Optimal. Leaf size=239 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{5/4}}+\frac {\left (5 a^2-b\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x \sqrt [4]{a x^4-b}}{\sqrt {a x^4-b}-\sqrt {a} x^2}\right )}{2 \sqrt {2} a^{5/4} b}+\frac {\left (5 a^2-b\right ) \tanh ^{-1}\left (\frac {\sqrt {a x^4-b}+\sqrt {a} x^2}{\sqrt {2} \sqrt [4]{a} x \sqrt [4]{a x^4-b}}\right )}{2 \sqrt {2} a^{5/4} b}+\frac {2 \left (a x^4-b\right )^{3/4}}{3 b x^3} \]
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Rubi [A] time = 0.89, antiderivative size = 346, normalized size of antiderivative = 1.45, number of steps used = 17, number of rules used = 13, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6725, 240, 212, 206, 203, 264, 377, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 a^{5/4}}-\frac {\left (5 a^2-b\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt {2} a^{5/4} b}+\frac {\left (5 a^2-b\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}+1\right )}{2 \sqrt {2} a^{5/4} b}-\frac {\left (5 a^2-b\right ) \log \left (-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}+\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}+1\right )}{4 \sqrt {2} a^{5/4} b}+\frac {\left (5 a^2-b\right ) \log \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}+\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}+1\right )}{4 \sqrt {2} a^{5/4} b}+\frac {2 \left (a x^4-b\right )^{3/4}}{3 b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 204
Rule 206
Rule 211
Rule 212
Rule 240
Rule 264
Rule 377
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 6725
Rubi steps
\begin {align*} \int \frac {-2 b-a x^4+2 x^8}{x^4 \sqrt [4]{-b+a x^4} \left (-b+2 a x^4\right )} \, dx &=\int \left (\frac {1}{a \sqrt [4]{-b+a x^4}}+\frac {2}{x^4 \sqrt [4]{-b+a x^4}}+\frac {-5 a^2+b}{a \sqrt [4]{-b+a x^4} \left (-b+2 a x^4\right )}\right ) \, dx\\ &=2 \int \frac {1}{x^4 \sqrt [4]{-b+a x^4}} \, dx+\frac {\int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx}{a}+\frac {\left (-5 a^2+b\right ) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-b+2 a x^4\right )} \, dx}{a}\\ &=\frac {2 \left (-b+a x^4\right )^{3/4}}{3 b x^3}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{a}+\frac {\left (-5 a^2+b\right ) \operatorname {Subst}\left (\int \frac {1}{-b-a b x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{a}\\ &=\frac {2 \left (-b+a x^4\right )^{3/4}}{3 b x^3}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 a}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 a}-\frac {\left (5 a^2-b\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt {a} x^2}{-b-a b x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 a}-\frac {\left (5 a^2-b\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt {a} x^2}{-b-a b x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 a}\\ &=\frac {2 \left (-b+a x^4\right )^{3/4}}{3 b x^3}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{5/4}}+\frac {\left (5 a^2-b\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {a}}-\frac {\sqrt {2} x}{\sqrt [4]{a}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 a^{3/2} b}+\frac {\left (5 a^2-b\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {a}}+\frac {\sqrt {2} x}{\sqrt [4]{a}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 a^{3/2} b}-\frac {\left (5 a^2-b\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2}}{\sqrt [4]{a}}+2 x}{-\frac {1}{\sqrt {a}}-\frac {\sqrt {2} x}{\sqrt [4]{a}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {2} a^{5/4} b}-\frac {\left (5 a^2-b\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2}}{\sqrt [4]{a}}-2 x}{-\frac {1}{\sqrt {a}}+\frac {\sqrt {2} x}{\sqrt [4]{a}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {2} a^{5/4} b}\\ &=\frac {2 \left (-b+a x^4\right )^{3/4}}{3 b x^3}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{5/4}}-\frac {\left (5 a^2-b\right ) \log \left (1+\frac {\sqrt {a} x^2}{\sqrt {-b+a x^4}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {2} a^{5/4} b}+\frac {\left (5 a^2-b\right ) \log \left (1+\frac {\sqrt {a} x^2}{\sqrt {-b+a x^4}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {2} a^{5/4} b}+\frac {\left (5 a^2-b\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} a^{5/4} b}-\frac {\left (5 a^2-b\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} a^{5/4} b}\\ &=\frac {2 \left (-b+a x^4\right )^{3/4}}{3 b x^3}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{5/4}}-\frac {\left (5 a^2-b\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} a^{5/4} b}+\frac {\left (5 a^2-b\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} a^{5/4} b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{5/4}}-\frac {\left (5 a^2-b\right ) \log \left (1+\frac {\sqrt {a} x^2}{\sqrt {-b+a x^4}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {2} a^{5/4} b}+\frac {\left (5 a^2-b\right ) \log \left (1+\frac {\sqrt {a} x^2}{\sqrt {-b+a x^4}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{4 \sqrt {2} a^{5/4} b}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 292, normalized size = 1.22 \begin {gather*} \frac {16 a^{5/4} \left (a x^4-b\right )^{3/4}-6 \sqrt {2} x^3 \left (5 a^2-b\right ) \left (\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )-\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}+1\right )\right )-3 \sqrt {2} x^3 \left (5 a^2-b\right ) \left (\log \left (-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}+\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}+1\right )-\log \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}+\frac {\sqrt {a} x^2}{\sqrt {a x^4-b}}+1\right )\right )+12 b x^3 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+12 b x^3 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{24 a^{5/4} b x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.02, size = 239, normalized size = 1.00 \begin {gather*} \frac {2 \left (-b+a x^4\right )^{3/4}}{3 b x^3}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{5/4}}+\frac {\left (5 a^2-b\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} x \sqrt [4]{-b+a x^4}}{-\sqrt {a} x^2+\sqrt {-b+a x^4}}\right )}{2 \sqrt {2} a^{5/4} b}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{2 a^{5/4}}+\frac {\left (5 a^2-b\right ) \tanh ^{-1}\left (\frac {\sqrt {a} x^2+\sqrt {-b+a x^4}}{\sqrt {2} \sqrt [4]{a} x \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {2} a^{5/4} b} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{8} - a x^{4} - 2 \, b}{{\left (2 \, a x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}} 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 \[\int \frac {2 x^{8}-a \,x^{4}-2 b}{x^{4} \left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (2 a \,x^{4}-b \right )}\, dx\]
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*} \int \frac {2 \, x^{8} - a x^{4} - 2 \, b}{{\left (2 \, a x^{4} - b\right )} {\left (a x^{4} - b\right )}^{\frac {1}{4}} 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 = -0.00 \begin {gather*} \int \frac {-2\,x^8+a\,x^4+2\,b}{x^4\,{\left (a\,x^4-b\right )}^{1/4}\,\left (b-2\,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 {- a x^{4} - 2 b + 2 x^{8}}{x^{4} \sqrt [4]{a x^{4} - b} \left (2 a x^{4} - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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