Optimal. Leaf size=104 \[ \frac {a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt [4]{a x^4+b}}\right )}{4\ 2^{3/4} b}+\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt [4]{a x^4+b}}\right )}{4\ 2^{3/4} b}-\frac {\left (a x^4+b\right )^{3/4}}{6 b x^3} \]
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Rubi [C] time = 0.07, antiderivative size = 46, normalized size of antiderivative = 0.44, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {511, 510} \begin {gather*} -\frac {\left (a x^4+b\right )^{3/4} \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};\frac {a x^4}{2 \left (a x^4+b\right )}\right )}{6 b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rubi steps
\begin {align*} \int \frac {\left (b+a x^4\right )^{3/4}}{x^4 \left (2 b+a x^4\right )} \, dx &=\frac {\left (b+a x^4\right )^{3/4} \int \frac {\left (1+\frac {a x^4}{b}\right )^{3/4}}{x^4 \left (2 b+a x^4\right )} \, dx}{\left (1+\frac {a x^4}{b}\right )^{3/4}}\\ &=-\frac {\left (b+a x^4\right )^{3/4} \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};\frac {a x^4}{2 \left (b+a x^4\right )}\right )}{6 b x^3}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 77, normalized size = 0.74 \begin {gather*} -\frac {\left (a x^4+b\right )^{3/4} \left (\frac {a x^4}{b}+2\right )^{3/4} \, _2F_1\left (-\frac {3}{4},-\frac {3}{4};\frac {1}{4};-\frac {a x^4}{a x^4+2 b}\right )}{6 b x^3 \left (\frac {2 a x^4}{b}+2\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 104, normalized size = 1.00 \begin {gather*} -\frac {\left (b+a x^4\right )^{3/4}}{6 b x^3}+\frac {a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt [4]{b+a x^4}}\right )}{4\ 2^{3/4} b}+\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt [4]{b+a x^4}}\right )}{4\ 2^{3/4} b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 63.20, size = 474, normalized size = 4.56 \begin {gather*} -\frac {12 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \arctan \left (-\frac {4 \, {\left (\left (\frac {1}{8}\right )^{\frac {1}{4}} {\left (a x^{4} + b\right )}^{\frac {1}{4}} a^{4} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} + 4 \, \left (\frac {1}{8}\right )^{\frac {3}{4}} {\left (a x^{4} + b\right )}^{\frac {3}{4}} a^{2} b^{3} x \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} - 2 \, \sqrt {\frac {1}{2}} {\left (\left (\frac {1}{8}\right )^{\frac {1}{4}} \sqrt {a x^{4} + b} a^{2} b x^{2} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} + \left (\frac {1}{8}\right )^{\frac {3}{4}} {\left (3 \, a b^{3} x^{4} + 2 \, b^{4}\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}}\right )} \sqrt {\sqrt {\frac {1}{2}} a^{2} b^{2} \sqrt {\frac {a^{3}}{b^{4}}}}\right )}}{a^{5} x^{4} + 2 \, a^{4} b}\right ) - 3 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {2 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {a^{3}}{b^{4}}} + 8 \, \left (\frac {1}{8}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} b^{3} x^{2} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} + 2 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} a^{2} x + \left (\frac {1}{8}\right )^{\frac {1}{4}} {\left (3 \, a^{2} b x^{4} + 2 \, a b^{2}\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + 2 \, b\right )}}\right ) + 3 \, \left (\frac {1}{8}\right )^{\frac {1}{4}} b x^{3} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}} \log \left (\frac {2 \, \sqrt {\frac {1}{2}} {\left (a x^{4} + b\right )}^{\frac {1}{4}} a b^{2} x^{3} \sqrt {\frac {a^{3}}{b^{4}}} - 8 \, \left (\frac {1}{8}\right )^{\frac {3}{4}} \sqrt {a x^{4} + b} b^{3} x^{2} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {3}{4}} + 2 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}} a^{2} x - \left (\frac {1}{8}\right )^{\frac {1}{4}} {\left (3 \, a^{2} b x^{4} + 2 \, a b^{2}\right )} \left (\frac {a^{3}}{b^{4}}\right )^{\frac {1}{4}}}{2 \, {\left (a x^{4} + 2 \, b\right )}}\right ) + 8 \, {\left (a x^{4} + b\right )}^{\frac {3}{4}}}{48 \, b x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{4} + b\right )}^{\frac {3}{4}}}{{\left (a x^{4} + 2 \, b\right )} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a \,x^{4}+b \right )^{\frac {3}{4}}}{x^{4} \left (a \,x^{4}+2 b \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{4} + b\right )}^{\frac {3}{4}}}{{\left (a x^{4} + 2 \, b\right )} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a\,x^4+b\right )}^{3/4}}{x^4\,\left (a\,x^4+2\,b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a x^{4} + b\right )^{\frac {3}{4}}}{x^{4} \left (a x^{4} + 2 b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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