Optimal. Leaf size=75 \[ -\frac {\sqrt [4]{a+b x^4}}{4 x^4}-\frac {b \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {272, 43, 65,
218, 212, 209} \begin {gather*} -\frac {b \text {ArcTan}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac {\sqrt [4]{a+b x^4}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a+b x^4}}{x^5} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {\sqrt [4]{a+b x}}{x^2} \, dx,x,x^4\right )\\ &=-\frac {\sqrt [4]{a+b x^4}}{4 x^4}+\frac {1}{16} b \text {Subst}\left (\int \frac {1}{x (a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt [4]{a+b x^4}}{4 x^4}+\frac {1}{4} \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )\\ &=-\frac {\sqrt [4]{a+b x^4}}{4 x^4}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{8 \sqrt {a}}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{8 \sqrt {a}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{4 x^4}-\frac {b \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 75, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [4]{a+b x^4}}{4 x^4}-\frac {b \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac {b \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{x^{5}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 73, normalized size = 0.97 \begin {gather*} -\frac {b \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{8 \, 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 )}{16 \, a^{\frac {3}{4}}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 180 vs.
\(2 (55) = 110\).
time = 0.37, size = 180, normalized size = 2.40 \begin {gather*} \frac {4 \, \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x^{4} \arctan \left (-\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} a^{2} b \left (\frac {b^{4}}{a^{3}}\right )^{\frac {3}{4}} - \sqrt {\sqrt {b x^{4} + a} b^{2} + a^{2} \sqrt {\frac {b^{4}}{a^{3}}}} a^{2} \left (\frac {b^{4}}{a^{3}}\right )^{\frac {3}{4}}}{b^{4}}\right ) - \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x^{4} \log \left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} b + a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}}\right ) + \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}} x^{4} \log \left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} b - a \left (\frac {b^{4}}{a^{3}}\right )^{\frac {1}{4}}\right ) - 4 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}}{16 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.64, size = 41, normalized size = 0.55 \begin {gather*} - \frac {\sqrt [4]{b} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac {7}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 207 vs.
\(2 (55) = 110\).
time = 1.41, size = 207, normalized size = 2.76 \begin {gather*} \frac {\frac {2 \, \sqrt {2} 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 )}{\left (-a\right )^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} 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 )}{\left (-a\right )^{\frac {3}{4}}} + \frac {\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}}} + \frac {\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} - \frac {8 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b}{x^{4}}}{32 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.28, size = 55, normalized size = 0.73 \begin {gather*} -\frac {{\left (b\,x^4+a\right )}^{1/4}}{4\,x^4}-\frac {b\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{3/4}}-\frac {b\,\mathrm {atanh}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{3/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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