Optimal. Leaf size=104 \[ -\frac {\left (-b+2 a x^4\right )^{3/4}}{3 x^3}+\frac {a^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+2 a x^4}}\right )}{2 \sqrt [4]{2}}+\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+2 a x^4}}\right )}{2 \sqrt [4]{2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {462, 246, 218,
212, 209} \begin {gather*} \frac {a^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{2 a x^4-b}}\right )}{2 \sqrt [4]{2}}+\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{2 a x^4-b}}\right )}{2 \sqrt [4]{2}}-\frac {\left (2 a x^4-b\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 462
Rubi steps
\begin {align*} \int \frac {-b+a x^4}{x^4 \sqrt [4]{-b+2 a x^4}} \, dx &=-\frac {\left (-b+2 a x^4\right )^{3/4}}{3 x^3}+a \int \frac {1}{\sqrt [4]{-b+2 a x^4}} \, dx\\ &=-\frac {\left (-b+2 a x^4\right )^{3/4}}{3 x^3}+a \text {Subst}\left (\int \frac {1}{1-2 a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+2 a x^4}}\right )\\ &=-\frac {\left (-b+2 a x^4\right )^{3/4}}{3 x^3}+\frac {1}{2} a \text {Subst}\left (\int \frac {1}{1-\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+2 a x^4}}\right )+\frac {1}{2} a \text {Subst}\left (\int \frac {1}{1+\sqrt {2} \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+2 a x^4}}\right )\\ &=-\frac {\left (-b+2 a x^4\right )^{3/4}}{3 x^3}+\frac {a^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+2 a x^4}}\right )}{2 \sqrt [4]{2}}+\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+2 a x^4}}\right )}{2 \sqrt [4]{2}}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 104, normalized size = 1.00 \begin {gather*} -\frac {\left (-b+2 a x^4\right )^{3/4}}{3 x^3}+\frac {a^{3/4} \text {ArcTan}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+2 a x^4}}\right )}{2 \sqrt [4]{2}}+\frac {a^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{-b+2 a x^4}}\right )}{2 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}-b}{x^{4} \left (2 a \,x^{4}-b \right )^{\frac {1}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 115, normalized size = 1.11 \begin {gather*} -\frac {1}{8} \, {\left (\frac {2 \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2 \, a x^{4} - b\right )}^{\frac {1}{4}}}{2 \, a^{\frac {1}{4}} x}\right )}{a^{\frac {1}{4}}} + \frac {2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {1}{4}} a^{\frac {1}{4}} - \frac {{\left (2 \, a x^{4} - b\right )}^{\frac {1}{4}}}{x}}{2^{\frac {1}{4}} a^{\frac {1}{4}} + \frac {{\left (2 \, a x^{4} - b\right )}^{\frac {1}{4}}}{x}}\right )}{a^{\frac {1}{4}}}\right )} a - \frac {{\left (2 \, a x^{4} - b\right )}^{\frac {3}{4}}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.24, size = 139, normalized size = 1.34 \begin {gather*} \frac {a x e^{- \frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {2 a x^{4}}{b}} \right )}}{4 \sqrt [4]{b} \Gamma \left (\frac {5}{4}\right )} - b \left (\begin {cases} - \frac {2^{\frac {3}{4}} a^{\frac {3}{4}} \left (-1 + \frac {b}{2 a x^{4}}\right )^{\frac {3}{4}} e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {3}{4}\right )}{4 b \Gamma \left (\frac {1}{4}\right )} & \text {for}\: \left |{\frac {b}{a x^{4}}}\right | > 2 \\- \frac {2^{\frac {3}{4}} a^{\frac {3}{4}} \left (1 - \frac {b}{2 a x^{4}}\right )^{\frac {3}{4}} \Gamma \left (- \frac {3}{4}\right )}{4 b \Gamma \left (\frac {1}{4}\right )} & \text {otherwise} \end {cases}\right ) \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.17, size = 60, normalized size = 0.58 \begin {gather*} \frac {a\,x\,{\left (1-\frac {2\,a\,x^4}{b}\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ \frac {2\,a\,x^4}{b}\right )}{{\left (2\,a\,x^4-b\right )}^{1/4}}-\frac {{\left (2\,a\,x^4-b\right )}^{3/4}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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