Integrand size = 15, antiderivative size = 71 \[ \int \frac {\sqrt {a+c x^4}}{x^9} \, dx=-\frac {\sqrt {a+c x^4}}{8 x^8}-\frac {c \sqrt {a+c x^4}}{16 a x^4}+\frac {c^2 \text {arctanh}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{16 a^{3/2}} \] Output:
-1/8*(c*x^4+a)^(1/2)/x^8-1/16*c*(c*x^4+a)^(1/2)/a/x^4+1/16*c^2*arctanh((c* x^4+a)^(1/2)/a^(1/2))/a^(3/2)
Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {a+c x^4}}{x^9} \, dx=\frac {\left (-2 a-c x^4\right ) \sqrt {a+c x^4}}{16 a x^8}+\frac {c^2 \text {arctanh}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{16 a^{3/2}} \] Input:
Integrate[Sqrt[a + c*x^4]/x^9,x]
Output:
((-2*a - c*x^4)*Sqrt[a + c*x^4])/(16*a*x^8) + (c^2*ArcTanh[Sqrt[a + c*x^4] /Sqrt[a]])/(16*a^(3/2))
Time = 0.31 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {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.
\(\displaystyle \int \frac {\sqrt {a+c x^4}}{x^9} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int \frac {\sqrt {c x^4+a}}{x^{12}}dx^4\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{4} c \int \frac {1}{x^8 \sqrt {c x^4+a}}dx^4-\frac {\sqrt {a+c x^4}}{2 x^8}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{4} c \left (-\frac {c \int \frac {1}{x^4 \sqrt {c x^4+a}}dx^4}{2 a}-\frac {\sqrt {a+c x^4}}{a x^4}\right )-\frac {\sqrt {a+c x^4}}{2 x^8}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{4} c \left (-\frac {\int \frac {1}{\frac {x^8}{c}-\frac {a}{c}}d\sqrt {c x^4+a}}{a}-\frac {\sqrt {a+c x^4}}{a x^4}\right )-\frac {\sqrt {a+c x^4}}{2 x^8}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{4} \left (\frac {1}{4} c \left (\frac {c \text {arctanh}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+c x^4}}{a x^4}\right )-\frac {\sqrt {a+c x^4}}{2 x^8}\right )\) |
Input:
Int[Sqrt[a + c*x^4]/x^9,x]
Output:
(-1/2*Sqrt[a + c*x^4]/x^8 + (c*(-(Sqrt[a + c*x^4]/(a*x^4)) + (c*ArcTanh[Sq rt[a + c*x^4]/Sqrt[a]])/a^(3/2)))/4)/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 + 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 = 0.69 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\frac {\operatorname {arctanh}\left (\frac {\sqrt {c \,x^{4}+a}}{\sqrt {a}}\right ) c^{2} x^{8}-\left (c \,x^{4} \sqrt {a}+2 a^{\frac {3}{2}}\right ) \sqrt {c \,x^{4}+a}}{16 a^{\frac {3}{2}} x^{8}}\) | \(56\) |
risch | \(-\frac {\sqrt {c \,x^{4}+a}\, \left (c \,x^{4}+2 a \right )}{16 x^{8} a}+\frac {c^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}\) | \(59\) |
default | \(-\frac {\left (c \,x^{4}+a \right )^{\frac {3}{2}}}{8 a \,x^{8}}+\frac {c \left (c \,x^{4}+a \right )^{\frac {3}{2}}}{16 a^{2} x^{4}}+\frac {c^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}-\frac {c^{2} \sqrt {c \,x^{4}+a}}{16 a^{2}}\) | \(85\) |
elliptic | \(-\frac {\left (c \,x^{4}+a \right )^{\frac {3}{2}}}{8 a \,x^{8}}+\frac {c \left (c \,x^{4}+a \right )^{\frac {3}{2}}}{16 a^{2} x^{4}}+\frac {c^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{16 a^{\frac {3}{2}}}-\frac {c^{2} \sqrt {c \,x^{4}+a}}{16 a^{2}}\) | \(85\) |
Input:
int((c*x^4+a)^(1/2)/x^9,x,method=_RETURNVERBOSE)
Output:
1/16*(arctanh((c*x^4+a)^(1/2)/a^(1/2))*c^2*x^8-(c*x^4*a^(1/2)+2*a^(3/2))*( c*x^4+a)^(1/2))/a^(3/2)/x^8
Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.83 \[ \int \frac {\sqrt {a+c x^4}}{x^9} \, dx=\left [\frac {\sqrt {a} c^{2} x^{8} \log \left (\frac {c x^{4} + 2 \, \sqrt {c x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 2 \, {\left (a c x^{4} + 2 \, a^{2}\right )} \sqrt {c x^{4} + a}}{32 \, a^{2} x^{8}}, -\frac {\sqrt {-a} c^{2} x^{8} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{4} + a}}\right ) + {\left (a c x^{4} + 2 \, a^{2}\right )} \sqrt {c x^{4} + a}}{16 \, a^{2} x^{8}}\right ] \] Input:
integrate((c*x^4+a)^(1/2)/x^9,x, algorithm="fricas")
Output:
[1/32*(sqrt(a)*c^2*x^8*log((c*x^4 + 2*sqrt(c*x^4 + a)*sqrt(a) + 2*a)/x^4) - 2*(a*c*x^4 + 2*a^2)*sqrt(c*x^4 + a))/(a^2*x^8), -1/16*(sqrt(-a)*c^2*x^8* arctan(sqrt(-a)/sqrt(c*x^4 + a)) + (a*c*x^4 + 2*a^2)*sqrt(c*x^4 + a))/(a^2 *x^8)]
Time = 2.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {a+c x^4}}{x^9} \, dx=- \frac {a}{8 \sqrt {c} x^{10} \sqrt {\frac {a}{c x^{4}} + 1}} - \frac {3 \sqrt {c}}{16 x^{6} \sqrt {\frac {a}{c x^{4}} + 1}} - \frac {c^{\frac {3}{2}}}{16 a x^{2} \sqrt {\frac {a}{c x^{4}} + 1}} + \frac {c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x^{2}} \right )}}{16 a^{\frac {3}{2}}} \] Input:
integrate((c*x**4+a)**(1/2)/x**9,x)
Output:
-a/(8*sqrt(c)*x**10*sqrt(a/(c*x**4) + 1)) - 3*sqrt(c)/(16*x**6*sqrt(a/(c*x **4) + 1)) - c**(3/2)/(16*a*x**2*sqrt(a/(c*x**4) + 1)) + c**2*asinh(sqrt(a )/(sqrt(c)*x**2))/(16*a**(3/2))
Time = 0.10 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {a+c x^4}}{x^9} \, dx=-\frac {c^{2} \log \left (\frac {\sqrt {c x^{4} + a} - \sqrt {a}}{\sqrt {c x^{4} + a} + \sqrt {a}}\right )}{32 \, a^{\frac {3}{2}}} - \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}} c^{2} + \sqrt {c x^{4} + a} a c^{2}}{16 \, {\left ({\left (c x^{4} + a\right )}^{2} a - 2 \, {\left (c x^{4} + a\right )} a^{2} + a^{3}\right )}} \] Input:
integrate((c*x^4+a)^(1/2)/x^9,x, algorithm="maxima")
Output:
-1/32*c^2*log((sqrt(c*x^4 + a) - sqrt(a))/(sqrt(c*x^4 + a) + sqrt(a)))/a^( 3/2) - 1/16*((c*x^4 + a)^(3/2)*c^2 + sqrt(c*x^4 + a)*a*c^2)/((c*x^4 + a)^2 *a - 2*(c*x^4 + a)*a^2 + a^3)
Time = 0.12 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.01 \[ \int \frac {\sqrt {a+c x^4}}{x^9} \, dx=-\frac {\frac {c^{3} \arctan \left (\frac {\sqrt {c x^{4} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}} c^{3} + \sqrt {c x^{4} + a} a c^{3}}{a c^{2} x^{8}}}{16 \, c} \] Input:
integrate((c*x^4+a)^(1/2)/x^9,x, algorithm="giac")
Output:
-1/16*(c^3*arctan(sqrt(c*x^4 + a)/sqrt(-a))/(sqrt(-a)*a) + ((c*x^4 + a)^(3 /2)*c^3 + sqrt(c*x^4 + a)*a*c^3)/(a*c^2*x^8))/c
Time = 0.64 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a+c x^4}}{x^9} \, dx=\frac {c^2\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^4+a}}{\sqrt {a}}\right )}{16\,a^{3/2}}-\frac {\sqrt {c\,x^4+a}}{16\,x^8}-\frac {{\left (c\,x^4+a\right )}^{3/2}}{16\,a\,x^8} \] Input:
int((a + c*x^4)^(1/2)/x^9,x)
Output:
(c^2*atanh((a + c*x^4)^(1/2)/a^(1/2)))/(16*a^(3/2)) - (a + c*x^4)^(1/2)/(1 6*x^8) - (a + c*x^4)^(3/2)/(16*a*x^8)
Time = 0.27 (sec) , antiderivative size = 556, normalized size of antiderivative = 7.83 \[ \int \frac {\sqrt {a+c x^4}}{x^9} \, dx=\frac {-2 \sqrt {a}\, \sqrt {c \,x^{4}+a}\, a^{3}-17 \sqrt {a}\, \sqrt {c \,x^{4}+a}\, a^{2} c \,x^{4}-24 \sqrt {a}\, \sqrt {c \,x^{4}+a}\, a \,c^{2} x^{8}-8 \sqrt {a}\, \sqrt {c \,x^{4}+a}\, c^{3} x^{12}-4 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}-\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a \,c^{2} x^{10}-8 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}-\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) c^{3} x^{14}+4 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a \,c^{2} x^{10}+8 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) c^{3} x^{14}-8 \sqrt {c}\, \sqrt {a}\, a^{3} x^{2}-28 \sqrt {c}\, \sqrt {a}\, a^{2} c \,x^{6}-28 \sqrt {c}\, \sqrt {a}\, a \,c^{2} x^{10}-8 \sqrt {c}\, \sqrt {a}\, c^{3} x^{14}-\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}-\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a^{2} c^{2} x^{8}-8 \,\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}-\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a \,c^{3} x^{12}-8 \,\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}-\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) c^{4} x^{16}+\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a^{2} c^{2} x^{8}+8 \,\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a \,c^{3} x^{12}+8 \,\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) c^{4} x^{16}}{16 \sqrt {a}\, a \,x^{8} \left (4 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a \,x^{2}+8 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, c \,x^{6}+a^{2}+8 a c \,x^{4}+8 c^{2} x^{8}\right )} \] Input:
int((c*x^4+a)^(1/2)/x^9,x)
Output:
( - 2*sqrt(a)*sqrt(a + c*x**4)*a**3 - 17*sqrt(a)*sqrt(a + c*x**4)*a**2*c*x **4 - 24*sqrt(a)*sqrt(a + c*x**4)*a*c**2*x**8 - 8*sqrt(a)*sqrt(a + c*x**4) *c**3*x**12 - 4*sqrt(c)*sqrt(a + c*x**4)*log((sqrt(a + c*x**4) - sqrt(a) + sqrt(c)*x**2)/sqrt(a))*a*c**2*x**10 - 8*sqrt(c)*sqrt(a + c*x**4)*log((sqr t(a + c*x**4) - sqrt(a) + sqrt(c)*x**2)/sqrt(a))*c**3*x**14 + 4*sqrt(c)*sq rt(a + c*x**4)*log((sqrt(a + c*x**4) + sqrt(a) + sqrt(c)*x**2)/sqrt(a))*a* c**2*x**10 + 8*sqrt(c)*sqrt(a + c*x**4)*log((sqrt(a + c*x**4) + sqrt(a) + sqrt(c)*x**2)/sqrt(a))*c**3*x**14 - 8*sqrt(c)*sqrt(a)*a**3*x**2 - 28*sqrt( c)*sqrt(a)*a**2*c*x**6 - 28*sqrt(c)*sqrt(a)*a*c**2*x**10 - 8*sqrt(c)*sqrt( a)*c**3*x**14 - log((sqrt(a + c*x**4) - sqrt(a) + sqrt(c)*x**2)/sqrt(a))*a **2*c**2*x**8 - 8*log((sqrt(a + c*x**4) - sqrt(a) + sqrt(c)*x**2)/sqrt(a)) *a*c**3*x**12 - 8*log((sqrt(a + c*x**4) - sqrt(a) + sqrt(c)*x**2)/sqrt(a)) *c**4*x**16 + log((sqrt(a + c*x**4) + sqrt(a) + sqrt(c)*x**2)/sqrt(a))*a** 2*c**2*x**8 + 8*log((sqrt(a + c*x**4) + sqrt(a) + sqrt(c)*x**2)/sqrt(a))*a *c**3*x**12 + 8*log((sqrt(a + c*x**4) + sqrt(a) + sqrt(c)*x**2)/sqrt(a))*c **4*x**16)/(16*sqrt(a)*a*x**8*(4*sqrt(c)*sqrt(a + c*x**4)*a*x**2 + 8*sqrt( c)*sqrt(a + c*x**4)*c*x**6 + a**2 + 8*a*c*x**4 + 8*c**2*x**8))