Integrand size = 21, antiderivative size = 101 \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx=-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{9 x^9}-\frac {b c^3 \sqrt {a+b \left (c x^3\right )^{3/2}}}{18 a \left (c x^3\right )^{3/2}}+\frac {b^2 c^3 \text {arctanh}\left (\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{\sqrt {a}}\right )}{18 a^{3/2}} \]
1/18*b^2*c^3*arctanh((a+b*(c*x^3)^(3/2))^(1/2)/a^(1/2))/a^(3/2)-1/9*(a+b*( c*x^3)^(3/2))^(1/2)/x^9-1/18*b*c^3*(a+b*(c*x^3)^(3/2))^(1/2)/a/(c*x^3)^(3/ 2)
Time = 1.79 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx=-\frac {\sqrt {a+b \left (c x^3\right )^{3/2}} \left (2 a+b \left (c x^3\right )^{3/2}\right )}{18 a x^9}+\frac {b^2 c^3 \text {arctanh}\left (\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{\sqrt {a}}\right )}{18 a^{3/2}} \]
-1/18*(Sqrt[a + b*(c*x^3)^(3/2)]*(2*a + b*(c*x^3)^(3/2)))/(a*x^9) + (b^2*c ^3*ArcTanh[Sqrt[a + b*(c*x^3)^(3/2)]/Sqrt[a]])/(18*a^(3/2))
Time = 0.23 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.05, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {893, 798, 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.
\(\displaystyle \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx\) |
\(\Big \downarrow \) 893 |
\(\displaystyle \int \frac {\sqrt {a+b c^{3/2} x^{9/2}}}{x^{10}}dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {2}{9} \int \frac {\sqrt {b c^{3/2} x^{9/2}+a}}{x^{27/2}}dx^{9/2}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {2}{9} \left (\frac {1}{4} b c^{3/2} \int \frac {1}{x^9 \sqrt {b c^{3/2} x^{9/2}+a}}dx^{9/2}-\frac {\sqrt {a+b c^{3/2} x^{9/2}}}{2 x^9}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {2}{9} \left (\frac {1}{4} b c^{3/2} \left (-\frac {b c^{3/2} \int \frac {1}{x^{9/2} \sqrt {b c^{3/2} x^{9/2}+a}}dx^{9/2}}{2 a}-\frac {\sqrt {a+b c^{3/2} x^{9/2}}}{a x^{9/2}}\right )-\frac {\sqrt {a+b c^{3/2} x^{9/2}}}{2 x^9}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2}{9} \left (\frac {1}{4} b c^{3/2} \left (-\frac {\int \frac {1}{\frac {x^9}{b c^{3/2}}-\frac {a}{b c^{3/2}}}d\sqrt {b c^{3/2} x^{9/2}+a}}{a}-\frac {\sqrt {a+b c^{3/2} x^{9/2}}}{a x^{9/2}}\right )-\frac {\sqrt {a+b c^{3/2} x^{9/2}}}{2 x^9}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2}{9} \left (\frac {1}{4} b c^{3/2} \left (\frac {b c^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b c^{3/2} x^{9/2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b c^{3/2} x^{9/2}}}{a x^{9/2}}\right )-\frac {\sqrt {a+b c^{3/2} x^{9/2}}}{2 x^9}\right )\) |
(2*(-1/2*Sqrt[a + b*c^(3/2)*x^(9/2)]/x^9 + (b*c^(3/2)*(-(Sqrt[a + b*c^(3/2 )*x^(9/2)]/(a*x^(9/2))) + (b*c^(3/2)*ArcTanh[Sqrt[a + b*c^(3/2)*x^(9/2)]/S qrt[a]])/a^(3/2)))/4))/9
3.30.73.3.1 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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo l] :> With[{k = Denominator[n]}, Subst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x ], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b, c, d, m, p, q}, x] && FractionQ[n]
\[\int \frac {\sqrt {a +b \left (c \,x^{3}\right )^{\frac {3}{2}}}}{x^{10}}d x\]
Timed out. \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx=\int \frac {\sqrt {a + b \left (c x^{3}\right )^{\frac {3}{2}}}}{x^{10}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.27 \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx=-\frac {1}{36} \, {\left (\frac {b^{2} \log \left (\frac {\sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} - \sqrt {a}}{\sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}}} + \frac {2 \, {\left ({\left (\left (c x^{3}\right )^{\frac {3}{2}} b + a\right )}^{\frac {3}{2}} b^{2} + \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} a b^{2}\right )}}{{\left (\left (c x^{3}\right )^{\frac {3}{2}} b + a\right )}^{2} a - 2 \, {\left (\left (c x^{3}\right )^{\frac {3}{2}} b + a\right )} a^{2} + a^{3}}\right )} c^{3} \]
-1/36*(b^2*log((sqrt((c*x^3)^(3/2)*b + a) - sqrt(a))/(sqrt((c*x^3)^(3/2)*b + a) + sqrt(a)))/a^(3/2) + 2*(((c*x^3)^(3/2)*b + a)^(3/2)*b^2 + sqrt((c*x ^3)^(3/2)*b + a)*a*b^2)/(((c*x^3)^(3/2)*b + a)^2*a - 2*((c*x^3)^(3/2)*b + a)*a^2 + a^3))*c^3
Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx=-\frac {{\left (\frac {b^{3} c^{6} \arctan \left (\frac {\sqrt {\sqrt {c x} b c^{5} x^{4} + a c^{4}}}{\sqrt {-a} c^{2}}\right )}{\sqrt {-a} a} + \frac {\sqrt {\sqrt {c x} b c^{5} x^{4} + a c^{4}} a b^{3} c^{12} + {\left (\sqrt {c x} b c^{5} x^{4} + a c^{4}\right )}^{\frac {3}{2}} b^{3} c^{8}}{a b^{2} c^{11} x^{9}}\right )} {\left | c \right |}^{2}}{18 \, b c^{5}} \]
-1/18*(b^3*c^6*arctan(sqrt(sqrt(c*x)*b*c^5*x^4 + a*c^4)/(sqrt(-a)*c^2))/(s qrt(-a)*a) + (sqrt(sqrt(c*x)*b*c^5*x^4 + a*c^4)*a*b^3*c^12 + (sqrt(c*x)*b* c^5*x^4 + a*c^4)^(3/2)*b^3*c^8)/(a*b^2*c^11*x^9))*abs(c)^2/(b*c^5)
Timed out. \[ \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x^{10}} \, dx=\int \frac {\sqrt {a+b\,{\left (c\,x^3\right )}^{3/2}}}{x^{10}} \,d x \]