Optimal. Leaf size=81 \[ -\frac {\sqrt [4]{a-b x^4}}{4 a x^4}-\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {272, 44, 65,
218, 212, 209} \begin {gather*} -\frac {3 b \text {ArcTan}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}-\frac {\sqrt [4]{a-b x^4}}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a-b x^4\right )^{3/4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x^2 (a-b x)^{3/4}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt [4]{a-b x^4}}{4 a x^4}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{x (a-b x)^{3/4}} \, dx,x,x^4\right )}{16 a}\\ &=-\frac {\sqrt [4]{a-b x^4}}{4 a x^4}-\frac {3 \text {Subst}\left (\int \frac {1}{\frac {a}{b}-\frac {x^4}{b}} \, dx,x,\sqrt [4]{a-b x^4}\right )}{4 a}\\ &=-\frac {\sqrt [4]{a-b x^4}}{4 a x^4}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a-b x^4}\right )}{8 a^{3/2}}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a-b x^4}\right )}{8 a^{3/2}}\\ &=-\frac {\sqrt [4]{a-b x^4}}{4 a x^4}-\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 81, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [4]{a-b x^4}}{4 a x^4}-\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{5} \left (-b \,x^{4}+a \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 98, normalized size = 1.21 \begin {gather*} -\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} b}{4 \, {\left ({\left (b x^{4} - a\right )} a + a^{2}\right )}} - \frac {3 \, {\left (\frac {2 \, b \arctan \left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}} - \frac {b \log \left (\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (-b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{a^{\frac {3}{4}}}\right )}}{16 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 199 vs.
\(2 (61) = 122\).
time = 0.37, size = 199, normalized size = 2.46 \begin {gather*} \frac {12 \, a x^{4} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{5} b \left (\frac {b^{4}}{a^{7}}\right )^{\frac {3}{4}} - \sqrt {a^{4} \sqrt {\frac {b^{4}}{a^{7}}} + \sqrt {-b x^{4} + a} b^{2}} a^{5} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {3}{4}}}{b^{4}}\right ) - 3 \, a x^{4} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} \log \left (3 \, a^{2} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} + 3 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b\right ) + 3 \, a x^{4} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} \log \left (-3 \, a^{2} \left (\frac {b^{4}}{a^{7}}\right )^{\frac {1}{4}} + 3 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b\right ) - 4 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{16 \, a x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.77, size = 39, normalized size = 0.48 \begin {gather*} \frac {e^{\frac {i \pi }{4}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {a}{b x^{4}}} \right )}}{4 b^{\frac {3}{4}} x^{7} \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 228 vs.
\(2 (61) = 122\).
time = 2.18, size = 228, normalized size = 2.81 \begin {gather*} -\frac {\frac {6 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{2}} + \frac {6 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a^{2}} + \frac {3 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {-b x^{4} + a} + \sqrt {-a}\right )}{a^{2}} + \frac {3 \, \sqrt {2} b^{2} \log \left (-\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {-b x^{4} + a} + \sqrt {-a}\right )}{\left (-a\right )^{\frac {3}{4}} a} + \frac {8 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b}{a x^{4}}}{32 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.30, size = 61, normalized size = 0.75 \begin {gather*} -\frac {{\left (a-b\,x^4\right )}^{1/4}}{4\,a\,x^4}-\frac {3\,b\,\mathrm {atan}\left (\frac {{\left (a-b\,x^4\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{7/4}}-\frac {3\,b\,\mathrm {atanh}\left (\frac {{\left (a-b\,x^4\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{7/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________