Integrand size = 13, antiderivative size = 141 \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=-\frac {21 b^4 \sqrt {a+b x}}{256 x^2}-\frac {21 b^5 \sqrt {a+b x}}{512 a x}-\frac {7 b^3 (a+b x)^{3/2}}{64 x^3}-\frac {21 b^2 (a+b x)^{5/2}}{160 x^4}-\frac {3 b (a+b x)^{7/2}}{20 x^5}-\frac {(a+b x)^{9/2}}{6 x^6}+\frac {21 b^6 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{3/2}} \]
-7/64*b^3*(b*x+a)^(3/2)/x^3-21/160*b^2*(b*x+a)^(5/2)/x^4-3/20*b*(b*x+a)^(7 /2)/x^5-1/6*(b*x+a)^(9/2)/x^6+21/512*b^6*arctanh((b*x+a)^(1/2)/a^(1/2))/a^ (3/2)-21/256*b^4*(b*x+a)^(1/2)/x^2-21/512*b^5*(b*x+a)^(1/2)/a/x
Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.71 \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=-\frac {\sqrt {a+b x} \left (1280 a^5+6272 a^4 b x+12144 a^3 b^2 x^2+11432 a^2 b^3 x^3+4910 a b^4 x^4+315 b^5 x^5\right )}{7680 a x^6}+\frac {21 b^6 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{3/2}} \]
-1/7680*(Sqrt[a + b*x]*(1280*a^5 + 6272*a^4*b*x + 12144*a^3*b^2*x^2 + 1143 2*a^2*b^3*x^3 + 4910*a*b^4*x^4 + 315*b^5*x^5))/(a*x^6) + (21*b^6*ArcTanh[S qrt[a + b*x]/Sqrt[a]])/(512*a^(3/2))
Time = 0.21 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {51, 51, 51, 51, 51, 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.
| \(\displaystyle \int \frac {(a+b x)^{9/2}}{x^7} \, dx\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {3}{4} b \int \frac {(a+b x)^{7/2}}{x^6}dx-\frac {(a+b x)^{9/2}}{6 x^6}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {3}{4} b \left (\frac {7}{10} b \int \frac {(a+b x)^{5/2}}{x^5}dx-\frac {(a+b x)^{7/2}}{5 x^5}\right )-\frac {(a+b x)^{9/2}}{6 x^6}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {3}{4} b \left (\frac {7}{10} b \left (\frac {5}{8} b \int \frac {(a+b x)^{3/2}}{x^4}dx-\frac {(a+b x)^{5/2}}{4 x^4}\right )-\frac {(a+b x)^{7/2}}{5 x^5}\right )-\frac {(a+b x)^{9/2}}{6 x^6}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {3}{4} b \left (\frac {7}{10} b \left (\frac {5}{8} b \left (\frac {1}{2} b \int \frac {\sqrt {a+b x}}{x^3}dx-\frac {(a+b x)^{3/2}}{3 x^3}\right )-\frac {(a+b x)^{5/2}}{4 x^4}\right )-\frac {(a+b x)^{7/2}}{5 x^5}\right )-\frac {(a+b x)^{9/2}}{6 x^6}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {3}{4} b \left (\frac {7}{10} b \left (\frac {5}{8} b \left (\frac {1}{2} b \left (\frac {1}{4} b \int \frac {1}{x^2 \sqrt {a+b x}}dx-\frac {\sqrt {a+b x}}{2 x^2}\right )-\frac {(a+b x)^{3/2}}{3 x^3}\right )-\frac {(a+b x)^{5/2}}{4 x^4}\right )-\frac {(a+b x)^{7/2}}{5 x^5}\right )-\frac {(a+b x)^{9/2}}{6 x^6}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle \frac {3}{4} b \left (\frac {7}{10} b \left (\frac {5}{8} b \left (\frac {1}{2} b \left (\frac {1}{4} b \left (-\frac {b \int \frac {1}{x \sqrt {a+b x}}dx}{2 a}-\frac {\sqrt {a+b x}}{a x}\right )-\frac {\sqrt {a+b x}}{2 x^2}\right )-\frac {(a+b x)^{3/2}}{3 x^3}\right )-\frac {(a+b x)^{5/2}}{4 x^4}\right )-\frac {(a+b x)^{7/2}}{5 x^5}\right )-\frac {(a+b x)^{9/2}}{6 x^6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {3}{4} b \left (\frac {7}{10} b \left (\frac {5}{8} b \left (\frac {1}{2} b \left (\frac {1}{4} b \left (-\frac {\int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{a}-\frac {\sqrt {a+b x}}{a x}\right )-\frac {\sqrt {a+b x}}{2 x^2}\right )-\frac {(a+b x)^{3/2}}{3 x^3}\right )-\frac {(a+b x)^{5/2}}{4 x^4}\right )-\frac {(a+b x)^{7/2}}{5 x^5}\right )-\frac {(a+b x)^{9/2}}{6 x^6}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {3}{4} b \left (\frac {7}{10} b \left (\frac {5}{8} b \left (\frac {1}{2} b \left (\frac {1}{4} b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b x}}{a x}\right )-\frac {\sqrt {a+b x}}{2 x^2}\right )-\frac {(a+b x)^{3/2}}{3 x^3}\right )-\frac {(a+b x)^{5/2}}{4 x^4}\right )-\frac {(a+b x)^{7/2}}{5 x^5}\right )-\frac {(a+b x)^{9/2}}{6 x^6}\) |
-1/6*(a + b*x)^(9/2)/x^6 + (3*b*(-1/5*(a + b*x)^(7/2)/x^5 + (7*b*(-1/4*(a + b*x)^(5/2)/x^4 + (5*b*(-1/3*(a + b*x)^(3/2)/x^3 + (b*(-1/2*Sqrt[a + b*x] /x^2 + (b*(-(Sqrt[a + b*x]/(a*x)) + (b*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/a^( 3/2)))/4))/2))/8))/10))/4
3.4.23.3.1 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 = 0.15 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.63
| method | result | size |
| risch | \(-\frac {\sqrt {b x +a}\, \left (315 b^{5} x^{5}+4910 a \,b^{4} x^{4}+11432 a^{2} b^{3} x^{3}+12144 a^{3} b^{2} x^{2}+6272 a^{4} b x +1280 a^{5}\right )}{7680 x^{6} a}+\frac {21 b^{6} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{512 a^{\frac {3}{2}}}\) | \(89\) |
| pseudoelliptic | \(-\frac {49 \left (-\frac {45 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{6} x^{6}}{896}+\sqrt {b x +a}\, \left (\frac {45 \sqrt {a}\, b^{5} x^{5}}{896}+\frac {2455 a^{\frac {3}{2}} b^{4} x^{4}}{3136}+\frac {1429 a^{\frac {5}{2}} b^{3} x^{3}}{784}+\frac {759 a^{\frac {7}{2}} b^{2} x^{2}}{392}+a^{\frac {9}{2}} b x +\frac {10 a^{\frac {11}{2}}}{49}\right )\right )}{60 a^{\frac {3}{2}} x^{6}}\) | \(94\) |
| derivativedivides | \(2 b^{6} \left (-\frac {\frac {21 \left (b x +a \right )^{\frac {11}{2}}}{1024 a}+\frac {667 \left (b x +a \right )^{\frac {9}{2}}}{3072}-\frac {843 a \left (b x +a \right )^{\frac {7}{2}}}{2560}+\frac {693 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{2560}-\frac {119 a^{3} \left (b x +a \right )^{\frac {3}{2}}}{1024}+\frac {21 a^{4} \sqrt {b x +a}}{1024}}{b^{6} x^{6}}+\frac {21 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {3}{2}}}\right )\) | \(100\) |
| default | \(2 b^{6} \left (-\frac {\frac {21 \left (b x +a \right )^{\frac {11}{2}}}{1024 a}+\frac {667 \left (b x +a \right )^{\frac {9}{2}}}{3072}-\frac {843 a \left (b x +a \right )^{\frac {7}{2}}}{2560}+\frac {693 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{2560}-\frac {119 a^{3} \left (b x +a \right )^{\frac {3}{2}}}{1024}+\frac {21 a^{4} \sqrt {b x +a}}{1024}}{b^{6} x^{6}}+\frac {21 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {3}{2}}}\right )\) | \(100\) |
-1/7680*(b*x+a)^(1/2)*(315*b^5*x^5+4910*a*b^4*x^4+11432*a^2*b^3*x^3+12144* a^3*b^2*x^2+6272*a^4*b*x+1280*a^5)/x^6/a+21/512*b^6*arctanh((b*x+a)^(1/2)/ a^(1/2))/a^(3/2)
Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.50 \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=\left [\frac {315 \, \sqrt {a} b^{6} x^{6} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) - 2 \, {\left (315 \, a b^{5} x^{5} + 4910 \, a^{2} b^{4} x^{4} + 11432 \, a^{3} b^{3} x^{3} + 12144 \, a^{4} b^{2} x^{2} + 6272 \, a^{5} b x + 1280 \, a^{6}\right )} \sqrt {b x + a}}{15360 \, a^{2} x^{6}}, -\frac {315 \, \sqrt {-a} b^{6} x^{6} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (315 \, a b^{5} x^{5} + 4910 \, a^{2} b^{4} x^{4} + 11432 \, a^{3} b^{3} x^{3} + 12144 \, a^{4} b^{2} x^{2} + 6272 \, a^{5} b x + 1280 \, a^{6}\right )} \sqrt {b x + a}}{7680 \, a^{2} x^{6}}\right ] \]
[1/15360*(315*sqrt(a)*b^6*x^6*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) - 2*(315*a*b^5*x^5 + 4910*a^2*b^4*x^4 + 11432*a^3*b^3*x^3 + 12144*a^4*b^2 *x^2 + 6272*a^5*b*x + 1280*a^6)*sqrt(b*x + a))/(a^2*x^6), -1/7680*(315*sqr t(-a)*b^6*x^6*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (315*a*b^5*x^5 + 4910*a^2 *b^4*x^4 + 11432*a^3*b^3*x^3 + 12144*a^4*b^2*x^2 + 6272*a^5*b*x + 1280*a^6 )*sqrt(b*x + a))/(a^2*x^6)]
Time = 63.65 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.48 \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=- \frac {a^{5}}{6 \sqrt {b} x^{\frac {13}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {59 a^{4} \sqrt {b}}{60 x^{\frac {11}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {1151 a^{3} b^{\frac {3}{2}}}{480 x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {2947 a^{2} b^{\frac {5}{2}}}{960 x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {8171 a b^{\frac {7}{2}}}{3840 x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {1045 b^{\frac {9}{2}}}{1536 x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {21 b^{\frac {11}{2}}}{512 a \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {21 b^{6} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{512 a^{\frac {3}{2}}} \]
-a**5/(6*sqrt(b)*x**(13/2)*sqrt(a/(b*x) + 1)) - 59*a**4*sqrt(b)/(60*x**(11 /2)*sqrt(a/(b*x) + 1)) - 1151*a**3*b**(3/2)/(480*x**(9/2)*sqrt(a/(b*x) + 1 )) - 2947*a**2*b**(5/2)/(960*x**(7/2)*sqrt(a/(b*x) + 1)) - 8171*a*b**(7/2) /(3840*x**(5/2)*sqrt(a/(b*x) + 1)) - 1045*b**(9/2)/(1536*x**(3/2)*sqrt(a/( b*x) + 1)) - 21*b**(11/2)/(512*a*sqrt(x)*sqrt(a/(b*x) + 1)) + 21*b**6*asin h(sqrt(a)/(sqrt(b)*sqrt(x)))/(512*a**(3/2))
Time = 0.29 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.40 \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=-\frac {21 \, b^{6} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{1024 \, a^{\frac {3}{2}}} - \frac {315 \, {\left (b x + a\right )}^{\frac {11}{2}} b^{6} + 3335 \, {\left (b x + a\right )}^{\frac {9}{2}} a b^{6} - 5058 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} b^{6} + 4158 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} b^{6} - 1785 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} b^{6} + 315 \, \sqrt {b x + a} a^{5} b^{6}}{7680 \, {\left ({\left (b x + a\right )}^{6} a - 6 \, {\left (b x + a\right )}^{5} a^{2} + 15 \, {\left (b x + a\right )}^{4} a^{3} - 20 \, {\left (b x + a\right )}^{3} a^{4} + 15 \, {\left (b x + a\right )}^{2} a^{5} - 6 \, {\left (b x + a\right )} a^{6} + a^{7}\right )}} \]
-21/1024*b^6*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/a^(3 /2) - 1/7680*(315*(b*x + a)^(11/2)*b^6 + 3335*(b*x + a)^(9/2)*a*b^6 - 5058 *(b*x + a)^(7/2)*a^2*b^6 + 4158*(b*x + a)^(5/2)*a^3*b^6 - 1785*(b*x + a)^( 3/2)*a^4*b^6 + 315*sqrt(b*x + a)*a^5*b^6)/((b*x + a)^6*a - 6*(b*x + a)^5*a ^2 + 15*(b*x + a)^4*a^3 - 20*(b*x + a)^3*a^4 + 15*(b*x + a)^2*a^5 - 6*(b*x + a)*a^6 + a^7)
Time = 0.31 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=-\frac {\frac {315 \, b^{7} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {315 \, {\left (b x + a\right )}^{\frac {11}{2}} b^{7} + 3335 \, {\left (b x + a\right )}^{\frac {9}{2}} a b^{7} - 5058 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} b^{7} + 4158 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} b^{7} - 1785 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} b^{7} + 315 \, \sqrt {b x + a} a^{5} b^{7}}{a b^{6} x^{6}}}{7680 \, b} \]
-1/7680*(315*b^7*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a) + (315*(b*x + a)^(11/2)*b^7 + 3335*(b*x + a)^(9/2)*a*b^7 - 5058*(b*x + a)^(7/2)*a^2*b^7 + 4158*(b*x + a)^(5/2)*a^3*b^7 - 1785*(b*x + a)^(3/2)*a^4*b^7 + 315*sqrt( b*x + a)*a^5*b^7)/(a*b^6*x^6))/b
Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b x)^{9/2}}{x^7} \, dx=\frac {119\,a^3\,{\left (a+b\,x\right )}^{3/2}}{512\,x^6}-\frac {21\,a^4\,\sqrt {a+b\,x}}{512\,x^6}-\frac {667\,{\left (a+b\,x\right )}^{9/2}}{1536\,x^6}-\frac {693\,a^2\,{\left (a+b\,x\right )}^{5/2}}{1280\,x^6}-\frac {21\,{\left (a+b\,x\right )}^{11/2}}{512\,a\,x^6}+\frac {843\,a\,{\left (a+b\,x\right )}^{7/2}}{1280\,x^6}-\frac {b^6\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,21{}\mathrm {i}}{512\,a^{3/2}} \]