Integrand size = 15, antiderivative size = 95 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=-\frac {\sqrt {a+b x^3}}{9 x^9}-\frac {b \sqrt {a+b x^3}}{36 a x^6}+\frac {b^2 \sqrt {a+b x^3}}{24 a^2 x^3}-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{24 a^{5/2}} \] Output:
-1/9*(b*x^3+a)^(1/2)/x^9-1/36*b*(b*x^3+a)^(1/2)/a/x^6+1/24*b^2*(b*x^3+a)^( 1/2)/a^2/x^3-1/24*b^3*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(5/2)
Time = 0.11 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=\frac {\sqrt {a+b x^3} \left (-8 a^2-2 a b x^3+3 b^2 x^6\right )}{72 a^2 x^9}-\frac {b^3 \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{24 a^{5/2}} \] Input:
Integrate[Sqrt[a + b*x^3]/x^10,x]
Output:
(Sqrt[a + b*x^3]*(-8*a^2 - 2*a*b*x^3 + 3*b^2*x^6))/(72*a^2*x^9) - (b^3*Arc Tanh[Sqrt[a + b*x^3]/Sqrt[a]])/(24*a^(5/2))
Time = 0.31 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {798, 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.
\(\displaystyle \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {\sqrt {b x^3+a}}{x^{12}}dx^3\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{6} b \int \frac {1}{x^9 \sqrt {b x^3+a}}dx^3-\frac {\sqrt {a+b x^3}}{3 x^9}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{6} b \left (-\frac {3 b \int \frac {1}{x^6 \sqrt {b x^3+a}}dx^3}{4 a}-\frac {\sqrt {a+b x^3}}{2 a x^6}\right )-\frac {\sqrt {a+b x^3}}{3 x^9}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{6} b \left (-\frac {3 b \left (-\frac {b \int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3}{2 a}-\frac {\sqrt {a+b x^3}}{a x^3}\right )}{4 a}-\frac {\sqrt {a+b x^3}}{2 a x^6}\right )-\frac {\sqrt {a+b x^3}}{3 x^9}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{6} b \left (-\frac {3 b \left (-\frac {\int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}}{a}-\frac {\sqrt {a+b x^3}}{a x^3}\right )}{4 a}-\frac {\sqrt {a+b x^3}}{2 a x^6}\right )-\frac {\sqrt {a+b x^3}}{3 x^9}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{6} b \left (-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b x^3}}{a x^3}\right )}{4 a}-\frac {\sqrt {a+b x^3}}{2 a x^6}\right )-\frac {\sqrt {a+b x^3}}{3 x^9}\right )\) |
Input:
Int[Sqrt[a + b*x^3]/x^10,x]
Output:
(-1/3*Sqrt[a + b*x^3]/x^9 + (b*(-1/2*Sqrt[a + b*x^3]/(a*x^6) - (3*b*(-(Sqr t[a + b*x^3]/(a*x^3)) + (b*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/a^(3/2)))/(4* a)))/6)/3
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]]
Time = 0.49 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.65
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{3}+a}\, \left (-3 b^{2} x^{6}+2 a b \,x^{3}+8 a^{2}\right )}{72 x^{9} a^{2}}-\frac {b^{3} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{24 a^{\frac {5}{2}}}\) | \(62\) |
default | \(-\frac {\sqrt {b \,x^{3}+a}}{9 x^{9}}-\frac {b \sqrt {b \,x^{3}+a}}{36 a \,x^{6}}+\frac {b^{2} \sqrt {b \,x^{3}+a}}{24 a^{2} x^{3}}-\frac {b^{3} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{24 a^{\frac {5}{2}}}\) | \(76\) |
elliptic | \(-\frac {\sqrt {b \,x^{3}+a}}{9 x^{9}}-\frac {b \sqrt {b \,x^{3}+a}}{36 a \,x^{6}}+\frac {b^{2} \sqrt {b \,x^{3}+a}}{24 a^{2} x^{3}}-\frac {b^{3} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{24 a^{\frac {5}{2}}}\) | \(76\) |
pseudoelliptic | \(\frac {-3 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right ) b^{3} x^{9}+3 b^{2} x^{6} \sqrt {b \,x^{3}+a}\, \sqrt {a}-2 a^{\frac {3}{2}} b \,x^{3} \sqrt {b \,x^{3}+a}-8 a^{\frac {5}{2}} \sqrt {b \,x^{3}+a}}{72 a^{\frac {5}{2}} x^{9}}\) | \(84\) |
Input:
int((b*x^3+a)^(1/2)/x^10,x,method=_RETURNVERBOSE)
Output:
-1/72*(b*x^3+a)^(1/2)*(-3*b^2*x^6+2*a*b*x^3+8*a^2)/x^9/a^2-1/24*b^3*arctan h((b*x^3+a)^(1/2)/a^(1/2))/a^(5/2)
Time = 0.09 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=\left [\frac {3 \, \sqrt {a} b^{3} x^{9} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, {\left (3 \, a b^{2} x^{6} - 2 \, a^{2} b x^{3} - 8 \, a^{3}\right )} \sqrt {b x^{3} + a}}{144 \, a^{3} x^{9}}, \frac {3 \, \sqrt {-a} b^{3} x^{9} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{3} + a}}\right ) + {\left (3 \, a b^{2} x^{6} - 2 \, a^{2} b x^{3} - 8 \, a^{3}\right )} \sqrt {b x^{3} + a}}{72 \, a^{3} x^{9}}\right ] \] Input:
integrate((b*x^3+a)^(1/2)/x^10,x, algorithm="fricas")
Output:
[1/144*(3*sqrt(a)*b^3*x^9*log((b*x^3 - 2*sqrt(b*x^3 + a)*sqrt(a) + 2*a)/x^ 3) + 2*(3*a*b^2*x^6 - 2*a^2*b*x^3 - 8*a^3)*sqrt(b*x^3 + a))/(a^3*x^9), 1/7 2*(3*sqrt(-a)*b^3*x^9*arctan(sqrt(-a)/sqrt(b*x^3 + a)) + (3*a*b^2*x^6 - 2* a^2*b*x^3 - 8*a^3)*sqrt(b*x^3 + a))/(a^3*x^9)]
Time = 5.28 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=- \frac {a}{9 \sqrt {b} x^{\frac {21}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {5 \sqrt {b}}{36 x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {b^{\frac {3}{2}}}{72 a x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {b^{\frac {5}{2}}}{24 a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{24 a^{\frac {5}{2}}} \] Input:
integrate((b*x**3+a)**(1/2)/x**10,x)
Output:
-a/(9*sqrt(b)*x**(21/2)*sqrt(a/(b*x**3) + 1)) - 5*sqrt(b)/(36*x**(15/2)*sq rt(a/(b*x**3) + 1)) + b**(3/2)/(72*a*x**(9/2)*sqrt(a/(b*x**3) + 1)) + b**( 5/2)/(24*a**2*x**(3/2)*sqrt(a/(b*x**3) + 1)) - b**3*asinh(sqrt(a)/(sqrt(b) *x**(3/2)))/(24*a**(5/2))
Time = 0.11 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=\frac {b^{3} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{48 \, a^{\frac {5}{2}}} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} b^{3} - 8 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a b^{3} - 3 \, \sqrt {b x^{3} + a} a^{2} b^{3}}{72 \, {\left ({\left (b x^{3} + a\right )}^{3} a^{2} - 3 \, {\left (b x^{3} + a\right )}^{2} a^{3} + 3 \, {\left (b x^{3} + a\right )} a^{4} - a^{5}\right )}} \] Input:
integrate((b*x^3+a)^(1/2)/x^10,x, algorithm="maxima")
Output:
1/48*b^3*log((sqrt(b*x^3 + a) - sqrt(a))/(sqrt(b*x^3 + a) + sqrt(a)))/a^(5 /2) + 1/72*(3*(b*x^3 + a)^(5/2)*b^3 - 8*(b*x^3 + a)^(3/2)*a*b^3 - 3*sqrt(b *x^3 + a)*a^2*b^3)/((b*x^3 + a)^3*a^2 - 3*(b*x^3 + a)^2*a^3 + 3*(b*x^3 + a )*a^4 - a^5)
Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=\frac {1}{72} \, b^{3} {\left (\frac {3 \, \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {5}{2}} - 8 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} a - 3 \, \sqrt {b x^{3} + a} a^{2}}{a^{2} b^{3} x^{9}}\right )} \] Input:
integrate((b*x^3+a)^(1/2)/x^10,x, algorithm="giac")
Output:
1/72*b^3*(3*arctan(sqrt(b*x^3 + a)/sqrt(-a))/(sqrt(-a)*a^2) + (3*(b*x^3 + a)^(5/2) - 8*(b*x^3 + a)^(3/2)*a - 3*sqrt(b*x^3 + a)*a^2)/(a^2*b^3*x^9))
Time = 0.69 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=\frac {b^3\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{48\,a^{5/2}}-\frac {\sqrt {b\,x^3+a}}{9\,x^9}-\frac {b\,\sqrt {b\,x^3+a}}{36\,a\,x^6}+\frac {b^2\,\sqrt {b\,x^3+a}}{24\,a^2\,x^3} \] Input:
int((a + b*x^3)^(1/2)/x^10,x)
Output:
(b^3*log((((a + b*x^3)^(1/2) - a^(1/2))^3*((a + b*x^3)^(1/2) + a^(1/2)))/x ^6))/(48*a^(5/2)) - (a + b*x^3)^(1/2)/(9*x^9) - (b*(a + b*x^3)^(1/2))/(36* a*x^6) + (b^2*(a + b*x^3)^(1/2))/(24*a^2*x^3)
Time = 0.21 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {a+b x^3}}{x^{10}} \, dx=\frac {-16 \sqrt {b \,x^{3}+a}\, a^{3}-4 \sqrt {b \,x^{3}+a}\, a^{2} b \,x^{3}+6 \sqrt {b \,x^{3}+a}\, a \,b^{2} x^{6}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {a}\right ) b^{3} x^{9}-3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {a}\right ) b^{3} x^{9}}{144 a^{3} x^{9}} \] Input:
int((b*x^3+a)^(1/2)/x^10,x)
Output:
( - 16*sqrt(a + b*x**3)*a**3 - 4*sqrt(a + b*x**3)*a**2*b*x**3 + 6*sqrt(a + b*x**3)*a*b**2*x**6 + 3*sqrt(a)*log(sqrt(a + b*x**3) - sqrt(a))*b**3*x**9 - 3*sqrt(a)*log(sqrt(a + b*x**3) + sqrt(a))*b**3*x**9)/(144*a**3*x**9)