Optimal. Leaf size=194 \[ \frac {x \left (-b+a x^4\right )^{3/4}}{6 a^2}+\frac {\left (-6 a^2-b\right ) \text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{36 a^{9/4}}+\frac {\left (6 a^2+b\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{9 \sqrt {2} a^{9/4}}+\frac {\left (-6 a^2-b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{36 a^{9/4}}+\frac {\left (6 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{9 \sqrt {2} a^{9/4}} \]
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Rubi [A]
time = 0.30, antiderivative size = 256, normalized size of antiderivative = 1.32, number of steps
used = 15, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {6857, 246,
218, 212, 209, 327, 385} \begin {gather*} \frac {b \text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{12 a^{9/4}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{12 a^{9/4}}+\frac {x \left (a x^4-b\right )^{3/4}}{6 a^2}-\frac {\left (3 a^2+2 b\right ) \text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{18 a^{9/4}}+\frac {\left (6 a^2+b\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{9 \sqrt {2} a^{9/4}}-\frac {\left (3 a^2+2 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{18 a^{9/4}}+\frac {\left (6 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{9 \sqrt {2} a^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 327
Rule 385
Rule 6857
Rubi steps
\begin {align*} \int \frac {b-a x^4+2 x^8}{\sqrt [4]{-b+a x^4} \left (b+3 a x^4\right )} \, dx &=\int \left (\frac {-3-\frac {2 b}{a^2}}{9 \sqrt [4]{-b+a x^4}}+\frac {2 x^4}{3 a \sqrt [4]{-b+a x^4}}+\frac {2 \left (6 a^2 b+b^2\right )}{9 a^2 \sqrt [4]{-b+a x^4} \left (b+3 a x^4\right )}\right ) \, dx\\ &=\frac {2 \int \frac {x^4}{\sqrt [4]{-b+a x^4}} \, dx}{3 a}+\frac {\left (2 b \left (6 a^2+b\right )\right ) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (b+3 a x^4\right )} \, dx}{9 a^2}+\frac {1}{9} \left (-3-\frac {2 b}{a^2}\right ) \int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx\\ &=\frac {x \left (-b+a x^4\right )^{3/4}}{6 a^2}+\frac {b \int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx}{6 a^2}+\frac {\left (2 b \left (6 a^2+b\right )\right ) \text {Subst}\left (\int \frac {1}{b-4 a b x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{9 a^2}+\frac {1}{9} \left (-3-\frac {2 b}{a^2}\right ) \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=\frac {x \left (-b+a x^4\right )^{3/4}}{6 a^2}+\frac {b \text {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{6 a^2}+\frac {\left (6 a^2+b\right ) \text {Subst}\left (\int \frac {1}{1-2 \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{9 a^2}+\frac {\left (6 a^2+b\right ) \text {Subst}\left (\int \frac {1}{1+2 \sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{9 a^2}+\frac {1}{18} \left (-3-\frac {2 b}{a^2}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\frac {1}{18} \left (-3-\frac {2 b}{a^2}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )\\ &=\frac {x \left (-b+a x^4\right )^{3/4}}{6 a^2}-\frac {\left (3 a^2+2 b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{18 a^{9/4}}+\frac {\left (6 a^2+b\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{9 \sqrt {2} a^{9/4}}-\frac {\left (3 a^2+2 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{18 a^{9/4}}+\frac {\left (6 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{9 \sqrt {2} a^{9/4}}+\frac {b \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{12 a^2}+\frac {b \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{12 a^2}\\ &=\frac {x \left (-b+a x^4\right )^{3/4}}{6 a^2}+\frac {b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{12 a^{9/4}}-\frac {\left (3 a^2+2 b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{18 a^{9/4}}+\frac {\left (6 a^2+b\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{9 \sqrt {2} a^{9/4}}+\frac {b \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{12 a^{9/4}}-\frac {\left (3 a^2+2 b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{18 a^{9/4}}+\frac {\left (6 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{9 \sqrt {2} a^{9/4}}\\ \end {align*}
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Mathematica [A]
time = 0.77, size = 171, normalized size = 0.88 \begin {gather*} \frac {6 \sqrt [4]{a} x \left (-b+a x^4\right )^{3/4}-\left (6 a^2+b\right ) \text {ArcTan}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )+2 \sqrt {2} \left (6 a^2+b\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )-\left (6 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )+2 \sqrt {2} \left (6 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{36 a^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {2 x^{8}-a \,x^{4}+b}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (3 a \,x^{4}+b \right )}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1231 vs.
\(2 (146) = 292\).
time = 0.42, size = 1231, normalized size = 6.35 \begin {gather*} \frac {16 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} \left (\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}\right )^{\frac {1}{4}} \arctan \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \sqrt {\frac {2 \, {\left (1296 \, a^{13} + 864 \, a^{11} b + 216 \, a^{9} b^{2} + 24 \, a^{7} b^{3} + a^{5} b^{4}\right )} x^{2} \sqrt {\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}} + {\left (46656 \, a^{12} + 46656 \, a^{10} b + 19440 \, a^{8} b^{2} + 4320 \, a^{6} b^{3} + 540 \, a^{4} b^{4} + 36 \, a^{2} b^{5} + b^{6}\right )} \sqrt {a x^{4} - b}}{x^{2}}} \left (\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}\right )^{\frac {1}{4}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (216 \, a^{8} + 108 \, a^{6} b + 18 \, a^{4} b^{2} + a^{2} b^{3}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}} \left (\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}\right )^{\frac {1}{4}}}{{\left (1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}\right )} x}\right ) + 4 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} \left (\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{7} x \left (\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}\right )^{\frac {3}{4}} + {\left (216 \, a^{6} + 108 \, a^{4} b + 18 \, a^{2} b^{2} + b^{3}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} \left (\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{7} x \left (\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}\right )^{\frac {3}{4}} - {\left (216 \, a^{6} + 108 \, a^{4} b + 18 \, a^{2} b^{2} + b^{3}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, a^{2} \left (\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}\right )^{\frac {1}{4}} \arctan \left (\frac {a^{2} x \sqrt {\frac {{\left (1296 \, a^{13} + 864 \, a^{11} b + 216 \, a^{9} b^{2} + 24 \, a^{7} b^{3} + a^{5} b^{4}\right )} x^{2} \sqrt {\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}} + {\left (46656 \, a^{12} + 46656 \, a^{10} b + 19440 \, a^{8} b^{2} + 4320 \, a^{6} b^{3} + 540 \, a^{4} b^{4} + 36 \, a^{2} b^{5} + b^{6}\right )} \sqrt {a x^{4} - b}}{x^{2}}} \left (\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}\right )^{\frac {1}{4}} - {\left (216 \, a^{8} + 108 \, a^{6} b + 18 \, a^{4} b^{2} + a^{2} b^{3}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}} \left (\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}\right )^{\frac {1}{4}}}{{\left (1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}\right )} x}\right ) - a^{2} \left (\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}\right )^{\frac {1}{4}} \log \left (\frac {a^{7} x \left (\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}\right )^{\frac {3}{4}} + {\left (216 \, a^{6} + 108 \, a^{4} b + 18 \, a^{2} b^{2} + b^{3}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) + a^{2} \left (\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {a^{7} x \left (\frac {1296 \, a^{8} + 864 \, a^{6} b + 216 \, a^{4} b^{2} + 24 \, a^{2} b^{3} + b^{4}}{a^{9}}\right )^{\frac {3}{4}} - {\left (216 \, a^{6} + 108 \, a^{4} b + 18 \, a^{2} b^{2} + b^{3}\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}{x}\right ) + 12 \, {\left (a x^{4} - b\right )}^{\frac {3}{4}} x}{72 \, a^{2}} \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 {- a x^{4} + b + 2 x^{8}}{\sqrt [4]{a x^{4} - b} \left (3 a x^{4} + b\right )}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {2\,x^8-a\,x^4+b}{{\left (a\,x^4-b\right )}^{1/4}\,\left (3\,a\,x^4+b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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