Integrand size = 16, antiderivative size = 72 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/2}} \, dx=\frac {3 b}{4 a^2 \sqrt {a-b x^4}}-\frac {1}{4 a x^4 \sqrt {a-b x^4}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )}{4 a^{5/2}} \] Output:
3/4*b/a^2/(-b*x^4+a)^(1/2)-1/4/a/x^4/(-b*x^4+a)^(1/2)-3/4*b*arctanh((-b*x^ 4+a)^(1/2)/a^(1/2))/a^(5/2)
Time = 0.09 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/2}} \, dx=\frac {-a+3 b x^4}{4 a^2 x^4 \sqrt {a-b x^4}}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )}{4 a^{5/2}} \] Input:
Integrate[1/(x^5*(a - b*x^4)^(3/2)),x]
Output:
(-a + 3*b*x^4)/(4*a^2*x^4*Sqrt[a - b*x^4]) - (3*b*ArcTanh[Sqrt[a - b*x^4]/ Sqrt[a]])/(4*a^(5/2))
Time = 0.32 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {798, 52, 61, 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.
\(\displaystyle \int \frac {1}{x^5 \left (a-b x^4\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int \frac {1}{x^8 \left (a-b x^4\right )^{3/2}}dx^4\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{4} \left (\frac {3 b \int \frac {1}{x^4 \left (a-b x^4\right )^{3/2}}dx^4}{2 a}-\frac {1}{a x^4 \sqrt {a-b x^4}}\right )\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{4} \left (\frac {3 b \left (\frac {\int \frac {1}{x^4 \sqrt {a-b x^4}}dx^4}{a}+\frac {2}{a \sqrt {a-b x^4}}\right )}{2 a}-\frac {1}{a x^4 \sqrt {a-b x^4}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{4} \left (\frac {3 b \left (\frac {2}{a \sqrt {a-b x^4}}-\frac {2 \int \frac {1}{\frac {a}{b}-\frac {x^8}{b}}d\sqrt {a-b x^4}}{a b}\right )}{2 a}-\frac {1}{a x^4 \sqrt {a-b x^4}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{4} \left (\frac {3 b \left (\frac {2}{a \sqrt {a-b x^4}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )}{a^{3/2}}\right )}{2 a}-\frac {1}{a x^4 \sqrt {a-b x^4}}\right )\) |
Input:
Int[1/(x^5*(a - b*x^4)^(3/2)),x]
Output:
(-(1/(a*x^4*Sqrt[a - b*x^4])) + (3*b*(2/(a*Sqrt[a - b*x^4]) - (2*ArcTanh[S qrt[a - b*x^4]/Sqrt[a]])/a^(3/2)))/(2*a))/4
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] :> 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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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 = 57, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(-\frac {b \left (\frac {\sqrt {-b \,x^{4}+a}}{x^{4} b}+\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {-b \,x^{4}+a}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {2}{\sqrt {-b \,x^{4}+a}}\right )}{4 a^{2}}\) | \(57\) |
risch | \(-\frac {\sqrt {-b \,x^{4}+a}}{4 a^{2} x^{4}}-\frac {3 b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{4 a^{\frac {5}{2}}}+\frac {b}{2 a^{2} \sqrt {-b \,x^{4}+a}}\) | \(66\) |
default | \(-\frac {\sqrt {-b \,x^{4}+a}}{4 a^{2} x^{4}}-\frac {3 b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{4 a^{\frac {5}{2}}}-\frac {b \sqrt {-\left (x^{2}-\frac {\sqrt {a b}}{b}\right )^{2} b -2 \sqrt {a b}\, \left (x^{2}-\frac {\sqrt {a b}}{b}\right )}}{4 a^{2} \sqrt {a b}\, \left (x^{2}-\frac {\sqrt {a b}}{b}\right )}+\frac {b \sqrt {-\left (x^{2}+\frac {\sqrt {a b}}{b}\right )^{2} b +2 \sqrt {a b}\, \left (x^{2}+\frac {\sqrt {a b}}{b}\right )}}{4 a^{2} \sqrt {a b}\, \left (x^{2}+\frac {\sqrt {a b}}{b}\right )}\) | \(187\) |
elliptic | \(-\frac {\sqrt {-b \,x^{4}+a}}{4 a^{2} x^{4}}-\frac {3 b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{4 a^{\frac {5}{2}}}-\frac {b \sqrt {-\left (x^{2}-\frac {\sqrt {a b}}{b}\right )^{2} b -2 \sqrt {a b}\, \left (x^{2}-\frac {\sqrt {a b}}{b}\right )}}{4 a^{2} \sqrt {a b}\, \left (x^{2}-\frac {\sqrt {a b}}{b}\right )}+\frac {b \sqrt {-\left (x^{2}+\frac {\sqrt {a b}}{b}\right )^{2} b +2 \sqrt {a b}\, \left (x^{2}+\frac {\sqrt {a b}}{b}\right )}}{4 a^{2} \sqrt {a b}\, \left (x^{2}+\frac {\sqrt {a b}}{b}\right )}\) | \(187\) |
Input:
int(1/x^5/(-b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/4*b/a^2*((-b*x^4+a)^(1/2)/x^4/b+3*arctanh((-b*x^4+a)^(1/2)/a^(1/2))/a^( 1/2)-2/(-b*x^4+a)^(1/2))
Time = 0.09 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.68 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (b^{2} x^{8} - a b x^{4}\right )} \sqrt {a} \log \left (\frac {b x^{4} + 2 \, \sqrt {-b x^{4} + a} \sqrt {a} - 2 \, a}{x^{4}}\right ) - 2 \, {\left (3 \, a b x^{4} - a^{2}\right )} \sqrt {-b x^{4} + a}}{8 \, {\left (a^{3} b x^{8} - a^{4} x^{4}\right )}}, -\frac {3 \, {\left (b^{2} x^{8} - a b x^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-b x^{4} + a} \sqrt {-a}}{b x^{4} - a}\right ) + {\left (3 \, a b x^{4} - a^{2}\right )} \sqrt {-b x^{4} + a}}{4 \, {\left (a^{3} b x^{8} - a^{4} x^{4}\right )}}\right ] \] Input:
integrate(1/x^5/(-b*x^4+a)^(3/2),x, algorithm="fricas")
Output:
[1/8*(3*(b^2*x^8 - a*b*x^4)*sqrt(a)*log((b*x^4 + 2*sqrt(-b*x^4 + a)*sqrt(a ) - 2*a)/x^4) - 2*(3*a*b*x^4 - a^2)*sqrt(-b*x^4 + a))/(a^3*b*x^8 - a^4*x^4 ), -1/4*(3*(b^2*x^8 - a*b*x^4)*sqrt(-a)*arctan(sqrt(-b*x^4 + a)*sqrt(-a)/( b*x^4 - a)) + (3*a*b*x^4 - a^2)*sqrt(-b*x^4 + a))/(a^3*b*x^8 - a^4*x^4)]
Result contains complex when optimal does not.
Time = 1.80 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.29 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/2}} \, dx=\begin {cases} - \frac {1}{4 a \sqrt {b} x^{6} \sqrt {\frac {a}{b x^{4}} - 1}} + \frac {3 \sqrt {b}}{4 a^{2} x^{2} \sqrt {\frac {a}{b x^{4}} - 1}} - \frac {3 b \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4 a^{\frac {5}{2}}} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\\frac {i}{4 a \sqrt {b} x^{6} \sqrt {- \frac {a}{b x^{4}} + 1}} - \frac {3 i \sqrt {b}}{4 a^{2} x^{2} \sqrt {- \frac {a}{b x^{4}} + 1}} + \frac {3 i b \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4 a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/x**5/(-b*x**4+a)**(3/2),x)
Output:
Piecewise((-1/(4*a*sqrt(b)*x**6*sqrt(a/(b*x**4) - 1)) + 3*sqrt(b)/(4*a**2* x**2*sqrt(a/(b*x**4) - 1)) - 3*b*acosh(sqrt(a)/(sqrt(b)*x**2))/(4*a**(5/2) ), Abs(a/(b*x**4)) > 1), (I/(4*a*sqrt(b)*x**6*sqrt(-a/(b*x**4) + 1)) - 3*I *sqrt(b)/(4*a**2*x**2*sqrt(-a/(b*x**4) + 1)) + 3*I*b*asin(sqrt(a)/(sqrt(b) *x**2))/(4*a**(5/2)), True))
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/2}} \, dx=-\frac {3 \, {\left (b x^{4} - a\right )} b + 2 \, a b}{4 \, {\left ({\left (-b x^{4} + a\right )}^{\frac {3}{2}} a^{2} - \sqrt {-b x^{4} + a} a^{3}\right )}} + \frac {3 \, b \log \left (\frac {\sqrt {-b x^{4} + a} - \sqrt {a}}{\sqrt {-b x^{4} + a} + \sqrt {a}}\right )}{8 \, a^{\frac {5}{2}}} \] Input:
integrate(1/x^5/(-b*x^4+a)^(3/2),x, algorithm="maxima")
Output:
-1/4*(3*(b*x^4 - a)*b + 2*a*b)/((-b*x^4 + a)^(3/2)*a^2 - sqrt(-b*x^4 + a)* a^3) + 3/8*b*log((sqrt(-b*x^4 + a) - sqrt(a))/(sqrt(-b*x^4 + a) + sqrt(a)) )/a^(5/2)
Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/2}} \, dx=\frac {3 \, b \arctan \left (\frac {\sqrt {-b x^{4} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{2}} - \frac {3 \, {\left (b x^{4} - a\right )} b + 2 \, a b}{4 \, {\left ({\left (-b x^{4} + a\right )}^{\frac {3}{2}} - \sqrt {-b x^{4} + a} a\right )} a^{2}} \] Input:
integrate(1/x^5/(-b*x^4+a)^(3/2),x, algorithm="giac")
Output:
3/4*b*arctan(sqrt(-b*x^4 + a)/sqrt(-a))/(sqrt(-a)*a^2) - 1/4*(3*(b*x^4 - a )*b + 2*a*b)/(((-b*x^4 + a)^(3/2) - sqrt(-b*x^4 + a)*a)*a^2)
Time = 0.61 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/2}} \, dx=\frac {3\,b}{4\,a^2\,\sqrt {a-b\,x^4}}-\frac {1}{4\,a\,x^4\,\sqrt {a-b\,x^4}}-\frac {3\,b\,\mathrm {atanh}\left (\frac {\sqrt {a-b\,x^4}}{\sqrt {a}}\right )}{4\,a^{5/2}} \] Input:
int(1/(x^5*(a - b*x^4)^(3/2)),x)
Output:
(3*b)/(4*a^2*(a - b*x^4)^(1/2)) - 1/(4*a*x^4*(a - b*x^4)^(1/2)) - (3*b*ata nh((a - b*x^4)^(1/2)/a^(1/2)))/(4*a^(5/2))
Time = 0.21 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^5 \left (a-b x^4\right )^{3/2}} \, dx=\frac {12 \sqrt {a}\, \sqrt {-b \,x^{4}+a}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}\right )}{2}\right )\right ) b \,x^{4}-9 \sqrt {a}\, \sqrt {-b \,x^{4}+a}\, b \,x^{4}-4 a^{2}+12 a b \,x^{4}}{16 \sqrt {-b \,x^{4}+a}\, a^{3} x^{4}} \] Input:
int(1/x^5/(-b*x^4+a)^(3/2),x)
Output:
(12*sqrt(a)*sqrt(a - b*x**4)*log(tan(asin((sqrt(b)*x**2)/sqrt(a))/2))*b*x* *4 - 9*sqrt(a)*sqrt(a - b*x**4)*b*x**4 - 4*a**2 + 12*a*b*x**4)/(16*sqrt(a - b*x**4)*a**3*x**4)