Optimal. Leaf size=75 \[ -\frac {\left (a+b x^4\right )^{3/4}}{4 x^4}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}} \]
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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,
304, 209, 212} \begin {gather*} \frac {3 b \text {ArcTan}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}}-\frac {\left (a+b x^4\right )^{3/4}}{4 x^4}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 209
Rule 212
Rule 272
Rule 304
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{3/4}}{x^5} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {(a+b x)^{3/4}}{x^2} \, dx,x,x^4\right )\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{4 x^4}+\frac {1}{16} (3 b) \text {Subst}\left (\int \frac {1}{x \sqrt [4]{a+b x}} \, dx,x,x^4\right )\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{4 x^4}+\frac {3}{4} \text {Subst}\left (\int \frac {x^2}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{4 x^4}-\frac {1}{8} (3 b) \text {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )+\frac {1}{8} (3 b) \text {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{4 x^4}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 75, normalized size = 1.00 \begin {gather*} -\frac {\left (a+b x^4\right )^{3/4}}{4 x^4}+\frac {3 b \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}}-\frac {3 b \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}}}{x^{5}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 74, normalized size = 0.99 \begin {gather*} \frac {3}{16} \, b {\left (\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}}}\right )} - \frac {{\left (b x^{4} + a\right )}^{\frac {3}{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. 185 vs.
\(2 (55) = 110\).
time = 0.39, size = 185, normalized size = 2.47 \begin {gather*} -\frac {12 \, \left (\frac {b^{4}}{a}\right )^{\frac {1}{4}} x^{4} \arctan \left (-\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (\frac {b^{4}}{a}\right )^{\frac {1}{4}} b^{3} - \sqrt {\sqrt {b x^{4} + a} b^{6} + \sqrt {\frac {b^{4}}{a}} a b^{4}} \left (\frac {b^{4}}{a}\right )^{\frac {1}{4}}}{b^{4}}\right ) + 3 \, \left (\frac {b^{4}}{a}\right )^{\frac {1}{4}} x^{4} \log \left (27 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{3} + 27 \, \left (\frac {b^{4}}{a}\right )^{\frac {3}{4}} a\right ) - 3 \, \left (\frac {b^{4}}{a}\right )^{\frac {1}{4}} x^{4} \log \left (27 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{3} - 27 \, \left (\frac {b^{4}}{a}\right )^{\frac {3}{4}} a\right ) + 4 \, {\left (b x^{4} + a\right )}^{\frac {3}{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 = 0.70, size = 39, normalized size = 0.52 \begin {gather*} - \frac {b^{\frac {3}{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 x \Gamma \left (\frac {5}{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. 209 vs.
\(2 (55) = 110\).
time = 1.07, size = 209, normalized size = 2.79 \begin {gather*} \frac {\frac {6 \, \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 {1}{4}}} + \frac {6 \, \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 {1}{4}}} + \frac {3 \, \sqrt {2} \left (-a\right )^{\frac {3}{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 {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 {1}{4}}} - \frac {8 \, {\left (b x^{4} + a\right )}^{\frac {3}{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 = 1.26, size = 55, normalized size = 0.73 \begin {gather*} \frac {3\,b\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{1/4}}-\frac {{\left (b\,x^4+a\right )}^{3/4}}{4\,x^4}-\frac {3\,b\,\mathrm {atanh}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{8\,a^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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