Integrand size = 15, antiderivative size = 74 \[ \int \frac {1}{x^7 \sqrt {a+b x^3}} \, dx=-\frac {\sqrt {a+b x^3}}{6 a x^6}+\frac {b \sqrt {a+b x^3}}{4 a^2 x^3}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 a^{5/2}} \] Output:
-1/6*(b*x^3+a)^(1/2)/a/x^6+1/4*b*(b*x^3+a)^(1/2)/a^2/x^3-1/4*b^2*arctanh(( b*x^3+a)^(1/2)/a^(1/2))/a^(5/2)
Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^7 \sqrt {a+b x^3}} \, dx=\frac {\sqrt {a+b x^3} \left (-2 a+3 b x^3\right )}{12 a^2 x^6}-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{4 a^{5/2}} \] Input:
Integrate[1/(x^7*Sqrt[a + b*x^3]),x]
Output:
(Sqrt[a + b*x^3]*(-2*a + 3*b*x^3))/(12*a^2*x^6) - (b^2*ArcTanh[Sqrt[a + b* x^3]/Sqrt[a]])/(4*a^(5/2))
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {798, 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.
\(\displaystyle \int \frac {1}{x^7 \sqrt {a+b x^3}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{3} \int \frac {1}{x^9 \sqrt {b x^3+a}}dx^3\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (-\frac {3 b \int \frac {1}{x^6 \sqrt {b x^3+a}}dx^3}{4 a}-\frac {\sqrt {a+b x^3}}{2 a x^6}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{3} \left (-\frac {3 b \left (-\frac {b \int \frac {1}{x^3 \sqrt {b x^3+a}}dx^3}{2 a}-\frac {\sqrt {a+b x^3}}{a x^3}\right )}{4 a}-\frac {\sqrt {a+b x^3}}{2 a x^6}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (-\frac {3 b \left (-\frac {\int \frac {1}{\frac {x^6}{b}-\frac {a}{b}}d\sqrt {b x^3+a}}{a}-\frac {\sqrt {a+b x^3}}{a x^3}\right )}{4 a}-\frac {\sqrt {a+b x^3}}{2 a x^6}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{3} \left (-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b x^3}}{a x^3}\right )}{4 a}-\frac {\sqrt {a+b x^3}}{2 a x^6}\right )\) |
Input:
Int[1/(x^7*Sqrt[a + b*x^3]),x]
Output:
(-1/2*Sqrt[a + b*x^3]/(a*x^6) - (3*b*(-(Sqrt[a + b*x^3]/(a*x^3)) + (b*ArcT anh[Sqrt[a + b*x^3]/Sqrt[a]])/a^(3/2)))/(4*a))/3
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 = 0.51 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.69
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{3}+a}\, \left (-3 b \,x^{3}+2 a \right )}{12 a^{2} x^{6}}-\frac {b^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{4 a^{\frac {5}{2}}}\) | \(51\) |
default | \(-\frac {\sqrt {b \,x^{3}+a}}{6 a \,x^{6}}+\frac {b \sqrt {b \,x^{3}+a}}{4 a^{2} x^{3}}-\frac {b^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{4 a^{\frac {5}{2}}}\) | \(59\) |
elliptic | \(-\frac {\sqrt {b \,x^{3}+a}}{6 a \,x^{6}}+\frac {b \sqrt {b \,x^{3}+a}}{4 a^{2} x^{3}}-\frac {b^{2} \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{4 a^{\frac {5}{2}}}\) | \(59\) |
pseudoelliptic | \(\frac {-3 \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right ) b^{2} x^{6}+3 b \,x^{3} \sqrt {b \,x^{3}+a}\, \sqrt {a}-2 a^{\frac {3}{2}} \sqrt {b \,x^{3}+a}}{12 a^{\frac {5}{2}} x^{6}}\) | \(64\) |
Input:
int(1/x^7/(b*x^3+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/12*(b*x^3+a)^(1/2)*(-3*b*x^3+2*a)/a^2/x^6-1/4*b^2*arctanh((b*x^3+a)^(1/ 2)/a^(1/2))/a^(5/2)
Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.81 \[ \int \frac {1}{x^7 \sqrt {a+b x^3}} \, dx=\left [\frac {3 \, \sqrt {a} b^{2} x^{6} \log \left (\frac {b x^{3} - 2 \, \sqrt {b x^{3} + a} \sqrt {a} + 2 \, a}{x^{3}}\right ) + 2 \, {\left (3 \, a b x^{3} - 2 \, a^{2}\right )} \sqrt {b x^{3} + a}}{24 \, a^{3} x^{6}}, \frac {3 \, \sqrt {-a} b^{2} x^{6} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{3} + a}}\right ) + {\left (3 \, a b x^{3} - 2 \, a^{2}\right )} \sqrt {b x^{3} + a}}{12 \, a^{3} x^{6}}\right ] \] Input:
integrate(1/x^7/(b*x^3+a)^(1/2),x, algorithm="fricas")
Output:
[1/24*(3*sqrt(a)*b^2*x^6*log((b*x^3 - 2*sqrt(b*x^3 + a)*sqrt(a) + 2*a)/x^3 ) + 2*(3*a*b*x^3 - 2*a^2)*sqrt(b*x^3 + a))/(a^3*x^6), 1/12*(3*sqrt(-a)*b^2 *x^6*arctan(sqrt(-a)/sqrt(b*x^3 + a)) + (3*a*b*x^3 - 2*a^2)*sqrt(b*x^3 + a ))/(a^3*x^6)]
Time = 2.63 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.41 \[ \int \frac {1}{x^7 \sqrt {a+b x^3}} \, dx=- \frac {1}{6 \sqrt {b} x^{\frac {15}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {\sqrt {b}}{12 a x^{\frac {9}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} + \frac {b^{\frac {3}{2}}}{4 a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x^{3}} + 1}} - \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{\frac {3}{2}}} \right )}}{4 a^{\frac {5}{2}}} \] Input:
integrate(1/x**7/(b*x**3+a)**(1/2),x)
Output:
-1/(6*sqrt(b)*x**(15/2)*sqrt(a/(b*x**3) + 1)) + sqrt(b)/(12*a*x**(9/2)*sqr t(a/(b*x**3) + 1)) + b**(3/2)/(4*a**2*x**(3/2)*sqrt(a/(b*x**3) + 1)) - b** 2*asinh(sqrt(a)/(sqrt(b)*x**(3/2)))/(4*a**(5/2))
Time = 0.11 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.41 \[ \int \frac {1}{x^7 \sqrt {a+b x^3}} \, dx=\frac {b^{2} \log \left (\frac {\sqrt {b x^{3} + a} - \sqrt {a}}{\sqrt {b x^{3} + a} + \sqrt {a}}\right )}{8 \, a^{\frac {5}{2}}} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{2} - 5 \, \sqrt {b x^{3} + a} a b^{2}}{12 \, {\left ({\left (b x^{3} + a\right )}^{2} a^{2} - 2 \, {\left (b x^{3} + a\right )} a^{3} + a^{4}\right )}} \] Input:
integrate(1/x^7/(b*x^3+a)^(1/2),x, algorithm="maxima")
Output:
1/8*b^2*log((sqrt(b*x^3 + a) - sqrt(a))/(sqrt(b*x^3 + a) + sqrt(a)))/a^(5/ 2) + 1/12*(3*(b*x^3 + a)^(3/2)*b^2 - 5*sqrt(b*x^3 + a)*a*b^2)/((b*x^3 + a) ^2*a^2 - 2*(b*x^3 + a)*a^3 + a^4)
Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^7 \sqrt {a+b x^3}} \, dx=\frac {\frac {3 \, b^{3} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{3} - 5 \, \sqrt {b x^{3} + a} a b^{3}}{a^{2} b^{2} x^{6}}}{12 \, b} \] Input:
integrate(1/x^7/(b*x^3+a)^(1/2),x, algorithm="giac")
Output:
1/12*(3*b^3*arctan(sqrt(b*x^3 + a)/sqrt(-a))/(sqrt(-a)*a^2) + (3*(b*x^3 + a)^(3/2)*b^3 - 5*sqrt(b*x^3 + a)*a*b^3)/(a^2*b^2*x^6))/b
Time = 0.52 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^7 \sqrt {a+b x^3}} \, dx=\frac {b^2\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{8\,a^{5/2}}-\frac {\sqrt {b\,x^3+a}}{6\,a\,x^6}+\frac {b\,\sqrt {b\,x^3+a}}{4\,a^2\,x^3} \] Input:
int(1/(x^7*(a + b*x^3)^(1/2)),x)
Output:
(b^2*log((((a + b*x^3)^(1/2) - a^(1/2))^3*((a + b*x^3)^(1/2) + a^(1/2)))/x ^6))/(8*a^(5/2)) - (a + b*x^3)^(1/2)/(6*a*x^6) + (b*(a + b*x^3)^(1/2))/(4* a^2*x^3)
Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^7 \sqrt {a+b x^3}} \, dx=\frac {-4 \sqrt {b \,x^{3}+a}\, a^{2}+6 \sqrt {b \,x^{3}+a}\, a b \,x^{3}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}-\sqrt {a}\right ) b^{2} x^{6}-3 \sqrt {a}\, \mathrm {log}\left (\sqrt {b \,x^{3}+a}+\sqrt {a}\right ) b^{2} x^{6}}{24 a^{3} x^{6}} \] Input:
int(1/x^7/(b*x^3+a)^(1/2),x)
Output:
( - 4*sqrt(a + b*x**3)*a**2 + 6*sqrt(a + b*x**3)*a*b*x**3 + 3*sqrt(a)*log( sqrt(a + b*x**3) - sqrt(a))*b**2*x**6 - 3*sqrt(a)*log(sqrt(a + b*x**3) + s qrt(a))*b**2*x**6)/(24*a**3*x**6)