Optimal. Leaf size=83 \[ -\frac {1}{2} a^{5/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac {1}{2} a^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )+a \sqrt [4]{a+b x^4}+\frac {1}{5} \left (a+b x^4\right )^{5/4} \]
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Rubi [A] time = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {266, 50, 63, 212, 206, 203} \[ -\frac {1}{2} a^{5/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac {1}{2} a^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )+a \sqrt [4]{a+b x^4}+\frac {1}{5} \left (a+b x^4\right )^{5/4} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 206
Rule 212
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{5/4}}{x} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {(a+b x)^{5/4}}{x} \, dx,x,x^4\right )\\ &=\frac {1}{5} \left (a+b x^4\right )^{5/4}+\frac {1}{4} a \operatorname {Subst}\left (\int \frac {\sqrt [4]{a+b x}}{x} \, dx,x,x^4\right )\\ &=a \sqrt [4]{a+b x^4}+\frac {1}{5} \left (a+b x^4\right )^{5/4}+\frac {1}{4} a^2 \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=a \sqrt [4]{a+b x^4}+\frac {1}{5} \left (a+b x^4\right )^{5/4}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )}{b}\\ &=a \sqrt [4]{a+b x^4}+\frac {1}{5} \left (a+b x^4\right )^{5/4}-\frac {1}{2} a^{3/2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )-\frac {1}{2} a^{3/2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )\\ &=a \sqrt [4]{a+b x^4}+\frac {1}{5} \left (a+b x^4\right )^{5/4}-\frac {1}{2} a^{5/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac {1}{2} a^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 77, normalized size = 0.93 \[ \frac {1}{10} \left (-5 a^{5/4} \tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-5 a^{5/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )+2 \sqrt [4]{a+b x^4} \left (6 a+b x^4\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 130, normalized size = 1.57 \[ \frac {1}{5} \, {\left (b x^{4} + 6 \, a\right )} {\left (b x^{4} + a\right )}^{\frac {1}{4}} + {\left (a^{5}\right )}^{\frac {1}{4}} \arctan \left (-\frac {{\left (a^{5}\right )}^{\frac {3}{4}} {\left (b x^{4} + a\right )}^{\frac {1}{4}} a - {\left (a^{5}\right )}^{\frac {3}{4}} \sqrt {\sqrt {b x^{4} + a} a^{2} + \sqrt {a^{5}}}}{a^{5}}\right ) - \frac {1}{4} \, {\left (a^{5}\right )}^{\frac {1}{4}} \log \left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} a + {\left (a^{5}\right )}^{\frac {1}{4}}\right ) + \frac {1}{4} \, {\left (a^{5}\right )}^{\frac {1}{4}} \log \left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} a - {\left (a^{5}\right )}^{\frac {1}{4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 200, normalized size = 2.41 \[ -\frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} a \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{4} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} a \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right ) - \frac {1}{8} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} a \log \left (\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right ) + \frac {1}{8} \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} a \log \left (-\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right ) + \frac {1}{5} \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}} a \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{4}+a \right )^{\frac {5}{4}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.95, size = 79, normalized size = 0.95 \[ -\frac {1}{2} \, a^{\frac {5}{4}} \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right ) + \frac {1}{4} \, a^{\frac {5}{4}} \log \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right ) + \frac {1}{5} \, {\left (b x^{4} + a\right )}^{\frac {5}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 64, normalized size = 0.77 \[ a\,{\left (b\,x^4+a\right )}^{1/4}-\frac {a^{5/4}\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{2}+\frac {{\left (b\,x^4+a\right )}^{5/4}}{5}+\frac {a^{5/4}\,\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}\,1{}\mathrm {i}}{a^{1/4}}\right )\,1{}\mathrm {i}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.76, size = 48, normalized size = 0.58 \[ - \frac {b^{\frac {5}{4}} x^{5} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {5}{4} \\ - \frac {1}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 \Gamma \left (- \frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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