Integrand size = 15, antiderivative size = 69 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^9} \, dx=-\frac {a \sqrt {a+c x^4}}{8 x^8}-\frac {5 c \sqrt {a+c x^4}}{16 x^4}-\frac {3 c^2 \text {arctanh}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}} \] Output:
-1/8*a*(c*x^4+a)^(1/2)/x^8-5/16*c*(c*x^4+a)^(1/2)/x^4-3/16*c^2*arctanh((c* x^4+a)^(1/2)/a^(1/2))/a^(1/2)
Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^9} \, dx=\frac {\left (-2 a-5 c x^4\right ) \sqrt {a+c x^4}}{16 x^8}-\frac {3 c^2 \text {arctanh}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{16 \sqrt {a}} \] Input:
Integrate[(a + c*x^4)^(3/2)/x^9,x]
Output:
((-2*a - 5*c*x^4)*Sqrt[a + c*x^4])/(16*x^8) - (3*c^2*ArcTanh[Sqrt[a + c*x^ 4]/Sqrt[a]])/(16*Sqrt[a])
Time = 0.30 (sec) , antiderivative size = 71, 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, 51, 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.
\(\displaystyle \int \frac {\left (a+c x^4\right )^{3/2}}{x^9} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int \frac {\left (c x^4+a\right )^{3/2}}{x^{12}}dx^4\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{4} \left (\frac {3}{4} c \int \frac {\sqrt {c x^4+a}}{x^8}dx^4-\frac {\left (a+c x^4\right )^{3/2}}{2 x^8}\right )\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{4} \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{x^4 \sqrt {c x^4+a}}dx^4-\frac {\sqrt {a+c x^4}}{x^4}\right )-\frac {\left (a+c x^4\right )^{3/2}}{2 x^8}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (\frac {3}{4} c \left (\int \frac {1}{\frac {x^8}{c}-\frac {a}{c}}d\sqrt {c x^4+a}-\frac {\sqrt {a+c x^4}}{x^4}\right )-\frac {\left (a+c x^4\right )^{3/2}}{2 x^8}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{4} \left (\frac {3}{4} c \left (-\frac {c \text {arctanh}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+c x^4}}{x^4}\right )-\frac {\left (a+c x^4\right )^{3/2}}{2 x^8}\right )\) |
Input:
Int[(a + c*x^4)^(3/2)/x^9,x]
Output:
(-1/2*(a + c*x^4)^(3/2)/x^8 + (3*c*(-(Sqrt[a + c*x^4]/x^4) - (c*ArcTanh[Sq rt[a + c*x^4]/Sqrt[a]])/Sqrt[a]))/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] :> 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 = 0.66 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83
method | result | size |
risch | \(-\frac {\sqrt {c \,x^{4}+a}\, \left (5 c \,x^{4}+2 a \right )}{16 x^{8}}-\frac {3 c^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}\) | \(57\) |
default | \(-\frac {a \sqrt {c \,x^{4}+a}}{8 x^{8}}-\frac {5 c \sqrt {c \,x^{4}+a}}{16 x^{4}}-\frac {3 c^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}\) | \(63\) |
elliptic | \(-\frac {a \sqrt {c \,x^{4}+a}}{8 x^{8}}-\frac {5 c \sqrt {c \,x^{4}+a}}{16 x^{4}}-\frac {3 c^{2} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{16 \sqrt {a}}\) | \(63\) |
pseudoelliptic | \(\frac {-3 \,\operatorname {arctanh}\left (\frac {\sqrt {c \,x^{4}+a}}{\sqrt {a}}\right ) c^{2} x^{8}-5 c \,x^{4} \sqrt {c \,x^{4}+a}\, \sqrt {a}-2 \sqrt {c \,x^{4}+a}\, a^{\frac {3}{2}}}{16 x^{8} \sqrt {a}}\) | \(64\) |
Input:
int((c*x^4+a)^(3/2)/x^9,x,method=_RETURNVERBOSE)
Output:
-1/16*(c*x^4+a)^(1/2)*(5*c*x^4+2*a)/x^8-3/16*c^2/a^(1/2)*ln((2*a+2*a^(1/2) *(c*x^4+a)^(1/2))/x^2)
Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.96 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^9} \, dx=\left [\frac {3 \, \sqrt {a} c^{2} x^{8} \log \left (\frac {c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) - 2 \, {\left (5 \, a c x^{4} + 2 \, a^{2}\right )} \sqrt {c x^{4} + a}}{32 \, a x^{8}}, \frac {3 \, \sqrt {-a} c^{2} x^{8} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{4} + a}}\right ) - {\left (5 \, a c x^{4} + 2 \, a^{2}\right )} \sqrt {c x^{4} + a}}{16 \, a x^{8}}\right ] \] Input:
integrate((c*x^4+a)^(3/2)/x^9,x, algorithm="fricas")
Output:
[1/32*(3*sqrt(a)*c^2*x^8*log((c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(a) + 2*a)/x^4 ) - 2*(5*a*c*x^4 + 2*a^2)*sqrt(c*x^4 + a))/(a*x^8), 1/16*(3*sqrt(-a)*c^2*x ^8*arctan(sqrt(-a)/sqrt(c*x^4 + a)) - (5*a*c*x^4 + 2*a^2)*sqrt(c*x^4 + a)) /(a*x^8)]
Time = 1.78 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^9} \, dx=- \frac {a \sqrt {c} \sqrt {\frac {a}{c x^{4}} + 1}}{8 x^{6}} - \frac {5 c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{4}} + 1}}{16 x^{2}} - \frac {3 c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x^{2}} \right )}}{16 \sqrt {a}} \] Input:
integrate((c*x**4+a)**(3/2)/x**9,x)
Output:
-a*sqrt(c)*sqrt(a/(c*x**4) + 1)/(8*x**6) - 5*c**(3/2)*sqrt(a/(c*x**4) + 1) /(16*x**2) - 3*c**2*asinh(sqrt(a)/(sqrt(c)*x**2))/(16*sqrt(a))
Time = 0.12 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.42 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^9} \, dx=\frac {3 \, c^{2} \log \left (\frac {\sqrt {c x^{4} + a} - \sqrt {a}}{\sqrt {c x^{4} + a} + \sqrt {a}}\right )}{32 \, \sqrt {a}} - \frac {5 \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} c^{2} - 3 \, \sqrt {c x^{4} + a} a c^{2}}{16 \, {\left ({\left (c x^{4} + a\right )}^{2} - 2 \, {\left (c x^{4} + a\right )} a + a^{2}\right )}} \] Input:
integrate((c*x^4+a)^(3/2)/x^9,x, algorithm="maxima")
Output:
3/32*c^2*log((sqrt(c*x^4 + a) - sqrt(a))/(sqrt(c*x^4 + a) + sqrt(a)))/sqrt (a) - 1/16*(5*(c*x^4 + a)^(3/2)*c^2 - 3*sqrt(c*x^4 + a)*a*c^2)/((c*x^4 + a )^2 - 2*(c*x^4 + a)*a + a^2)
Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^9} \, dx=\frac {\frac {3 \, c^{3} \arctan \left (\frac {\sqrt {c x^{4} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {5 \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} c^{3} - 3 \, \sqrt {c x^{4} + a} a c^{3}}{c^{2} x^{8}}}{16 \, c} \] Input:
integrate((c*x^4+a)^(3/2)/x^9,x, algorithm="giac")
Output:
1/16*(3*c^3*arctan(sqrt(c*x^4 + a)/sqrt(-a))/sqrt(-a) - (5*(c*x^4 + a)^(3/ 2)*c^3 - 3*sqrt(c*x^4 + a)*a*c^3)/(c^2*x^8))/c
Time = 0.77 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.75 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^9} \, dx=\frac {3\,a\,\sqrt {c\,x^4+a}}{16\,x^8}-\frac {3\,c^2\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^4+a}}{\sqrt {a}}\right )}{16\,\sqrt {a}}-\frac {5\,{\left (c\,x^4+a\right )}^{3/2}}{16\,x^8} \] Input:
int((a + c*x^4)^(3/2)/x^9,x)
Output:
(3*a*(a + c*x^4)^(1/2))/(16*x^8) - (3*c^2*atanh((a + c*x^4)^(1/2)/a^(1/2)) )/(16*a^(1/2)) - (5*(a + c*x^4)^(3/2))/(16*x^8)
Time = 0.23 (sec) , antiderivative size = 554, normalized size of antiderivative = 8.03 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x^9} \, dx=\frac {-2 \sqrt {a}\, \sqrt {c \,x^{4}+a}\, a^{3}-21 \sqrt {a}\, \sqrt {c \,x^{4}+a}\, a^{2} c \,x^{4}-56 \sqrt {a}\, \sqrt {c \,x^{4}+a}\, a \,c^{2} x^{8}-40 \sqrt {a}\, \sqrt {c \,x^{4}+a}\, c^{3} x^{12}+12 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}-\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a \,c^{2} x^{10}+24 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}-\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) c^{3} x^{14}-12 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a \,c^{2} x^{10}-24 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) c^{3} x^{14}-8 \sqrt {c}\, \sqrt {a}\, a^{3} x^{2}-44 \sqrt {c}\, \sqrt {a}\, a^{2} c \,x^{6}-76 \sqrt {c}\, \sqrt {a}\, a \,c^{2} x^{10}-40 \sqrt {c}\, \sqrt {a}\, c^{3} x^{14}+3 \,\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}-\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a^{2} c^{2} x^{8}+24 \,\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}-\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a \,c^{3} x^{12}+24 \,\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}-\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) c^{4} x^{16}-3 \,\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a^{2} c^{2} x^{8}-24 \,\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a \,c^{3} x^{12}-24 \,\mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) c^{4} x^{16}}{16 \sqrt {a}\, x^{8} \left (4 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a \,x^{2}+8 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, c \,x^{6}+a^{2}+8 a c \,x^{4}+8 c^{2} x^{8}\right )} \] Input:
int((c*x^4+a)^(3/2)/x^9,x)
Output:
( - 2*sqrt(a)*sqrt(a + c*x**4)*a**3 - 21*sqrt(a)*sqrt(a + c*x**4)*a**2*c*x **4 - 56*sqrt(a)*sqrt(a + c*x**4)*a*c**2*x**8 - 40*sqrt(a)*sqrt(a + c*x**4 )*c**3*x**12 + 12*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 + 24*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 - 12*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 - 24*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 - 44* sqrt(c)*sqrt(a)*a**2*c*x**6 - 76*sqrt(c)*sqrt(a)*a*c**2*x**10 - 40*sqrt(c) *sqrt(a)*c**3*x**14 + 3*log((sqrt(a + c*x**4) - sqrt(a) + sqrt(c)*x**2)/sq rt(a))*a**2*c**2*x**8 + 24*log((sqrt(a + c*x**4) - sqrt(a) + sqrt(c)*x**2) /sqrt(a))*a*c**3*x**12 + 24*log((sqrt(a + c*x**4) - sqrt(a) + sqrt(c)*x**2 )/sqrt(a))*c**4*x**16 - 3*log((sqrt(a + c*x**4) + sqrt(a) + sqrt(c)*x**2)/ sqrt(a))*a**2*c**2*x**8 - 24*log((sqrt(a + c*x**4) + sqrt(a) + sqrt(c)*x** 2)/sqrt(a))*a*c**3*x**12 - 24*log((sqrt(a + c*x**4) + sqrt(a) + sqrt(c)*x* *2)/sqrt(a))*c**4*x**16)/(16*sqrt(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))