Integrand size = 16, antiderivative size = 67 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^5} \, dx=-\frac {1}{2} b \sqrt {a-b x^4}-\frac {a \sqrt {a-b x^4}}{4 x^4}+\frac {3}{4} \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right ) \] Output:
-1/2*b*(-b*x^4+a)^(1/2)-1/4*a*(-b*x^4+a)^(1/2)/x^4+3/4*a^(1/2)*b*arctanh(( -b*x^4+a)^(1/2)/a^(1/2))
Time = 0.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^5} \, dx=\frac {\left (-a-2 b x^4\right ) \sqrt {a-b x^4}}{4 x^4}+\frac {3}{4} \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right ) \] Input:
Integrate[(a - b*x^4)^(3/2)/x^5,x]
Output:
((-a - 2*b*x^4)*Sqrt[a - b*x^4])/(4*x^4) + (3*Sqrt[a]*b*ArcTanh[Sqrt[a - b *x^4]/Sqrt[a]])/4
Time = 0.31 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {798, 51, 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.
\(\displaystyle \int \frac {\left (a-b x^4\right )^{3/2}}{x^5} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int \frac {\left (a-b x^4\right )^{3/2}}{x^8}dx^4\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {1}{4} \left (-\frac {3}{2} b \int \frac {\sqrt {a-b x^4}}{x^4}dx^4-\frac {\left (a-b x^4\right )^{3/2}}{x^4}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (-\frac {3}{2} b \left (a \int \frac {1}{x^4 \sqrt {a-b x^4}}dx^4+2 \sqrt {a-b x^4}\right )-\frac {\left (a-b x^4\right )^{3/2}}{x^4}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (-\frac {3}{2} b \left (2 \sqrt {a-b x^4}-\frac {2 a \int \frac {1}{\frac {a}{b}-\frac {x^8}{b}}d\sqrt {a-b x^4}}{b}\right )-\frac {\left (a-b x^4\right )^{3/2}}{x^4}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{4} \left (-\frac {3}{2} b \left (2 \sqrt {a-b x^4}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )\right )-\frac {\left (a-b x^4\right )^{3/2}}{x^4}\right )\) |
Input:
Int[(a - b*x^4)^(3/2)/x^5,x]
Output:
(-((a - b*x^4)^(3/2)/x^4) - (3*b*(2*Sqrt[a - b*x^4] - 2*Sqrt[a]*ArcTanh[Sq rt[a - b*x^4]/Sqrt[a]]))/2)/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/(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 = 0.64 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(\frac {3 \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {-b \,x^{4}+a}}{\sqrt {a}}\right ) b \,x^{4}-\sqrt {-b \,x^{4}+a}\, \left (2 b \,x^{4}+a \right )}{4 x^{4}}\) | \(51\) |
default | \(-\frac {b \sqrt {-b \,x^{4}+a}}{2}-\frac {a \sqrt {-b \,x^{4}+a}}{4 x^{4}}+\frac {3 \sqrt {a}\, b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{4}\) | \(61\) |
risch | \(-\frac {b \sqrt {-b \,x^{4}+a}}{2}-\frac {a \sqrt {-b \,x^{4}+a}}{4 x^{4}}+\frac {3 \sqrt {a}\, b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{4}\) | \(61\) |
elliptic | \(-\frac {b \sqrt {-b \,x^{4}+a}}{2}-\frac {a \sqrt {-b \,x^{4}+a}}{4 x^{4}}+\frac {3 \sqrt {a}\, b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{4}\) | \(61\) |
Input:
int((-b*x^4+a)^(3/2)/x^5,x,method=_RETURNVERBOSE)
Output:
1/4*(3*a^(1/2)*arctanh((-b*x^4+a)^(1/2)/a^(1/2))*b*x^4-(-b*x^4+a)^(1/2)*(2 *b*x^4+a))/x^4
Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^5} \, dx=\left [\frac {3 \, \sqrt {a} b x^{4} \log \left (\frac {b x^{4} - 2 \, \sqrt {-b x^{4} + a} \sqrt {a} - 2 \, a}{x^{4}}\right ) - 2 \, {\left (2 \, b x^{4} + a\right )} \sqrt {-b x^{4} + a}}{8 \, x^{4}}, \frac {3 \, \sqrt {-a} b x^{4} \arctan \left (\frac {\sqrt {-b x^{4} + a} \sqrt {-a}}{b x^{4} - a}\right ) - {\left (2 \, b x^{4} + a\right )} \sqrt {-b x^{4} + a}}{4 \, x^{4}}\right ] \] Input:
integrate((-b*x^4+a)^(3/2)/x^5,x, algorithm="fricas")
Output:
[1/8*(3*sqrt(a)*b*x^4*log((b*x^4 - 2*sqrt(-b*x^4 + a)*sqrt(a) - 2*a)/x^4) - 2*(2*b*x^4 + a)*sqrt(-b*x^4 + a))/x^4, 1/4*(3*sqrt(-a)*b*x^4*arctan(sqrt (-b*x^4 + a)*sqrt(-a)/(b*x^4 - a)) - (2*b*x^4 + a)*sqrt(-b*x^4 + a))/x^4]
Result contains complex when optimal does not.
Time = 1.69 (sec) , antiderivative size = 206, normalized size of antiderivative = 3.07 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^5} \, dx=\begin {cases} \frac {3 \sqrt {a} b \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4} - \frac {a^{2}}{4 \sqrt {b} x^{6} \sqrt {\frac {a}{b x^{4}} - 1}} - \frac {a \sqrt {b}}{4 x^{2} \sqrt {\frac {a}{b x^{4}} - 1}} + \frac {b^{\frac {3}{2}} x^{2}}{2 \sqrt {\frac {a}{b x^{4}} - 1}} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\- \frac {3 i \sqrt {a} b \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4} + \frac {i a^{2}}{4 \sqrt {b} x^{6} \sqrt {- \frac {a}{b x^{4}} + 1}} + \frac {i a \sqrt {b}}{4 x^{2} \sqrt {- \frac {a}{b x^{4}} + 1}} - \frac {i b^{\frac {3}{2}} x^{2}}{2 \sqrt {- \frac {a}{b x^{4}} + 1}} & \text {otherwise} \end {cases} \] Input:
integrate((-b*x**4+a)**(3/2)/x**5,x)
Output:
Piecewise((3*sqrt(a)*b*acosh(sqrt(a)/(sqrt(b)*x**2))/4 - a**2/(4*sqrt(b)*x **6*sqrt(a/(b*x**4) - 1)) - a*sqrt(b)/(4*x**2*sqrt(a/(b*x**4) - 1)) + b**( 3/2)*x**2/(2*sqrt(a/(b*x**4) - 1)), Abs(a/(b*x**4)) > 1), (-3*I*sqrt(a)*b* asin(sqrt(a)/(sqrt(b)*x**2))/4 + I*a**2/(4*sqrt(b)*x**6*sqrt(-a/(b*x**4) + 1)) + I*a*sqrt(b)/(4*x**2*sqrt(-a/(b*x**4) + 1)) - I*b**(3/2)*x**2/(2*sqr t(-a/(b*x**4) + 1)), True))
Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^5} \, dx=-\frac {3}{8} \, \sqrt {a} b \log \left (\frac {\sqrt {-b x^{4} + a} - \sqrt {a}}{\sqrt {-b x^{4} + a} + \sqrt {a}}\right ) - \frac {1}{2} \, \sqrt {-b x^{4} + a} b - \frac {\sqrt {-b x^{4} + a} a}{4 \, x^{4}} \] Input:
integrate((-b*x^4+a)^(3/2)/x^5,x, algorithm="maxima")
Output:
-3/8*sqrt(a)*b*log((sqrt(-b*x^4 + a) - sqrt(a))/(sqrt(-b*x^4 + a) + sqrt(a ))) - 1/2*sqrt(-b*x^4 + a)*b - 1/4*sqrt(-b*x^4 + a)*a/x^4
Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^5} \, dx=-\frac {\frac {3 \, a b^{2} \arctan \left (\frac {\sqrt {-b x^{4} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + 2 \, \sqrt {-b x^{4} + a} b^{2} + \frac {\sqrt {-b x^{4} + a} a b}{x^{4}}}{4 \, b} \] Input:
integrate((-b*x^4+a)^(3/2)/x^5,x, algorithm="giac")
Output:
-1/4*(3*a*b^2*arctan(sqrt(-b*x^4 + a)/sqrt(-a))/sqrt(-a) + 2*sqrt(-b*x^4 + a)*b^2 + sqrt(-b*x^4 + a)*a*b/x^4)/b
Time = 0.60 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^5} \, dx=\frac {3\,\sqrt {a}\,b\,\mathrm {atanh}\left (\frac {\sqrt {a-b\,x^4}}{\sqrt {a}}\right )}{4}-\frac {a\,\sqrt {a-b\,x^4}}{4\,x^4}-\frac {b\,\sqrt {a-b\,x^4}}{2} \] Input:
int((a - b*x^4)^(3/2)/x^5,x)
Output:
(3*a^(1/2)*b*atanh((a - b*x^4)^(1/2)/a^(1/2)))/4 - (a*(a - b*x^4)^(1/2))/( 4*x^4) - (b*(a - b*x^4)^(1/2))/2
Time = 0.21 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{x^5} \, dx=\frac {-4 \sqrt {-b \,x^{4}+a}\, a -8 \sqrt {-b \,x^{4}+a}\, b \,x^{4}-12 \sqrt {a}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}\right )}{2}\right )\right ) b \,x^{4}+9 \sqrt {a}\, b \,x^{4}}{16 x^{4}} \] Input:
int((-b*x^4+a)^(3/2)/x^5,x)
Output:
( - 4*sqrt(a - b*x**4)*a - 8*sqrt(a - b*x**4)*b*x**4 - 12*sqrt(a)*log(tan( asin((sqrt(b)*x**2)/sqrt(a))/2))*b*x**4 + 9*sqrt(a)*b*x**4)/(16*x**4)