Optimal. Leaf size=70 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}+\frac {1}{a \sqrt [4]{a+b x^4}} \]
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Rubi [A] time = 0.04, 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, 51, 63, 298, 203, 206} \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}+\frac {1}{a \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 203
Rule 206
Rule 266
Rule 298
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^4\right )^{5/4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{5/4}} \, dx,x,x^4\right )\\ &=\frac {1}{a \sqrt [4]{a+b x^4}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{a+b x}} \, dx,x,x^4\right )}{4 a}\\ &=\frac {1}{a \sqrt [4]{a+b x^4}}+\frac {\operatorname {Subst}\left (\int \frac {x^2}{-\frac {a}{b}+\frac {x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )}{a b}\\ &=\frac {1}{a \sqrt [4]{a+b x^4}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{2 a}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{2 a}\\ &=\frac {1}{a \sqrt [4]{a+b x^4}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 33, normalized size = 0.47 \[ \frac {\, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};\frac {b x^4}{a}+1\right )}{a \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 164, normalized size = 2.34 \[ -\frac {4 \, {\left (a b x^{4} + a^{2}\right )} \frac {1}{a^{5}}^{\frac {1}{4}} \arctan \left (\sqrt {a^{3} \sqrt {\frac {1}{a^{5}}} + \sqrt {b x^{4} + a}} a \frac {1}{a^{5}}^{\frac {1}{4}} - {\left (b x^{4} + a\right )}^{\frac {1}{4}} a \frac {1}{a^{5}}^{\frac {1}{4}}\right ) + {\left (a b x^{4} + a^{2}\right )} \frac {1}{a^{5}}^{\frac {1}{4}} \log \left (a^{4} \frac {1}{a^{5}}^{\frac {3}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right ) - {\left (a b x^{4} + a^{2}\right )} \frac {1}{a^{5}}^{\frac {1}{4}} \log \left (-a^{4} \frac {1}{a^{5}}^{\frac {3}{4}} + {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right ) - 4 \, {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{4 \, {\left (a b x^{4} + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 199, normalized size = 2.84 \[ -\frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{4 \, a^{2}} - \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (b x^{4} + a\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{4 \, a^{2}} + \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{8 \, a^{2}} - \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} {\left (b x^{4} + a\right )}^{\frac {1}{4}} \left (-a\right )^{\frac {1}{4}} + \sqrt {b x^{4} + a} + \sqrt {-a}\right )}{8 \, a^{2}} + \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} a} \]
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 \[ \int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {5}{4}} x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.09, size = 75, normalized size = 1.07 \[ \frac {\frac {2 \, \arctan \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}} - a^{\frac {1}{4}}}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} + a^{\frac {1}{4}}}\right )}{a^{\frac {1}{4}}}}{4 \, a} + \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 52, normalized size = 0.74 \[ \frac {\mathrm {atan}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{2\,a^{5/4}}-\frac {\mathrm {atanh}\left (\frac {{\left (b\,x^4+a\right )}^{1/4}}{a^{1/4}}\right )}{2\,a^{5/4}}+\frac {1}{a\,{\left (b\,x^4+a\right )}^{1/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.82, size = 39, normalized size = 0.56 \[ - \frac {\Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{4 b^{\frac {5}{4}} x^{5} \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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