Optimal. Leaf size=128 \[ \frac {\sqrt {2} \tanh ^{-1}\left (\frac {\frac {\sqrt {a x^5-b}}{\sqrt {2} \sqrt [4]{b}}+\frac {\sqrt [4]{b}}{\sqrt {2}}}{\sqrt [4]{a x^5-b}}\right )}{5 b^{3/4}}-\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}}{\sqrt {a x^5-b}-\sqrt {b}}\right )}{5 b^{3/4}} \]
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Rubi [A] time = 0.19, antiderivative size = 201, normalized size of antiderivative = 1.57, number of steps used = 11, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {266, 63, 211, 1165, 628, 1162, 617, 204} \begin {gather*} -\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}+\sqrt {a x^5-b}+\sqrt {b}\right )}{5 \sqrt {2} b^{3/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}+\sqrt {a x^5-b}+\sqrt {b}\right )}{5 \sqrt {2} b^{3/4}}-\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}\right )}{5 b^{3/4}}+\frac {\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}+1\right )}{5 b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 204
Rule 211
Rule 266
Rule 617
Rule 628
Rule 1162
Rule 1165
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 {\sqrt {b}-x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^5}\right )}{5 a \sqrt {b}}+\frac {2 \operatorname {Subst}\left (\int \frac {\sqrt {b}+x^2}{\frac {b}{a}+\frac {x^4}{a}} \, dx,x,\sqrt [4]{-b+a x^5}\right )}{5 a \sqrt {b}}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}+2 x}{-\sqrt {b}-\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^5}\right )}{5 \sqrt {2} b^{3/4}}-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt [4]{b}-2 x}{-\sqrt {b}+\sqrt {2} \sqrt [4]{b} x-x^2} \, dx,x,\sqrt [4]{-b+a x^5}\right )}{5 \sqrt {2} b^{3/4}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {b}-\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^5}\right )}{5 \sqrt {b}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {b}+\sqrt {2} \sqrt [4]{b} x+x^2} \, dx,x,\sqrt [4]{-b+a x^5}\right )}{5 \sqrt {b}}\\ &=-\frac {\log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}+\sqrt {-b+a x^5}\right )}{5 \sqrt {2} b^{3/4}}+\frac {\log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}+\sqrt {-b+a x^5}\right )}{5 \sqrt {2} b^{3/4}}+\frac {\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{-b+a x^5}}{\sqrt [4]{b}}\right )}{5 b^{3/4}}-\frac {\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{-b+a x^5}}{\sqrt [4]{b}}\right )}{5 b^{3/4}}\\ &=-\frac {\sqrt {2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{-b+a x^5}}{\sqrt [4]{b}}\right )}{5 b^{3/4}}+\frac {\sqrt {2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{-b+a x^5}}{\sqrt [4]{b}}\right )}{5 b^{3/4}}-\frac {\log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}+\sqrt {-b+a x^5}\right )}{5 \sqrt {2} b^{3/4}}+\frac {\log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}+\sqrt {-b+a x^5}\right )}{5 \sqrt {2} b^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 165, normalized size = 1.29 \begin {gather*} \frac {-\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}+\sqrt {a x^5-b}+\sqrt {b}\right )+\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}+\sqrt {a x^5-b}+\sqrt {b}\right )-2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}\right )+2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}+1\right )}{5 \sqrt {2} b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 127, normalized size = 0.99 \begin {gather*} \frac {\sqrt {2} \tan ^{-1}\left (\frac {-\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^5}}\right )}{5 b^{3/4}}+\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}}{\sqrt {b}+\sqrt {-b+a x^5}}\right )}{5 b^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 134, normalized size = 1.05 \begin {gather*} \frac {4}{5} \, \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} \arctan \left (\sqrt {b^{2} \sqrt {-\frac {1}{b^{3}}} + \sqrt {a x^{5} - b}} b^{2} \left (-\frac {1}{b^{3}}\right )^{\frac {3}{4}} - {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{2} \left (-\frac {1}{b^{3}}\right )^{\frac {3}{4}}\right ) + \frac {1}{5} \, \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} \log \left (b \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} + {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right ) - \frac {1}{5} \, \left (-\frac {1}{b^{3}}\right )^{\frac {1}{4}} \log \left (-b \left (-\frac {1}{b^{3}}\right )^{\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 [A] time = 0.32, size = 162, normalized size = 1.27 \begin {gather*} \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{5 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{5 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{10 \, b^{\frac {3}{4}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{10 \, b^{\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 \[\int \frac {1}{x \left (a \,x^{5}-b \right )^{\frac {3}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 162, normalized size = 1.27 \begin {gather*} \frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{5 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{5 \, b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{10 \, b^{\frac {3}{4}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{10 \, b^{\frac {3}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 51, normalized size = 0.40 \begin {gather*} -\frac {2\,\mathrm {atan}\left (\frac {{\left (a\,x^5-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{5\,{\left (-b\right )}^{3/4}}-\frac {2\,\mathrm {atanh}\left (\frac {{\left (a\,x^5-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{5\,{\left (-b\right )}^{3/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.05, size = 42, normalized size = 0.33 \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^{2 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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