Optimal. Leaf size=141 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a}} \]
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Rubi [A] time = 0.11, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {530, 240, 212, 206, 203, 377} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 240
Rule 377
Rule 530
Rubi steps
\begin {align*} \int \frac {-3 b+2 a x^4}{\left (-2 b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx &=2 \int \frac {1}{\sqrt [4]{b+a x^4}} \, dx+b \int \frac {1}{\left (-2 b+a x^4\right ) \sqrt [4]{b+a x^4}} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+b \operatorname {Subst}\left (\int \frac {1}{-2 b+3 a b x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {3} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {3} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{2 \sqrt {2}}+\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )+\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 128, normalized size = 0.91 \begin {gather*} \frac {12 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-\sqrt [4]{2} 3^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+12 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )-\sqrt [4]{2} 3^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{12 \sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 141, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{a} x}{\sqrt [4]{b+a x^4}}\right )}{2\ 2^{3/4} \sqrt [4]{3} \sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 241, normalized size = 1.71 \begin {gather*} -\frac {\left (\frac {1}{24}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, {\left (\frac {\left (\frac {1}{24}\right )^{\frac {1}{4}} x \sqrt {\frac {3 \, \sqrt {\frac {1}{6}} \sqrt {a} x^{2} + \sqrt {a x^{4} + b}}{x^{2}}}}{a^{\frac {1}{4}}} - \frac {\left (\frac {1}{24}\right )^{\frac {1}{4}} {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}\right )}}{x}\right )}{a^{\frac {1}{4}}} - \frac {\left (\frac {1}{24}\right )^{\frac {1}{4}} \log \left (\frac {12 \, \left (\frac {1}{24}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{4 \, a^{\frac {1}{4}}} + \frac {\left (\frac {1}{24}\right )^{\frac {1}{4}} \log \left (-\frac {12 \, \left (\frac {1}{24}\right )^{\frac {3}{4}} a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{4 \, a^{\frac {1}{4}}} + \frac {2 \, \arctan \left (\frac {\frac {x \sqrt {\frac {\sqrt {a} x^{2} + \sqrt {a x^{4} + b}}{x^{2}}}}{a^{\frac {1}{4}}} - \frac {{\left (a x^{4} + b\right )}^{\frac {1}{4}}}{a^{\frac {1}{4}}}}{x}\right )}{a^{\frac {1}{4}}} + \frac {\log \left (\frac {a^{\frac {1}{4}} x + {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} - \frac {\log \left (-\frac {a^{\frac {1}{4}} x - {\left (a x^{4} + b\right )}^{\frac {1}{4}}}{x}\right )}{2 \, a^{\frac {1}{4}}} \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 {2 \, a x^{4} - 3 \, b}{{\left (a x^{4} + b\right )}^{\frac {1}{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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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{4}-3 b}{\left (a \,x^{4}-2 b \right ) \left (a \,x^{4}+b \right )^{\frac {1}{4}}}\, dx\]
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 {2 \, a x^{4} - 3 \, b}{{\left (a x^{4} + b\right )}^{\frac {1}{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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {3\,b-2\,a\,x^4}{{\left (a\,x^4+b\right )}^{1/4}\,\left (2\,b-a\,x^4\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 {2 a x^{4} - 3 b}{\left (a x^{4} - 2 b\right ) \sqrt [4]{a x^{4} + b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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