Optimal. Leaf size=55 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )}{5 b^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )}{5 b^{3/4}} \]
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Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {266, 63, 212, 206, 203} \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )}{5 b^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )}{5 b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 206
Rule 212
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \left (b+a x^5\right )^{3/4}} \, dx &=\frac {1}{5} \operatorname {Subst}\left (\int \frac {1}{x (b+a x)^{3/4}} \, dx,x,x^5\right )\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{b+a x^5}\right )}{5 a}\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-x^2} \, dx,x,\sqrt [4]{b+a x^5}\right )}{5 \sqrt {b}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+x^2} \, dx,x,\sqrt [4]{b+a x^5}\right )}{5 \sqrt {b}}\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{5 b^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{b+a x^5}}{\sqrt [4]{b}}\right )}{5 b^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 0.84 \begin {gather*} -\frac {2 \left (\tan ^{-1}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )+\tanh ^{-1}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )\right )}{5 b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 55, normalized size = 1.00 \begin {gather*} -\frac {2 \tan ^{-1}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )}{5 b^{3/4}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt [4]{a x^5+b}}{\sqrt [4]{b}}\right )}{5 b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 110, normalized size = 2.00 \begin {gather*} \frac {4}{5} \, \frac {1}{b^{3}}^{\frac {1}{4}} \arctan \left (\sqrt {b^{2} \sqrt {\frac {1}{b^{3}}} + \sqrt {a x^{5} + b}} b^{2} \frac {1}{b^{3}}^{\frac {3}{4}} - {\left (a x^{5} + b\right )}^{\frac {1}{4}} b^{2} \frac {1}{b^{3}}^{\frac {3}{4}}\right ) - \frac {1}{5} \, \frac {1}{b^{3}}^{\frac {1}{4}} \log \left (b \frac {1}{b^{3}}^{\frac {1}{4}} + {\left (a x^{5} + b\right )}^{\frac {1}{4}}\right ) + \frac {1}{5} \, \frac {1}{b^{3}}^{\frac {1}{4}} \log \left (-b \frac {1}{b^{3}}^{\frac {1}{4}} + {\left (a x^{5} + b\right )}^{\frac {1}{4}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 186, normalized size = 3.38 \begin {gather*} -\frac {\sqrt {2} \left (-b\right )^{\frac {1}{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 )}{5 \, b} - \frac {\sqrt {2} \left (-b\right )^{\frac {1}{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 )}{5 \, b} - \frac {\sqrt {2} \left (-b\right )^{\frac {1}{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 )}{10 \, b} + \frac {\sqrt {2} \left (-b\right )^{\frac {1}{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 )}{10 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (a \,x^{5}+b \right )^{\frac {3}{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 57, normalized size = 1.04 \begin {gather*} -\frac {2 \, \arctan \left (\frac {{\left (a x^{5} + b\right )}^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right )}{5 \, b^{\frac {3}{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 )}{5 \, b^{\frac {3}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.71, size = 39, normalized size = 0.71 \begin {gather*} -\frac {2\,\mathrm {atan}\left (\frac {{\left (a\,x^5+b\right )}^{1/4}}{b^{1/4}}\right )}{5\,b^{3/4}}-\frac {2\,\mathrm {atanh}\left (\frac {{\left (a\,x^5+b\right )}^{1/4}}{b^{1/4}}\right )}{5\,b^{3/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.93, size = 41, normalized size = 0.75 \begin {gather*} - \frac {\Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{5}}} \right )}}{5 a^{\frac {3}{4}} x^{\frac {15}{4}} \Gamma \left (\frac {7}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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