Optimal. Leaf size=92 \[ -\frac {b \sqrt [4]{a+b x^4}}{x}-\frac {\left (a+b x^4\right )^{5/4}}{5 x^5}-\frac {1}{2} b^{5/4} \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac {1}{2} b^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {283, 338, 304,
209, 212} \begin {gather*} -\frac {1}{2} b^{5/4} \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac {1}{2} b^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )-\frac {b \sqrt [4]{a+b x^4}}{x}-\frac {\left (a+b x^4\right )^{5/4}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 283
Rule 304
Rule 338
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{5/4}}{x^6} \, dx &=-\frac {\left (a+b x^4\right )^{5/4}}{5 x^5}+b \int \frac {\sqrt [4]{a+b x^4}}{x^2} \, dx\\ &=-\frac {b \sqrt [4]{a+b x^4}}{x}-\frac {\left (a+b x^4\right )^{5/4}}{5 x^5}+b^2 \int \frac {x^2}{\left (a+b x^4\right )^{3/4}} \, dx\\ &=-\frac {b \sqrt [4]{a+b x^4}}{x}-\frac {\left (a+b x^4\right )^{5/4}}{5 x^5}+b^2 \text {Subst}\left (\int \frac {x^2}{1-b x^4} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )\\ &=-\frac {b \sqrt [4]{a+b x^4}}{x}-\frac {\left (a+b x^4\right )^{5/4}}{5 x^5}+\frac {1}{2} b^{3/2} \text {Subst}\left (\int \frac {1}{1-\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )-\frac {1}{2} b^{3/2} \text {Subst}\left (\int \frac {1}{1+\sqrt {b} x^2} \, dx,x,\frac {x}{\sqrt [4]{a+b x^4}}\right )\\ &=-\frac {b \sqrt [4]{a+b x^4}}{x}-\frac {\left (a+b x^4\right )^{5/4}}{5 x^5}-\frac {1}{2} b^{5/4} \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac {1}{2} b^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 81, normalized size = 0.88 \begin {gather*} \frac {1}{10} \left (-\frac {2 \sqrt [4]{a+b x^4} \left (a+6 b x^4\right )}{x^5}-5 b^{5/4} \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+5 b^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}}}{x^{6}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 97, normalized size = 1.05 \begin {gather*} \frac {1}{2} \, b^{\frac {5}{4}} \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}} x}\right ) - \frac {1}{4} \, b^{\frac {5}{4}} \log \left (-\frac {b^{\frac {1}{4}} - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}{b^{\frac {1}{4}} + \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x}}\right ) - \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} b}{x} - \frac {{\left (b x^{4} + a\right )}^{\frac {5}{4}}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 1.01, size = 46, normalized size = 0.50 \begin {gather*} \frac {a^{\frac {5}{4}} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {5}{4} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (b\,x^4+a\right )}^{5/4}}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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