Integrand size = 13, antiderivative size = 163 \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=-\frac {3 b^4 \sqrt {a+b x}}{128 x^3}-\frac {3 b^5 \sqrt {a+b x}}{512 a x^2}+\frac {9 b^6 \sqrt {a+b x}}{1024 a^2 x}-\frac {3 b^3 (a+b x)^{3/2}}{64 x^4}-\frac {3 b^2 (a+b x)^{5/2}}{40 x^5}-\frac {3 b (a+b x)^{7/2}}{28 x^6}-\frac {(a+b x)^{9/2}}{7 x^7}-\frac {9 b^7 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{5/2}} \]
-3/64*b^3*(b*x+a)^(3/2)/x^4-3/40*b^2*(b*x+a)^(5/2)/x^5-3/28*b*(b*x+a)^(7/2 )/x^6-1/7*(b*x+a)^(9/2)/x^7-9/1024*b^7*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(5 /2)-3/128*b^4*(b*x+a)^(1/2)/x^3-3/512*b^5*(b*x+a)^(1/2)/a/x^2+9/1024*b^6*( b*x+a)^(1/2)/a^2/x
Time = 0.23 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.68 \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=-\frac {\sqrt {a+b x} \left (5120 a^6+24320 a^5 b x+44928 a^4 b^2 x^2+39056 a^3 b^3 x^3+14168 a^2 b^4 x^4+210 a b^5 x^5-315 b^6 x^6\right )}{35840 a^2 x^7}-\frac {9 b^7 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{5/2}} \]
-1/35840*(Sqrt[a + b*x]*(5120*a^6 + 24320*a^5*b*x + 44928*a^4*b^2*x^2 + 39 056*a^3*b^3*x^3 + 14168*a^2*b^4*x^4 + 210*a*b^5*x^5 - 315*b^6*x^6))/(a^2*x ^7) - (9*b^7*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*a^(5/2))
Time = 0.22 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {51, 51, 51, 51, 51, 52, 52, 73, 221}
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.
| \(\displaystyle \int \frac {(a+b x)^{9/2}}{x^8} \, dx\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9}{14} b \int \frac {(a+b x)^{7/2}}{x^7}dx-\frac {(a+b x)^{9/2}}{7 x^7}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9}{14} b \left (\frac {7}{12} b \int \frac {(a+b x)^{5/2}}{x^6}dx-\frac {(a+b x)^{7/2}}{6 x^6}\right )-\frac {(a+b x)^{9/2}}{7 x^7}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9}{14} b \left (\frac {7}{12} b \left (\frac {1}{2} b \int \frac {(a+b x)^{3/2}}{x^5}dx-\frac {(a+b x)^{5/2}}{5 x^5}\right )-\frac {(a+b x)^{7/2}}{6 x^6}\right )-\frac {(a+b x)^{9/2}}{7 x^7}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9}{14} b \left (\frac {7}{12} b \left (\frac {1}{2} b \left (\frac {3}{8} b \int \frac {\sqrt {a+b x}}{x^4}dx-\frac {(a+b x)^{3/2}}{4 x^4}\right )-\frac {(a+b x)^{5/2}}{5 x^5}\right )-\frac {(a+b x)^{7/2}}{6 x^6}\right )-\frac {(a+b x)^{9/2}}{7 x^7}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {9}{14} b \left (\frac {7}{12} b \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \int \frac {1}{x^3 \sqrt {a+b x}}dx-\frac {\sqrt {a+b x}}{3 x^3}\right )-\frac {(a+b x)^{3/2}}{4 x^4}\right )-\frac {(a+b x)^{5/2}}{5 x^5}\right )-\frac {(a+b x)^{7/2}}{6 x^6}\right )-\frac {(a+b x)^{9/2}}{7 x^7}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {9}{14} b \left (\frac {7}{12} b \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \int \frac {1}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )-\frac {\sqrt {a+b x}}{3 x^3}\right )-\frac {(a+b x)^{3/2}}{4 x^4}\right )-\frac {(a+b x)^{5/2}}{5 x^5}\right )-\frac {(a+b x)^{7/2}}{6 x^6}\right )-\frac {(a+b x)^{9/2}}{7 x^7}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {9}{14} b \left (\frac {7}{12} b \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (-\frac {b \int \frac {1}{x \sqrt {a+b x}}dx}{2 a}-\frac {\sqrt {a+b x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )-\frac {\sqrt {a+b x}}{3 x^3}\right )-\frac {(a+b x)^{3/2}}{4 x^4}\right )-\frac {(a+b x)^{5/2}}{5 x^5}\right )-\frac {(a+b x)^{7/2}}{6 x^6}\right )-\frac {(a+b x)^{9/2}}{7 x^7}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {9}{14} b \left (\frac {7}{12} b \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (-\frac {\int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{a}-\frac {\sqrt {a+b x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )-\frac {\sqrt {a+b x}}{3 x^3}\right )-\frac {(a+b x)^{3/2}}{4 x^4}\right )-\frac {(a+b x)^{5/2}}{5 x^5}\right )-\frac {(a+b x)^{7/2}}{6 x^6}\right )-\frac {(a+b x)^{9/2}}{7 x^7}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {9}{14} b \left (\frac {7}{12} b \left (\frac {1}{2} b \left (\frac {3}{8} b \left (\frac {1}{6} b \left (-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )-\frac {\sqrt {a+b x}}{3 x^3}\right )-\frac {(a+b x)^{3/2}}{4 x^4}\right )-\frac {(a+b x)^{5/2}}{5 x^5}\right )-\frac {(a+b x)^{7/2}}{6 x^6}\right )-\frac {(a+b x)^{9/2}}{7 x^7}\) |
-1/7*(a + b*x)^(9/2)/x^7 + (9*b*(-1/6*(a + b*x)^(7/2)/x^6 + (7*b*(-1/5*(a + b*x)^(5/2)/x^5 + (b*(-1/4*(a + b*x)^(3/2)/x^4 + (3*b*(-1/3*Sqrt[a + b*x] /x^3 + (b*(-1/2*Sqrt[a + b*x]/(a*x^2) - (3*b*(-(Sqrt[a + b*x]/(a*x)) + (b* ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/a^(3/2)))/(4*a)))/6))/8))/2))/12))/14
3.4.24.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Time = 0.16 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.61
| method | result | size |
| risch | \(-\frac {\sqrt {b x +a}\, \left (-315 b^{6} x^{6}+210 a \,x^{5} b^{5}+14168 a^{2} x^{4} b^{4}+39056 a^{3} x^{3} b^{3}+44928 a^{4} x^{2} b^{2}+24320 a^{5} x b +5120 a^{6}\right )}{35840 x^{7} a^{2}}-\frac {9 b^{7} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {5}{2}}}\) | \(100\) |
| pseudoelliptic | \(-\frac {351 \left (\frac {35 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) x^{7} b^{7}}{4992}+\sqrt {b x +a}\, \left (-\frac {35 \sqrt {a}\, b^{6} x^{6}}{4992}+\frac {35 a^{\frac {3}{2}} b^{5} x^{5}}{7488}+\frac {1771 a^{\frac {5}{2}} b^{4} x^{4}}{5616}+\frac {2441 a^{\frac {7}{2}} b^{3} x^{3}}{2808}+a^{\frac {9}{2}} b^{2} x^{2}+\frac {190 a^{\frac {11}{2}} b x}{351}+\frac {40 a^{\frac {13}{2}}}{351}\right )\right )}{280 a^{\frac {5}{2}} x^{7}}\) | \(105\) |
| derivativedivides | \(2 b^{7} \left (-\frac {-\frac {9 \left (b x +a \right )^{\frac {13}{2}}}{2048 a^{2}}+\frac {15 \left (b x +a \right )^{\frac {11}{2}}}{512 a}+\frac {1199 \left (b x +a \right )^{\frac {9}{2}}}{10240}-\frac {9 a \left (b x +a \right )^{\frac {7}{2}}}{70}+\frac {849 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{10240}-\frac {15 a^{3} \left (b x +a \right )^{\frac {3}{2}}}{512}+\frac {9 a^{4} \sqrt {b x +a}}{2048}}{b^{7} x^{7}}-\frac {9 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2048 a^{\frac {5}{2}}}\right )\) | \(112\) |
| default | \(2 b^{7} \left (-\frac {-\frac {9 \left (b x +a \right )^{\frac {13}{2}}}{2048 a^{2}}+\frac {15 \left (b x +a \right )^{\frac {11}{2}}}{512 a}+\frac {1199 \left (b x +a \right )^{\frac {9}{2}}}{10240}-\frac {9 a \left (b x +a \right )^{\frac {7}{2}}}{70}+\frac {849 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{10240}-\frac {15 a^{3} \left (b x +a \right )^{\frac {3}{2}}}{512}+\frac {9 a^{4} \sqrt {b x +a}}{2048}}{b^{7} x^{7}}-\frac {9 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2048 a^{\frac {5}{2}}}\right )\) | \(112\) |
-1/35840*(b*x+a)^(1/2)*(-315*b^6*x^6+210*a*b^5*x^5+14168*a^2*b^4*x^4+39056 *a^3*b^3*x^3+44928*a^4*b^2*x^2+24320*a^5*b*x+5120*a^6)/x^7/a^2-9/1024*b^7* arctanh((b*x+a)^(1/2)/a^(1/2))/a^(5/2)
Time = 0.24 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=\left [\frac {315 \, \sqrt {a} b^{7} x^{7} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (315 \, a b^{6} x^{6} - 210 \, a^{2} b^{5} x^{5} - 14168 \, a^{3} b^{4} x^{4} - 39056 \, a^{4} b^{3} x^{3} - 44928 \, a^{5} b^{2} x^{2} - 24320 \, a^{6} b x - 5120 \, a^{7}\right )} \sqrt {b x + a}}{71680 \, a^{3} x^{7}}, \frac {315 \, \sqrt {-a} b^{7} x^{7} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (315 \, a b^{6} x^{6} - 210 \, a^{2} b^{5} x^{5} - 14168 \, a^{3} b^{4} x^{4} - 39056 \, a^{4} b^{3} x^{3} - 44928 \, a^{5} b^{2} x^{2} - 24320 \, a^{6} b x - 5120 \, a^{7}\right )} \sqrt {b x + a}}{35840 \, a^{3} x^{7}}\right ] \]
[1/71680*(315*sqrt(a)*b^7*x^7*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(315*a*b^6*x^6 - 210*a^2*b^5*x^5 - 14168*a^3*b^4*x^4 - 39056*a^4*b^3* x^3 - 44928*a^5*b^2*x^2 - 24320*a^6*b*x - 5120*a^7)*sqrt(b*x + a))/(a^3*x^ 7), 1/35840*(315*sqrt(-a)*b^7*x^7*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (315* a*b^6*x^6 - 210*a^2*b^5*x^5 - 14168*a^3*b^4*x^4 - 39056*a^4*b^3*x^3 - 4492 8*a^5*b^2*x^2 - 24320*a^6*b*x - 5120*a^7)*sqrt(b*x + a))/(a^3*x^7)]
Timed out. \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=\text {Timed out} \]
Time = 0.32 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=\frac {9 \, b^{7} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{2048 \, a^{\frac {5}{2}}} + \frac {315 \, {\left (b x + a\right )}^{\frac {13}{2}} b^{7} - 2100 \, {\left (b x + a\right )}^{\frac {11}{2}} a b^{7} - 8393 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} b^{7} + 9216 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} b^{7} - 5943 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} b^{7} + 2100 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} b^{7} - 315 \, \sqrt {b x + a} a^{6} b^{7}}{35840 \, {\left ({\left (b x + a\right )}^{7} a^{2} - 7 \, {\left (b x + a\right )}^{6} a^{3} + 21 \, {\left (b x + a\right )}^{5} a^{4} - 35 \, {\left (b x + a\right )}^{4} a^{5} + 35 \, {\left (b x + a\right )}^{3} a^{6} - 21 \, {\left (b x + a\right )}^{2} a^{7} + 7 \, {\left (b x + a\right )} a^{8} - a^{9}\right )}} \]
9/2048*b^7*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/a^(5/2 ) + 1/35840*(315*(b*x + a)^(13/2)*b^7 - 2100*(b*x + a)^(11/2)*a*b^7 - 8393 *(b*x + a)^(9/2)*a^2*b^7 + 9216*(b*x + a)^(7/2)*a^3*b^7 - 5943*(b*x + a)^( 5/2)*a^4*b^7 + 2100*(b*x + a)^(3/2)*a^5*b^7 - 315*sqrt(b*x + a)*a^6*b^7)/( (b*x + a)^7*a^2 - 7*(b*x + a)^6*a^3 + 21*(b*x + a)^5*a^4 - 35*(b*x + a)^4* a^5 + 35*(b*x + a)^3*a^6 - 21*(b*x + a)^2*a^7 + 7*(b*x + a)*a^8 - a^9)
Time = 0.31 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=\frac {\frac {315 \, b^{8} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {315 \, {\left (b x + a\right )}^{\frac {13}{2}} b^{8} - 2100 \, {\left (b x + a\right )}^{\frac {11}{2}} a b^{8} - 8393 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{2} b^{8} + 9216 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{3} b^{8} - 5943 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{4} b^{8} + 2100 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{5} b^{8} - 315 \, \sqrt {b x + a} a^{6} b^{8}}{a^{2} b^{7} x^{7}}}{35840 \, b} \]
1/35840*(315*b^8*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^2) + (315*(b*x + a)^(13/2)*b^8 - 2100*(b*x + a)^(11/2)*a*b^8 - 8393*(b*x + a)^(9/2)*a^2* b^8 + 9216*(b*x + a)^(7/2)*a^3*b^8 - 5943*(b*x + a)^(5/2)*a^4*b^8 + 2100*( b*x + a)^(3/2)*a^5*b^8 - 315*sqrt(b*x + a)*a^6*b^8)/(a^2*b^7*x^7))/b
Time = 0.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.76 \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=\frac {15\,a^3\,{\left (a+b\,x\right )}^{3/2}}{256\,x^7}-\frac {9\,a^4\,\sqrt {a+b\,x}}{1024\,x^7}-\frac {1199\,{\left (a+b\,x\right )}^{9/2}}{5120\,x^7}-\frac {849\,a^2\,{\left (a+b\,x\right )}^{5/2}}{5120\,x^7}-\frac {15\,{\left (a+b\,x\right )}^{11/2}}{256\,a\,x^7}+\frac {9\,{\left (a+b\,x\right )}^{13/2}}{1024\,a^2\,x^7}+\frac {9\,a\,{\left (a+b\,x\right )}^{7/2}}{35\,x^7}+\frac {b^7\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,9{}\mathrm {i}}{1024\,a^{5/2}} \]
(15*a^3*(a + b*x)^(3/2))/(256*x^7) - (9*a^4*(a + b*x)^(1/2))/(1024*x^7) - (1199*(a + b*x)^(9/2))/(5120*x^7) - (849*a^2*(a + b*x)^(5/2))/(5120*x^7) - (15*(a + b*x)^(11/2))/(256*a*x^7) + (9*(a + b*x)^(13/2))/(1024*a^2*x^7) + (b^7*atan(((a + b*x)^(1/2)*1i)/a^(1/2))*9i)/(1024*a^(5/2)) + (9*a*(a + b* x)^(7/2))/(35*x^7)
Time = 0.00 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{9/2}}{x^8} \, dx=\frac {-10240 \sqrt {b x +a}\, a^{7}-48640 \sqrt {b x +a}\, a^{6} b x -89856 \sqrt {b x +a}\, a^{5} b^{2} x^{2}-78112 \sqrt {b x +a}\, a^{4} b^{3} x^{3}-28336 \sqrt {b x +a}\, a^{3} b^{4} x^{4}-420 \sqrt {b x +a}\, a^{2} b^{5} x^{5}+630 \sqrt {b x +a}\, a \,b^{6} x^{6}+315 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b^{7} x^{7}-315 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b^{7} x^{7}}{71680 a^{3} x^{7}} \]
int((sqrt(a + b*x)*(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4))/x**8,x)
( - 10240*sqrt(a + b*x)*a**7 - 48640*sqrt(a + b*x)*a**6*b*x - 89856*sqrt(a + b*x)*a**5*b**2*x**2 - 78112*sqrt(a + b*x)*a**4*b**3*x**3 - 28336*sqrt(a + b*x)*a**3*b**4*x**4 - 420*sqrt(a + b*x)*a**2*b**5*x**5 + 630*sqrt(a + b *x)*a*b**6*x**6 + 315*sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*b**7*x**7 - 315 *sqrt(a)*log(sqrt(a + b*x) + sqrt(a))*b**7*x**7)/(71680*a**3*x**7)