Integrand size = 15, antiderivative size = 59 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x} \, dx=\frac {1}{2} a \sqrt {a+c x^4}+\frac {1}{6} \left (a+c x^4\right )^{3/2}-\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right ) \] Output:
1/2*a*(c*x^4+a)^(1/2)+1/6*(c*x^4+a)^(3/2)-1/2*a^(3/2)*arctanh((c*x^4+a)^(1 /2)/a^(1/2))
Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x} \, dx=\frac {1}{6} \sqrt {a+c x^4} \left (4 a+c x^4\right )-\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right ) \] Input:
Integrate[(a + c*x^4)^(3/2)/x,x]
Output:
(Sqrt[a + c*x^4]*(4*a + c*x^4))/6 - (a^(3/2)*ArcTanh[Sqrt[a + c*x^4]/Sqrt[ a]])/2
Time = 0.29 (sec) , antiderivative size = 61, 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, 60, 60, 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.
\(\displaystyle \int \frac {\left (a+c x^4\right )^{3/2}}{x} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int \frac {\left (c x^4+a\right )^{3/2}}{x^4}dx^4\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (a \int \frac {\sqrt {c x^4+a}}{x^4}dx^4+\frac {2}{3} \left (a+c x^4\right )^{3/2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (a \left (a \int \frac {1}{x^4 \sqrt {c x^4+a}}dx^4+2 \sqrt {a+c x^4}\right )+\frac {2}{3} \left (a+c x^4\right )^{3/2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (a \left (\frac {2 a \int \frac {1}{\frac {x^8}{c}-\frac {a}{c}}d\sqrt {c x^4+a}}{c}+2 \sqrt {a+c x^4}\right )+\frac {2}{3} \left (a+c x^4\right )^{3/2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{4} \left (a \left (2 \sqrt {a+c x^4}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a+c x^4\right )^{3/2}\right )\) |
Input:
Int[(a + c*x^4)^(3/2)/x,x]
Output:
((2*(a + c*x^4)^(3/2))/3 + a*(2*Sqrt[a + c*x^4] - 2*Sqrt[a]*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]]))/4
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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 = 0.63 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(-\frac {a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {c \,x^{4}+a}}{\sqrt {a}}\right )}{2}+\frac {\sqrt {c \,x^{4}+a}\, \left (c \,x^{4}+4 a \right )}{6}\) | \(41\) |
default | \(-\frac {a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{2}+\frac {\sqrt {c \,x^{4}+a}\, c \,x^{4}}{6}+\frac {2 a \sqrt {c \,x^{4}+a}}{3}\) | \(57\) |
elliptic | \(-\frac {a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{2}+\frac {\sqrt {c \,x^{4}+a}\, c \,x^{4}}{6}+\frac {2 a \sqrt {c \,x^{4}+a}}{3}\) | \(57\) |
Input:
int((c*x^4+a)^(3/2)/x,x,method=_RETURNVERBOSE)
Output:
-1/2*a^(3/2)*arctanh((c*x^4+a)^(1/2)/a^(1/2))+1/6*(c*x^4+a)^(1/2)*(c*x^4+4 *a)
Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x} \, dx=\left [\frac {1}{4} \, a^{\frac {3}{2}} \log \left (\frac {c x^{4} - 2 \, \sqrt {c x^{4} + a} \sqrt {a} + 2 \, a}{x^{4}}\right ) + \frac {1}{6} \, {\left (c x^{4} + 4 \, a\right )} \sqrt {c x^{4} + a}, \frac {1}{2} \, \sqrt {-a} a \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{4} + a}}\right ) + \frac {1}{6} \, {\left (c x^{4} + 4 \, a\right )} \sqrt {c x^{4} + a}\right ] \] Input:
integrate((c*x^4+a)^(3/2)/x,x, algorithm="fricas")
Output:
[1/4*a^(3/2)*log((c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(a) + 2*a)/x^4) + 1/6*(c*x ^4 + 4*a)*sqrt(c*x^4 + a), 1/2*sqrt(-a)*a*arctan(sqrt(-a)/sqrt(c*x^4 + a)) + 1/6*(c*x^4 + 4*a)*sqrt(c*x^4 + a)]
Time = 1.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x} \, dx=\frac {2 a^{\frac {3}{2}} \sqrt {1 + \frac {c x^{4}}{a}}}{3} + \frac {a^{\frac {3}{2}} \log {\left (\frac {c x^{4}}{a} \right )}}{4} - \frac {a^{\frac {3}{2}} \log {\left (\sqrt {1 + \frac {c x^{4}}{a}} + 1 \right )}}{2} + \frac {\sqrt {a} c x^{4} \sqrt {1 + \frac {c x^{4}}{a}}}{6} \] Input:
integrate((c*x**4+a)**(3/2)/x,x)
Output:
2*a**(3/2)*sqrt(1 + c*x**4/a)/3 + a**(3/2)*log(c*x**4/a)/4 - a**(3/2)*log( sqrt(1 + c*x**4/a) + 1)/2 + sqrt(a)*c*x**4*sqrt(1 + c*x**4/a)/6
Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x} \, dx=\frac {1}{4} \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {c x^{4} + a} - \sqrt {a}}{\sqrt {c x^{4} + a} + \sqrt {a}}\right ) + \frac {1}{6} \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {c x^{4} + a} a \] Input:
integrate((c*x^4+a)^(3/2)/x,x, algorithm="maxima")
Output:
1/4*a^(3/2)*log((sqrt(c*x^4 + a) - sqrt(a))/(sqrt(c*x^4 + a) + sqrt(a))) + 1/6*(c*x^4 + a)^(3/2) + 1/2*sqrt(c*x^4 + a)*a
Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x} \, dx=\frac {a^{2} \arctan \left (\frac {\sqrt {c x^{4} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a}} + \frac {1}{6} \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {c x^{4} + a} a \] Input:
integrate((c*x^4+a)^(3/2)/x,x, algorithm="giac")
Output:
1/2*a^2*arctan(sqrt(c*x^4 + a)/sqrt(-a))/sqrt(-a) + 1/6*(c*x^4 + a)^(3/2) + 1/2*sqrt(c*x^4 + a)*a
Time = 0.37 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x} \, dx=\frac {a\,\sqrt {c\,x^4+a}}{2}-\frac {a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {c\,x^4+a}}{\sqrt {a}}\right )}{2}+\frac {{\left (c\,x^4+a\right )}^{3/2}}{6} \] Input:
int((a + c*x^4)^(3/2)/x,x)
Output:
(a*(a + c*x^4)^(1/2))/2 - (a^(3/2)*atanh((a + c*x^4)^(1/2)/a^(1/2)))/2 + ( a + c*x^4)^(3/2)/6
Time = 0.20 (sec) , antiderivative size = 432, normalized size of antiderivative = 7.32 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{x} \, dx=\frac {3 \sqrt {a}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}-\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a^{2}+12 \sqrt {a}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}-\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a c \,x^{4}-3 \sqrt {a}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a^{2}-12 \sqrt {a}\, \sqrt {c \,x^{4}+a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a c \,x^{4}+12 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a^{2} x^{2}+19 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, a c \,x^{6}+4 \sqrt {c}\, \sqrt {c \,x^{4}+a}\, c^{2} x^{10}+9 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}-\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a^{2} x^{2}+12 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}-\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a c \,x^{6}-9 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a^{2} x^{2}-12 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{4}+a}+\sqrt {a}+\sqrt {c}\, x^{2}}{\sqrt {a}}\right ) a c \,x^{6}+4 a^{3}+21 a^{2} c \,x^{4}+21 a \,c^{2} x^{8}+4 c^{3} x^{12}}{6 \sqrt {c \,x^{4}+a}\, a +24 \sqrt {c \,x^{4}+a}\, c \,x^{4}+18 \sqrt {c}\, a \,x^{2}+24 \sqrt {c}\, c \,x^{6}} \] Input:
int((c*x^4+a)^(3/2)/x,x)
Output:
(3*sqrt(a)*sqrt(a + c*x**4)*log((sqrt(a + c*x**4) - sqrt(a) + sqrt(c)*x**2 )/sqrt(a))*a**2 + 12*sqrt(a)*sqrt(a + c*x**4)*log((sqrt(a + c*x**4) - sqrt (a) + sqrt(c)*x**2)/sqrt(a))*a*c*x**4 - 3*sqrt(a)*sqrt(a + c*x**4)*log((sq rt(a + c*x**4) + sqrt(a) + sqrt(c)*x**2)/sqrt(a))*a**2 - 12*sqrt(a)*sqrt(a + c*x**4)*log((sqrt(a + c*x**4) + sqrt(a) + sqrt(c)*x**2)/sqrt(a))*a*c*x* *4 + 12*sqrt(c)*sqrt(a + c*x**4)*a**2*x**2 + 19*sqrt(c)*sqrt(a + c*x**4)*a *c*x**6 + 4*sqrt(c)*sqrt(a + c*x**4)*c**2*x**10 + 9*sqrt(c)*sqrt(a)*log((s qrt(a + c*x**4) - sqrt(a) + sqrt(c)*x**2)/sqrt(a))*a**2*x**2 + 12*sqrt(c)* sqrt(a)*log((sqrt(a + c*x**4) - sqrt(a) + sqrt(c)*x**2)/sqrt(a))*a*c*x**6 - 9*sqrt(c)*sqrt(a)*log((sqrt(a + c*x**4) + sqrt(a) + sqrt(c)*x**2)/sqrt(a ))*a**2*x**2 - 12*sqrt(c)*sqrt(a)*log((sqrt(a + c*x**4) + sqrt(a) + sqrt(c )*x**2)/sqrt(a))*a*c*x**6 + 4*a**3 + 21*a**2*c*x**4 + 21*a*c**2*x**8 + 4*c **3*x**12)/(6*(sqrt(a + c*x**4)*a + 4*sqrt(a + c*x**4)*c*x**4 + 3*sqrt(c)* a*x**2 + 4*sqrt(c)*c*x**6))