Optimal. Leaf size=194 \[ \frac {x \left (a x^4-b\right )^{3/4}}{6 a^2}+\frac {\left (-6 a^2-b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{36 a^{9/4}}+\frac {\left (6 a^2+b\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{9 \sqrt {2} a^{9/4}}+\frac {\left (-6 a^2-b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{36 a^{9/4}}+\frac {\left (6 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{9 \sqrt {2} a^{9/4}} \]
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Rubi [A] time = 0.43, 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 = {6725, 240, 212, 206, 203, 321, 377} \begin {gather*} \frac {b \tan ^{-1}\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 ) \tan ^{-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 ) \tan ^{-1}\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 203
Rule 206
Rule 212
Rule 240
Rule 321
Rule 377
Rule 6725
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 ) \operatorname {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 ) \operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 \operatorname {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 \operatorname {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 = 0.17, size = 221, normalized size = 1.14 \begin {gather*} \frac {-6 a^2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+12 \sqrt {2} a^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )-\left (6 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+2 \sqrt {2} \left (6 a^2+b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+6 \sqrt [4]{a} x \left (a x^4-b\right )^{3/4}-b \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+2 \sqrt {2} b \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{36 a^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.17, size = 194, normalized size = 1.00 \begin {gather*} \frac {x \left (a x^4-b\right )^{3/4}}{6 a^2}+\frac {\left (-6 a^2-b\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{36 a^{9/4}}+\frac {\left (6 a^2+b\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{9 \sqrt {2} a^{9/4}}+\frac {\left (-6 a^2-b\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{36 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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fricas [B] time = 0.66, size = 1231, normalized size = 6.35
result too large to display
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 \, x^{8} - a x^{4} + b}{{\left (3 \, a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
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 \, x^{8} - a x^{4} + b}{{\left (3 \, a x^{4} + b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{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 {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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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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