Integrand size = 13, antiderivative size = 116 \[ \int \frac {(a+b x)^{9/2}}{x^3} \, dx=12 a^2 b^2 \sqrt {a+b x}-\frac {a^4 \sqrt {a+b x}}{2 x^2}-\frac {17 a^3 b \sqrt {a+b x}}{4 x}+2 a b^2 (a+b x)^{3/2}+\frac {2}{5} b^2 (a+b x)^{5/2}-\frac {63}{4} a^{5/2} b^2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \] Output:
12*a^2*b^2*(b*x+a)^(1/2)-1/2*a^4*(b*x+a)^(1/2)/x^2-17/4*a^3*b*(b*x+a)^(1/2 )/x+2*a*b^2*(b*x+a)^(3/2)+2/5*b^2*(b*x+a)^(5/2)-63/4*a^(5/2)*b^2*arctanh(( b*x+a)^(1/2)/a^(1/2))
Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b x)^{9/2}}{x^3} \, dx=\frac {\sqrt {a+b x} \left (-10 a^4-85 a^3 b x+288 a^2 b^2 x^2+56 a b^3 x^3+8 b^4 x^4\right )}{20 x^2}-\frac {63}{4} a^{5/2} b^2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) \] Input:
Integrate[(a + b*x)^(9/2)/x^3,x]
Output:
(Sqrt[a + b*x]*(-10*a^4 - 85*a^3*b*x + 288*a^2*b^2*x^2 + 56*a*b^3*x^3 + 8* b^4*x^4))/(20*x^2) - (63*a^(5/2)*b^2*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/4
Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {51, 51, 60, 60, 60, 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.
\(\displaystyle \int \frac {(a+b x)^{9/2}}{x^3} \, dx\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {9}{4} b \int \frac {(a+b x)^{7/2}}{x^2}dx-\frac {(a+b x)^{9/2}}{2 x^2}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {9}{4} b \left (\frac {7}{2} b \int \frac {(a+b x)^{5/2}}{x}dx-\frac {(a+b x)^{7/2}}{x}\right )-\frac {(a+b x)^{9/2}}{2 x^2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {9}{4} b \left (\frac {7}{2} b \left (a \int \frac {(a+b x)^{3/2}}{x}dx+\frac {2}{5} (a+b x)^{5/2}\right )-\frac {(a+b x)^{7/2}}{x}\right )-\frac {(a+b x)^{9/2}}{2 x^2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {9}{4} b \left (\frac {7}{2} b \left (a \left (a \int \frac {\sqrt {a+b x}}{x}dx+\frac {2}{3} (a+b x)^{3/2}\right )+\frac {2}{5} (a+b x)^{5/2}\right )-\frac {(a+b x)^{7/2}}{x}\right )-\frac {(a+b x)^{9/2}}{2 x^2}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {9}{4} b \left (\frac {7}{2} b \left (a \left (a \left (a \int \frac {1}{x \sqrt {a+b x}}dx+2 \sqrt {a+b x}\right )+\frac {2}{3} (a+b x)^{3/2}\right )+\frac {2}{5} (a+b x)^{5/2}\right )-\frac {(a+b x)^{7/2}}{x}\right )-\frac {(a+b x)^{9/2}}{2 x^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {9}{4} b \left (\frac {7}{2} b \left (a \left (a \left (\frac {2 a \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{b}+2 \sqrt {a+b x}\right )+\frac {2}{3} (a+b x)^{3/2}\right )+\frac {2}{5} (a+b x)^{5/2}\right )-\frac {(a+b x)^{7/2}}{x}\right )-\frac {(a+b x)^{9/2}}{2 x^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {9}{4} b \left (\frac {7}{2} b \left (a \left (a \left (2 \sqrt {a+b x}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )+\frac {2}{3} (a+b x)^{3/2}\right )+\frac {2}{5} (a+b x)^{5/2}\right )-\frac {(a+b x)^{7/2}}{x}\right )-\frac {(a+b x)^{9/2}}{2 x^2}\) |
Input:
Int[(a + b*x)^(9/2)/x^3,x]
Output:
-1/2*(a + b*x)^(9/2)/x^2 + (9*b*(-((a + b*x)^(7/2)/x) + (7*b*((2*(a + b*x) ^(5/2))/5 + a*((2*(a + b*x)^(3/2))/3 + a*(2*Sqrt[a + b*x] - 2*Sqrt[a]*ArcT anh[Sqrt[a + b*x]/Sqrt[a]]))))/2))/4
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/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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 = 0.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68
method | result | size |
risch | \(-\frac {a^{3} \sqrt {b x +a}\, \left (17 b x +2 a \right )}{4 x^{2}}+\frac {b^{2} \left (\frac {16 \left (b x +a \right )^{\frac {5}{2}}}{5}+16 a \left (b x +a \right )^{\frac {3}{2}}+96 a^{2} \sqrt {b x +a}-126 a^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )\right )}{8}\) | \(79\) |
pseudoelliptic | \(-\frac {63 \left (\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a^{3} b^{2} x^{2}-\frac {8 \sqrt {b x +a}\, \left (\sqrt {a}\, b^{4} x^{4}+7 a^{\frac {3}{2}} b^{3} x^{3}+36 a^{\frac {5}{2}} b^{2} x^{2}-\frac {85 a^{\frac {7}{2}} b x}{8}-\frac {5 a^{\frac {9}{2}}}{4}\right )}{315}\right )}{4 \sqrt {a}\, x^{2}}\) | \(86\) |
derivativedivides | \(2 b^{2} \left (\frac {\left (b x +a \right )^{\frac {5}{2}}}{5}+a \left (b x +a \right )^{\frac {3}{2}}+6 a^{2} \sqrt {b x +a}-a^{3} \left (\frac {\frac {17 \left (b x +a \right )^{\frac {3}{2}}}{8}-\frac {15 a \sqrt {b x +a}}{8}}{b^{2} x^{2}}+\frac {63 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}\right )\right )\) | \(87\) |
default | \(2 b^{2} \left (\frac {\left (b x +a \right )^{\frac {5}{2}}}{5}+a \left (b x +a \right )^{\frac {3}{2}}+6 a^{2} \sqrt {b x +a}-a^{3} \left (\frac {\frac {17 \left (b x +a \right )^{\frac {3}{2}}}{8}-\frac {15 a \sqrt {b x +a}}{8}}{b^{2} x^{2}}+\frac {63 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}\right )\right )\) | \(87\) |
Input:
int((b*x+a)^(9/2)/x^3,x,method=_RETURNVERBOSE)
Output:
-1/4*a^3*(b*x+a)^(1/2)*(17*b*x+2*a)/x^2+1/8*b^2*(16/5*(b*x+a)^(5/2)+16*a*( b*x+a)^(3/2)+96*a^2*(b*x+a)^(1/2)-126*a^(5/2)*arctanh((b*x+a)^(1/2)/a^(1/2 )))
Time = 0.08 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.53 \[ \int \frac {(a+b x)^{9/2}}{x^3} \, dx=\left [\frac {315 \, a^{\frac {5}{2}} b^{2} x^{2} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (8 \, b^{4} x^{4} + 56 \, a b^{3} x^{3} + 288 \, a^{2} b^{2} x^{2} - 85 \, a^{3} b x - 10 \, a^{4}\right )} \sqrt {b x + a}}{40 \, x^{2}}, \frac {315 \, \sqrt {-a} a^{2} b^{2} x^{2} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x + a}}\right ) + {\left (8 \, b^{4} x^{4} + 56 \, a b^{3} x^{3} + 288 \, a^{2} b^{2} x^{2} - 85 \, a^{3} b x - 10 \, a^{4}\right )} \sqrt {b x + a}}{20 \, x^{2}}\right ] \] Input:
integrate((b*x+a)^(9/2)/x^3,x, algorithm="fricas")
Output:
[1/40*(315*a^(5/2)*b^2*x^2*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(8*b^4*x^4 + 56*a*b^3*x^3 + 288*a^2*b^2*x^2 - 85*a^3*b*x - 10*a^4)*sqrt( b*x + a))/x^2, 1/20*(315*sqrt(-a)*a^2*b^2*x^2*arctan(sqrt(-a)/sqrt(b*x + a )) + (8*b^4*x^4 + 56*a*b^3*x^3 + 288*a^2*b^2*x^2 - 85*a^3*b*x - 10*a^4)*sq rt(b*x + a))/x^2]
Time = 8.56 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^{9/2}}{x^3} \, dx=- \frac {63 a^{\frac {5}{2}} b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4} - \frac {a^{5}}{2 \sqrt {b} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {19 a^{4} \sqrt {b}}{4 x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {203 a^{3} b^{\frac {3}{2}}}{20 \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {86 a^{2} b^{\frac {5}{2}} \sqrt {x}}{5 \sqrt {\frac {a}{b x} + 1}} + \frac {16 a b^{\frac {7}{2}} x^{\frac {3}{2}}}{5 \sqrt {\frac {a}{b x} + 1}} + \frac {2 b^{\frac {9}{2}} x^{\frac {5}{2}}}{5 \sqrt {\frac {a}{b x} + 1}} \] Input:
integrate((b*x+a)**(9/2)/x**3,x)
Output:
-63*a**(5/2)*b**2*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/4 - a**5/(2*sqrt(b)*x** (5/2)*sqrt(a/(b*x) + 1)) - 19*a**4*sqrt(b)/(4*x**(3/2)*sqrt(a/(b*x) + 1)) + 203*a**3*b**(3/2)/(20*sqrt(x)*sqrt(a/(b*x) + 1)) + 86*a**2*b**(5/2)*sqrt (x)/(5*sqrt(a/(b*x) + 1)) + 16*a*b**(7/2)*x**(3/2)/(5*sqrt(a/(b*x) + 1)) + 2*b**(9/2)*x**(5/2)/(5*sqrt(a/(b*x) + 1))
Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x)^{9/2}}{x^3} \, dx=\frac {63}{8} \, a^{\frac {5}{2}} b^{2} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right ) + \frac {2}{5} \, {\left (b x + a\right )}^{\frac {5}{2}} b^{2} + 2 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{2} + 12 \, \sqrt {b x + a} a^{2} b^{2} - \frac {17 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{2} - 15 \, \sqrt {b x + a} a^{4} b^{2}}{4 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a + a^{2}\right )}} \] Input:
integrate((b*x+a)^(9/2)/x^3,x, algorithm="maxima")
Output:
63/8*a^(5/2)*b^2*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a))) + 2/5*(b*x + a)^(5/2)*b^2 + 2*(b*x + a)^(3/2)*a*b^2 + 12*sqrt(b*x + a)*a^2 *b^2 - 1/4*(17*(b*x + a)^(3/2)*a^3*b^2 - 15*sqrt(b*x + a)*a^4*b^2)/((b*x + a)^2 - 2*(b*x + a)*a + a^2)
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{9/2}}{x^3} \, dx=\frac {\frac {315 \, a^{3} b^{3} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 8 \, {\left (b x + a\right )}^{\frac {5}{2}} b^{3} + 40 \, {\left (b x + a\right )}^{\frac {3}{2}} a b^{3} + 240 \, \sqrt {b x + a} a^{2} b^{3} - \frac {5 \, {\left (17 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{3} - 15 \, \sqrt {b x + a} a^{4} b^{3}\right )}}{b^{2} x^{2}}}{20 \, b} \] Input:
integrate((b*x+a)^(9/2)/x^3,x, algorithm="giac")
Output:
1/20*(315*a^3*b^3*arctan(sqrt(b*x + a)/sqrt(-a))/sqrt(-a) + 8*(b*x + a)^(5 /2)*b^3 + 40*(b*x + a)^(3/2)*a*b^3 + 240*sqrt(b*x + a)*a^2*b^3 - 5*(17*(b* x + a)^(3/2)*a^3*b^3 - 15*sqrt(b*x + a)*a^4*b^3)/(b^2*x^2))/b
Time = 0.03 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b x)^{9/2}}{x^3} \, dx=\frac {2\,b^2\,{\left (a+b\,x\right )}^{5/2}}{5}+\frac {\frac {15\,a^4\,b^2\,\sqrt {a+b\,x}}{4}-\frac {17\,a^3\,b^2\,{\left (a+b\,x\right )}^{3/2}}{4}}{{\left (a+b\,x\right )}^2-2\,a\,\left (a+b\,x\right )+a^2}+12\,a^2\,b^2\,\sqrt {a+b\,x}+2\,a\,b^2\,{\left (a+b\,x\right )}^{3/2}+\frac {a^{5/2}\,b^2\,\mathrm {atan}\left (\frac {\sqrt {a+b\,x}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,63{}\mathrm {i}}{4} \] Input:
int((a + b*x)^(9/2)/x^3,x)
Output:
(2*b^2*(a + b*x)^(5/2))/5 + ((15*a^4*b^2*(a + b*x)^(1/2))/4 - (17*a^3*b^2* (a + b*x)^(3/2))/4)/((a + b*x)^2 - 2*a*(a + b*x) + a^2) + 12*a^2*b^2*(a + b*x)^(1/2) + (a^(5/2)*b^2*atan(((a + b*x)^(1/2)*1i)/a^(1/2))*63i)/4 + 2*a* b^2*(a + b*x)^(3/2)
Time = 0.16 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b x)^{9/2}}{x^3} \, dx=\frac {-20 \sqrt {b x +a}\, a^{4}-170 \sqrt {b x +a}\, a^{3} b x +576 \sqrt {b x +a}\, a^{2} b^{2} x^{2}+112 \sqrt {b x +a}\, a \,b^{3} x^{3}+16 \sqrt {b x +a}\, b^{4} x^{4}+315 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a^{2} b^{2} x^{2}-315 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a^{2} b^{2} x^{2}}{40 x^{2}} \] Input:
int((b*x+a)^(9/2)/x^3,x)
Output:
( - 20*sqrt(a + b*x)*a**4 - 170*sqrt(a + b*x)*a**3*b*x + 576*sqrt(a + b*x) *a**2*b**2*x**2 + 112*sqrt(a + b*x)*a*b**3*x**3 + 16*sqrt(a + b*x)*b**4*x* *4 + 315*sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*a**2*b**2*x**2 - 315*sqrt(a) *log(sqrt(a + b*x) + sqrt(a))*a**2*b**2*x**2)/(40*x**2)