Optimal. Leaf size=70 \[ \frac {2}{5} b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )-\frac {2}{5} b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )+\frac {4}{15} \left (a x^5+b\right )^{3/4} \]
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Rubi [A] time = 0.09, 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 = {266, 50, 63, 298, 203, 206} \begin {gather*} \frac {2}{5} b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )-\frac {2}{5} b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )+\frac {4}{15} \left (a x^5+b\right )^{3/4} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 206
Rule 266
Rule 298
Rubi steps
\begin {align*} \int \frac {\left (b+a x^5\right )^{3/4}}{x} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {(b+a x)^{3/4}}{x} \, dx,x,x^5\right )\\ &=\frac {4}{15} \left (b+a x^5\right )^{3/4}+\frac {1}{5} b \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{b+a x}} \, dx,x,x^5\right )\\ &=\frac {4}{15} \left (b+a x^5\right )^{3/4}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {x^2}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^5}\right )}{5 a}\\ &=\frac {4}{15} \left (b+a x^5\right )^{3/4}-\frac {1}{5} (2 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^5}\right )+\frac {1}{5} (2 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^5}\right )\\ &=\frac {4}{15} \left (b+a x^5\right )^{3/4}+\frac {2}{5} b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )-\frac {2}{5} b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 70, normalized size = 1.00 \begin {gather*} \frac {2}{5} b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )-\frac {2}{5} b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )+\frac {4}{15} \left (a x^5+b\right )^{3/4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 70, normalized size = 1.00 \begin {gather*} \frac {4}{15} \left (b+a x^5\right )^{3/4}+\frac {2}{5} b^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )-\frac {2}{5} b^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 132, normalized size = 1.89 \begin {gather*} -\frac {4}{5} \, {\left (b^{3}\right )}^{\frac {1}{4}} \arctan \left (-\frac {{\left (a x^{5} + b\right )}^{\frac {1}{4}} {\left (b^{3}\right )}^{\frac {1}{4}} b^{2} - \sqrt {\sqrt {a x^{5} + b} b^{4} + \sqrt {b^{3}} b^{3}} {\left (b^{3}\right )}^{\frac {1}{4}}}{b^{3}}\right ) - \frac {1}{5} \, {\left (b^{3}\right )}^{\frac {1}{4}} \log \left ({\left (a x^{5} + b\right )}^{\frac {1}{4}} b^{2} + {\left (b^{3}\right )}^{\frac {3}{4}}\right ) + \frac {1}{5} \, {\left (b^{3}\right )}^{\frac {1}{4}} \log \left ({\left (a x^{5} + b\right )}^{\frac {1}{4}} b^{2} - {\left (b^{3}\right )}^{\frac {3}{4}}\right ) + \frac {4}{15} \, {\left (a x^{5} + b\right )}^{\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 185, normalized size = 2.64 \begin {gather*} -\frac {1}{5} \, \sqrt {2} \left (-b\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} + 2 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right ) - \frac {1}{5} \, \sqrt {2} \left (-b\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-b\right )^{\frac {1}{4}} - 2 \, {\left (a x^{5} + b\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-b\right )^{\frac {1}{4}}}\right ) + \frac {1}{10} \, \sqrt {2} \left (-b\right )^{\frac {3}{4}} \log \left (\sqrt {2} {\left (a x^{5} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{5} + b} + \sqrt {-b}\right ) - \frac {1}{10} \, \sqrt {2} \left (-b\right )^{\frac {3}{4}} \log \left (-\sqrt {2} {\left (a x^{5} + b\right )}^{\frac {1}{4}} \left (-b\right )^{\frac {1}{4}} + \sqrt {a x^{5} + b} + \sqrt {-b}\right ) + \frac {4}{15} \, {\left (a x^{5} + b\right )}^{\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{5}+b \right )^{\frac {3}{4}}}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 71, normalized size = 1.01 \begin {gather*} \frac {1}{5} \, b {\left (\frac {2 \, \arctan \left (\frac {{\left (a x^{5} + b\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {\log \left (\frac {{\left (a x^{5} + b\right )}^{\frac {1}{4}} - b^{\frac {1}{4}}}{{\left (a x^{5} + b\right )}^{\frac {1}{4}} + b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}}\right )} + \frac {4}{15} \, {\left (a x^{5} + b\right )}^{\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 50, normalized size = 0.71 \begin {gather*} \frac {2\,b^{3/4}\,\mathrm {atan}\left (\frac {{\left (a\,x^5+b\right )}^{1/4}}{b^{1/4}}\right )}{5}-\frac {2\,b^{3/4}\,\mathrm {atanh}\left (\frac {{\left (a\,x^5+b\right )}^{1/4}}{b^{1/4}}\right )}{5}+\frac {4\,{\left (a\,x^5+b\right )}^{3/4}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.06, size = 46, normalized size = 0.66 \begin {gather*} - \frac {a^{\frac {3}{4}} x^{\frac {15}{4}} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {3}{4} \\ \frac {1}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{5}}} \right )}}{5 \Gamma \left (\frac {1}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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