Optimal. Leaf size=52 \[ -\frac {\sqrt {a-b x^4}}{4 a x^4}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {272, 44, 65,
214} \begin {gather*} -\frac {b \tanh ^{-1}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}-\frac {\sqrt {a-b x^4}}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 65
Rule 214
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^5 \sqrt {a-b x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a-b x}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt {a-b x^4}}{4 a x^4}+\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {a-b x}} \, dx,x,x^4\right )}{8 a}\\ &=-\frac {\sqrt {a-b x^4}}{4 a x^4}-\frac {\text {Subst}\left (\int \frac {1}{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a-b x^4}\right )}{4 a}\\ &=-\frac {\sqrt {a-b x^4}}{4 a x^4}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 52, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {a-b x^4}}{4 a x^4}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {a-b x^4}}{\sqrt {a}}\right )}{4 a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 50, normalized size = 0.96
method | result | size |
default | \(-\frac {\sqrt {-b \,x^{4}+a}}{4 a \,x^{4}}-\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}\) | \(50\) |
risch | \(-\frac {\sqrt {-b \,x^{4}+a}}{4 a \,x^{4}}-\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}\) | \(50\) |
elliptic | \(-\frac {\sqrt {-b \,x^{4}+a}}{4 a \,x^{4}}-\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {-b \,x^{4}+a}}{x^{2}}\right )}{4 a^{\frac {3}{2}}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 71, normalized size = 1.37 \begin {gather*} -\frac {\sqrt {-b x^{4} + a} b}{4 \, {\left ({\left (b x^{4} - a\right )} a + a^{2}\right )}} + \frac {b \log \left (\frac {\sqrt {-b x^{4} + a} - \sqrt {a}}{\sqrt {-b x^{4} + a} + \sqrt {a}}\right )}{8 \, a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 112, normalized size = 2.15 \begin {gather*} \left [\frac {\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 \, \sqrt {-b x^{4} + a} a}{8 \, a^{2} x^{4}}, \frac {\sqrt {-a} b x^{4} \arctan \left (\frac {\sqrt {-b x^{4} + a} \sqrt {-a}}{a}\right ) - \sqrt {-b x^{4} + a} a}{4 \, a^{2} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 1.22, size = 129, normalized size = 2.48 \begin {gather*} \begin {cases} - \frac {\sqrt {b} \sqrt {\frac {a}{b x^{4}} - 1}}{4 a x^{2}} - \frac {b \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4 a^{\frac {3}{2}}} & \text {for}\: \left |{\frac {a}{b x^{4}}}\right | > 1 \\\frac {i}{4 \sqrt {b} x^{6} \sqrt {- \frac {a}{b x^{4}} + 1}} - \frac {i \sqrt {b}}{4 a x^{2} \sqrt {- \frac {a}{b x^{4}} + 1}} + \frac {i b \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} x^{2}} \right )}}{4 a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.65, size = 54, normalized size = 1.04 \begin {gather*} \frac {\frac {b^{2} \arctan \left (\frac {\sqrt {-b x^{4} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {\sqrt {-b x^{4} + a} b}{a x^{4}}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.29, size = 40, normalized size = 0.77 \begin {gather*} -\frac {\sqrt {a-b\,x^4}}{4\,a\,x^4}-\frac {b\,\mathrm {atanh}\left (\frac {\sqrt {a-b\,x^4}}{\sqrt {a}}\right )}{4\,a^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________