Optimal. Leaf size=70 \[ \frac {1}{a \sqrt [4]{a+b x^4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 70, 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, 53, 65,
304, 209, 212} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}+\frac {1}{a \sqrt [4]{a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 209
Rule 212
Rule 272
Rule 304
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^4\right )^{5/4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x (a+b x)^{5/4}} \, dx,x,x^4\right )\\ &=\frac {1}{a \sqrt [4]{a+b x^4}}+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt [4]{a+b x}} \, dx,x,x^4\right )}{4 a}\\ &=\frac {1}{a \sqrt [4]{a+b x^4}}+\frac {\text {Subst}\left (\int \frac {x^2}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )}{a b}\\ &=\frac {1}{a \sqrt [4]{a+b x^4}}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{2 a}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{2 a}\\ &=\frac {1}{a \sqrt [4]{a+b x^4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 66, normalized size = 0.94 \begin {gather*} \frac {\frac {2 \sqrt [4]{a}}{\sqrt [4]{a+b x^4}}+\tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (b \,x^{4}+a \right )^{\frac {5}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 75, normalized size = 1.07 \begin {gather*} \frac {\frac {2 \, \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}} + \frac {\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 {1}{4}}}}{4 \, a} + \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} a} \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. 164 vs.
\(2 (52) = 104\).
time = 0.39, size = 164, normalized size = 2.34 \begin {gather*} -\frac {4 \, {\left (a b x^{4} + a^{2}\right )} \frac {1}{a^{5}}^{\frac {1}{4}} \arctan \left (\sqrt {a^{3} \sqrt {\frac {1}{a^{5}}} + \sqrt {b x^{4} + a}} a \frac {1}{a^{5}}^{\frac {1}{4}} - {\left (b x^{4} + a\right )}^{\frac {1}{4}} a \frac {1}{a^{5}}^{\frac {1}{4}}\right ) + {\left (a b x^{4} + a^{2}\right )} \frac {1}{a^{5}}^{\frac {1}{4}} \log \left (a^{4} \frac {1}{a^{5}}^{\frac {3}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right ) - {\left (a b x^{4} + a^{2}\right )} \frac {1}{a^{5}}^{\frac {1}{4}} \log \left (-a^{4} \frac {1}{a^{5}}^{\frac {3}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right ) - 4 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{4 \, {\left (a b x^{4} + a^{2}\right )}} \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 = 0.53, size = 39, normalized size = 0.56 \begin {gather*} - \frac {\Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 b^{\frac {5}{4}} x^{5} \Gamma \left (\frac {9}{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. 199 vs.
\(2 (52) = 104\).
time = 0.95, size = 199, normalized size = 2.84 \begin {gather*} -\frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \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 )}{4 \, a^{2}} - \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \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 )}{4 \, a^{2}} + \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \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 )}{8 \, a^{2}} - \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \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 )}{8 \, a^{2}} + \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.23, size = 52, normalized size = 0.74 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{2\,a^{5/4}}-\frac {\mathrm {atanh}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{2\,a^{5/4}}+\frac {1}{a\,{\left (b\,x^4+a\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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