Integrand size = 16, antiderivative size = 62 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x} \, dx=\frac {1}{2} a \sqrt {a-b x^4}+\frac {1}{6} \left (a-b x^4\right )^{3/2}-\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right ) \] Output:
1/2*a*(-b*x^4+a)^(1/2)+1/6*(-b*x^4+a)^(3/2)-1/2*a^(3/2)*arctanh((-b*x^4+a) ^(1/2)/a^(1/2))
Time = 0.04 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x} \, dx=\frac {1}{6} \sqrt {a-b x^4} \left (4 a-b x^4\right )-\frac {1}{2} a^{3/2} \text {arctanh}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right ) \] Input:
Integrate[(a - b*x^4)^(3/2)/x,x]
Output:
(Sqrt[a - b*x^4]*(4*a - b*x^4))/6 - (a^(3/2)*ArcTanh[Sqrt[a - b*x^4]/Sqrt[ a]])/2
Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, 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.
\(\displaystyle \int \frac {\left (a-b x^4\right )^{3/2}}{x} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int \frac {\left (a-b x^4\right )^{3/2}}{x^4}dx^4\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (a \int \frac {\sqrt {a-b x^4}}{x^4}dx^4+\frac {2}{3} \left (a-b x^4\right )^{3/2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (a \left (a \int \frac {1}{x^4 \sqrt {a-b x^4}}dx^4+2 \sqrt {a-b x^4}\right )+\frac {2}{3} \left (a-b x^4\right )^{3/2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (a \left (2 \sqrt {a-b x^4}-\frac {2 a \int \frac {1}{\frac {a}{b}-\frac {x^8}{b}}d\sqrt {a-b x^4}}{b}\right )+\frac {2}{3} \left (a-b x^4\right )^{3/2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{4} \left (a \left (2 \sqrt {a-b x^4}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )\right )+\frac {2}{3} \left (a-b x^4\right )^{3/2}\right )\) |
Input:
Int[(a - b*x^4)^(3/2)/x,x]
Output:
((2*(a - b*x^4)^(3/2))/3 + a*(2*Sqrt[a - b*x^4] - 2*Sqrt[a]*ArcTanh[Sqrt[a - b*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 = 0.58 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.71
method | result | size |
pseudoelliptic | \(-\frac {a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {-b \,x^{4}+a}}{\sqrt {a}}\right )}{2}+\frac {\sqrt {-b \,x^{4}+a}\, \left (-b \,x^{4}+4 a \right )}{6}\) | \(44\) |
default | \(-\frac {a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{2}-\frac {x^{4} b \sqrt {-b \,x^{4}+a}}{6}+\frac {2 a \sqrt {-b \,x^{4}+a}}{3}\) | \(60\) |
elliptic | \(-\frac {a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{2}-\frac {x^{4} b \sqrt {-b \,x^{4}+a}}{6}+\frac {2 a \sqrt {-b \,x^{4}+a}}{3}\) | \(60\) |
Input:
int((-b*x^4+a)^(3/2)/x,x,method=_RETURNVERBOSE)
Output:
-1/2*a^(3/2)*arctanh((-b*x^4+a)^(1/2)/a^(1/2))+1/6*(-b*x^4+a)^(1/2)*(-b*x^ 4+4*a)
Time = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x} \, dx=\left [\frac {1}{4} \, a^{\frac {3}{2}} \log \left (\frac {b x^{4} + 2 \, \sqrt {-b x^{4} + a} \sqrt {a} - 2 \, a}{x^{4}}\right ) - \frac {1}{6} \, {\left (b x^{4} - 4 \, a\right )} \sqrt {-b x^{4} + a}, -\frac {1}{2} \, \sqrt {-a} a \arctan \left (\frac {\sqrt {-b x^{4} + a} \sqrt {-a}}{b x^{4} - a}\right ) - \frac {1}{6} \, {\left (b x^{4} - 4 \, a\right )} \sqrt {-b x^{4} + a}\right ] \] Input:
integrate((-b*x^4+a)^(3/2)/x,x, algorithm="fricas")
Output:
[1/4*a^(3/2)*log((b*x^4 + 2*sqrt(-b*x^4 + a)*sqrt(a) - 2*a)/x^4) - 1/6*(b* x^4 - 4*a)*sqrt(-b*x^4 + a), -1/2*sqrt(-a)*a*arctan(sqrt(-b*x^4 + a)*sqrt( -a)/(b*x^4 - a)) - 1/6*(b*x^4 - 4*a)*sqrt(-b*x^4 + a)]
Result contains complex when optimal does not.
Time = 1.44 (sec) , antiderivative size = 196, normalized size of antiderivative = 3.16 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x} \, dx=\begin {cases} \frac {2 i a^{\frac {3}{2}} \sqrt {-1 + \frac {b x^{4}}{a}}}{3} + \frac {a^{\frac {3}{2}} \log {\left (\frac {b x^{4}}{a} \right )}}{4} - \frac {a^{\frac {3}{2}} \log {\left (\frac {\sqrt {b} x^{2}}{\sqrt {a}} \right )}}{2} + \frac {i a^{\frac {3}{2}} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{2} - \frac {i \sqrt {a} b x^{4} \sqrt {-1 + \frac {b x^{4}}{a}}}{6} & \text {for}\: \left |{\frac {b x^{4}}{a}}\right | > 1 \\\frac {2 a^{\frac {3}{2}} \sqrt {1 - \frac {b x^{4}}{a}}}{3} + \frac {a^{\frac {3}{2}} \log {\left (\frac {b x^{4}}{a} \right )}}{4} - \frac {a^{\frac {3}{2}} \log {\left (\sqrt {1 - \frac {b x^{4}}{a}} + 1 \right )}}{2} - \frac {\sqrt {a} b x^{4} \sqrt {1 - \frac {b x^{4}}{a}}}{6} & \text {otherwise} \end {cases} \] Input:
integrate((-b*x**4+a)**(3/2)/x,x)
Output:
Piecewise((2*I*a**(3/2)*sqrt(-1 + b*x**4/a)/3 + a**(3/2)*log(b*x**4/a)/4 - a**(3/2)*log(sqrt(b)*x**2/sqrt(a))/2 + I*a**(3/2)*asin(sqrt(a)/(sqrt(b)*x **2))/2 - I*sqrt(a)*b*x**4*sqrt(-1 + b*x**4/a)/6, Abs(b*x**4/a) > 1), (2*a **(3/2)*sqrt(1 - b*x**4/a)/3 + a**(3/2)*log(b*x**4/a)/4 - a**(3/2)*log(sqr t(1 - b*x**4/a) + 1)/2 - sqrt(a)*b*x**4*sqrt(1 - b*x**4/a)/6, True))
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x} \, dx=\frac {1}{4} \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {-b x^{4} + a} - \sqrt {a}}{\sqrt {-b x^{4} + a} + \sqrt {a}}\right ) + \frac {1}{6} \, {\left (-b x^{4} + a\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {-b x^{4} + a} a \] Input:
integrate((-b*x^4+a)^(3/2)/x,x, algorithm="maxima")
Output:
1/4*a^(3/2)*log((sqrt(-b*x^4 + a) - sqrt(a))/(sqrt(-b*x^4 + a) + sqrt(a))) + 1/6*(-b*x^4 + a)^(3/2) + 1/2*sqrt(-b*x^4 + a)*a
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x} \, dx=\frac {a^{2} \arctan \left (\frac {\sqrt {-b x^{4} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a}} + \frac {1}{6} \, {\left (-b x^{4} + a\right )}^{\frac {3}{2}} + \frac {1}{2} \, \sqrt {-b x^{4} + a} a \] Input:
integrate((-b*x^4+a)^(3/2)/x,x, algorithm="giac")
Output:
1/2*a^2*arctan(sqrt(-b*x^4 + a)/sqrt(-a))/sqrt(-a) + 1/6*(-b*x^4 + a)^(3/2 ) + 1/2*sqrt(-b*x^4 + a)*a
Time = 0.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x} \, dx=\frac {a\,\sqrt {a-b\,x^4}}{2}-\frac {a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {a-b\,x^4}}{\sqrt {a}}\right )}{2}+\frac {{\left (a-b\,x^4\right )}^{3/2}}{6} \] Input:
int((a - b*x^4)^(3/2)/x,x)
Output:
(a*(a - b*x^4)^(1/2))/2 - (a^(3/2)*atanh((a - b*x^4)^(1/2)/a^(1/2)))/2 + ( a - b*x^4)^(3/2)/6
Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x} \, dx=\frac {2 \sqrt {-b \,x^{4}+a}\, a}{3}-\frac {\sqrt {-b \,x^{4}+a}\, b \,x^{4}}{6}+\frac {\sqrt {a}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}\right )}{2}\right )\right ) a}{2}-\frac {2 \sqrt {a}\, a}{3} \] Input:
int((-b*x^4+a)^(3/2)/x,x)
Output:
(4*sqrt(a - b*x**4)*a - sqrt(a - b*x**4)*b*x**4 + 3*sqrt(a)*log(tan(asin(( sqrt(b)*x**2)/sqrt(a))/2))*a - 4*sqrt(a)*a)/6