left (-\frac {5 a \left (-\frac {3 a \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{b^{3/2}}-\frac {3 \sqrt {a x+b x^{2/3}}}{b x^{2/3}}\right )}{4 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{2 b x}\right )}{6 b}-\frac {\sqrt {a x+b x^{2/3}}}{b x^{4/3}}\right )}{8 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{4 b x^{5/3}}\right )}{10 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 b x^2}\right )}{12 b}-\frac {\sqrt {a x+b x^{2/3}}}{2 b x^{7/3}}\right )}{14 b}-\frac {3 \sqrt {a x+b x^{2/3}}}{7 b x^{8/3}}\right )-\frac {3 \sqrt {a x+b x^{2/3}}}{8 x^3}\right )-\frac {\left (a x+b x^{2/3}\right )^{3/2}}{3 x^4}\) |
-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
3.2.83.3.1 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 = 1.85 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.62
| method | result | size |
| derivativedivides | \(\frac {\left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (45045 b^{\frac {31}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}-390390 b^{\frac {29}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}}-2633274 b^{\frac {27}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}}+4349826 b^{\frac {25}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}}-4685824 b^{\frac {23}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}}+3317886 b^{\frac {21}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {11}{2}}-1495494 b^{\frac {19}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {13}{2}}+390390 b^{\frac {17}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {15}{2}}-45045 b^{\frac {15}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {17}{2}}+45045 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{7} a^{9} x^{3}\right )}{3440640 x^{4} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {29}{2}}}\) | \(181\) |
| default | \(\frac {\left (b \,x^{\frac {2}{3}}+a x \right )^{\frac {3}{2}} \left (45045 b^{\frac {31}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}-390390 b^{\frac {29}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}}-2633274 b^{\frac {27}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}}+4349826 b^{\frac {25}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}}-4685824 b^{\frac {23}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}}+3317886 b^{\frac {21}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {11}{2}}-1495494 b^{\frac {19}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {13}{2}}+390390 b^{\frac {17}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {15}{2}}-45045 b^{\frac {15}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {17}{2}}+45045 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{7} a^{9} x^{3}\right )}{3440640 x^{4} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}} b^{\frac {29}{2}}}\) | \(181\) |
1/3440640*(b*x^(2/3)+a*x)^(3/2)*(45045*b^(31/2)*(b+a*x^(1/3))^(1/2)-390390 *b^(29/2)*(b+a*x^(1/3))^(3/2)-2633274*b^(27/2)*(b+a*x^(1/3))^(5/2)+4349826 *b^(25/2)*(b+a*x^(1/3))^(7/2)-4685824*b^(23/2)*(b+a*x^(1/3))^(9/2)+3317886 *b^(21/2)*(b+a*x^(1/3))^(11/2)-1495494*b^(19/2)*(b+a*x^(1/3))^(13/2)+39039 0*b^(17/2)*(b+a*x^(1/3))^(15/2)-45045*b^(15/2)*(b+a*x^(1/3))^(17/2)+45045* arctanh((b+a*x^(1/3))^(1/2)/b^(1/2))*b^7*a^9*x^3)/x^4/(b+a*x^(1/3))^(3/2)/ b^(29/2)
Timed out. \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\text {Timed out} \]
\[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\int \frac {\left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}}{x^{5}}\, dx \]
\[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\int { \frac {{\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}}}{x^{5}} \,d x } \]
Time = 0.54 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.67 \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=-\frac {\frac {45045 \, a^{10} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{7}} + \frac {45045 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} a^{10} - 390390 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} a^{10} b + 1495494 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} a^{10} b^{2} - 3317886 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} a^{10} b^{3} + 4685824 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{10} b^{4} - 4349826 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{10} b^{5} + 2633274 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{10} b^{6} + 390390 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{10} b^{7} - 45045 \, \sqrt {a x^{\frac {1}{3}} + b} a^{10} b^{8}}{a^{9} b^{7} x^{3}}}{3440640 \, a} \]
-1/3440640*(45045*a^10*arctan(sqrt(a*x^(1/3) + b)/sqrt(-b))/(sqrt(-b)*b^7) + (45045*(a*x^(1/3) + b)^(17/2)*a^10 - 390390*(a*x^(1/3) + b)^(15/2)*a^10 *b + 1495494*(a*x^(1/3) + b)^(13/2)*a^10*b^2 - 3317886*(a*x^(1/3) + b)^(11 /2)*a^10*b^3 + 4685824*(a*x^(1/3) + b)^(9/2)*a^10*b^4 - 4349826*(a*x^(1/3) + b)^(7/2)*a^10*b^5 + 2633274*(a*x^(1/3) + b)^(5/2)*a^10*b^6 + 390390*(a* x^(1/3) + b)^(3/2)*a^10*b^7 - 45045*sqrt(a*x^(1/3) + b)*a^10*b^8)/(a^9*b^7 *x^3))/a
Timed out. \[ \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^5} \, dx=\int \frac {{\left (a\,x+b\,x^{2/3}\right )}^{3/2}}{x^5} \,d x \]