Optimal. Leaf size=198 \[ \frac {3}{4} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a+\text {$\#$1}^4 c+a^2+a b-a c\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)-c \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+a \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )-a \log (x)+c \log (x)}{-2 \text {$\#$1}^5+2 \text {$\#$1} a-\text {$\#$1} c}\& \right ]+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a}} \]
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Rubi [B] time = 2.22, antiderivative size = 605, normalized size of antiderivative = 3.06, number of steps used = 16, number of rules used = 8, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6728, 240, 212, 206, 203, 377, 208, 205} \begin {gather*} \frac {3 \left (\frac {2 a b-c^2}{\sqrt {c^2-4 a b}}+c\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}{\sqrt [4]{\sqrt {c^2-4 a b}-c} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a} \left (\sqrt {c^2-4 a b}-c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}-\frac {3 \left (c-\frac {2 a b-c^2}{\sqrt {c^2-4 a b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}{\sqrt [4]{\sqrt {c^2-4 a b}+c} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a} \left (\sqrt {c^2-4 a b}+c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}+\frac {3 \left (\frac {2 a b-c^2}{\sqrt {c^2-4 a b}}+c\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}{\sqrt [4]{\sqrt {c^2-4 a b}-c} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a} \left (\sqrt {c^2-4 a b}-c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}-\frac {3 \left (c-\frac {2 a b-c^2}{\sqrt {c^2-4 a b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}{\sqrt [4]{\sqrt {c^2-4 a b}+c} \sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a} \left (\sqrt {c^2-4 a b}+c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{\sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 205
Rule 206
Rule 208
Rule 212
Rule 240
Rule 377
Rule 6728
Rubi steps
\begin {align*} \int \frac {-b+c x^4+2 a x^8}{\sqrt [4]{-b+a x^4} \left (b-c x^4+a x^8\right )} \, dx &=\int \left (\frac {2}{\sqrt [4]{-b+a x^4}}-\frac {3 \left (b-c x^4\right )}{\sqrt [4]{-b+a x^4} \left (b-c x^4+a x^8\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{-b+a x^4}} \, dx-3 \int \frac {b-c x^4}{\sqrt [4]{-b+a x^4} \left (b-c x^4+a x^8\right )} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{1-a x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )-3 \int \left (\frac {-c+\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}}{\sqrt [4]{-b+a x^4} \left (-c-\sqrt {-4 a b+c^2}+2 a x^4\right )}+\frac {-c-\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}}{\sqrt [4]{-b+a x^4} \left (-c+\sqrt {-4 a b+c^2}+2 a x^4\right )}\right ) \, dx\\ &=\left (3 \left (c-\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-c-\sqrt {-4 a b+c^2}+2 a x^4\right )} \, dx+\left (3 \left (c+\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \int \frac {1}{\sqrt [4]{-b+a x^4} \left (-c+\sqrt {-4 a b+c^2}+2 a x^4\right )} \, dx+\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 {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\left (3 \left (c-\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-c-\sqrt {-4 a b+c^2}-\left (2 a b+a \left (-c-\sqrt {-4 a b+c^2}\right )\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )+\left (3 \left (c+\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-c+\sqrt {-4 a b+c^2}-\left (2 a b+a \left (-c+\sqrt {-4 a b+c^2}\right )\right ) x^4} \, 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 {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\left (3 \left (c+\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-c+\sqrt {-4 a b+c^2}}-\sqrt {a} \sqrt {2 b-c+\sqrt {-4 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {-c+\sqrt {-4 a b+c^2}}}+\frac {\left (3 \left (c+\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-c+\sqrt {-4 a b+c^2}}+\sqrt {a} \sqrt {2 b-c+\sqrt {-4 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {-c+\sqrt {-4 a b+c^2}}}-\frac {\left (3 \left (c-\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\sqrt {-4 a b+c^2}}-\sqrt {a} \sqrt {-2 b+c+\sqrt {-4 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {c+\sqrt {-4 a b+c^2}}}-\frac {\left (3 \left (c-\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\sqrt {-4 a b+c^2}}+\sqrt {a} \sqrt {-2 b+c+\sqrt {-4 a b+c^2}} x^2} \, dx,x,\frac {x}{\sqrt [4]{-b+a x^4}}\right )}{2 \sqrt {c+\sqrt {-4 a b+c^2}}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {3 \left (c+\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{2 b-c+\sqrt {-4 a b+c^2}} x}{\sqrt [4]{-c+\sqrt {-4 a b+c^2}} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{a} \left (-c+\sqrt {-4 a b+c^2}\right )^{3/4} \sqrt [4]{2 b-c+\sqrt {-4 a b+c^2}}}-\frac {3 \left (c-\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-2 b+c+\sqrt {-4 a b+c^2}} x}{\sqrt [4]{c+\sqrt {-4 a b+c^2}} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{a} \left (c+\sqrt {-4 a b+c^2}\right )^{3/4} \sqrt [4]{-2 b+c+\sqrt {-4 a b+c^2}}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {3 \left (c+\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{2 b-c+\sqrt {-4 a b+c^2}} x}{\sqrt [4]{-c+\sqrt {-4 a b+c^2}} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{a} \left (-c+\sqrt {-4 a b+c^2}\right )^{3/4} \sqrt [4]{2 b-c+\sqrt {-4 a b+c^2}}}-\frac {3 \left (c-\frac {2 a b-c^2}{\sqrt {-4 a b+c^2}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [4]{-2 b+c+\sqrt {-4 a b+c^2}} x}{\sqrt [4]{c+\sqrt {-4 a b+c^2}} \sqrt [4]{-b+a x^4}}\right )}{2 \sqrt [4]{a} \left (c+\sqrt {-4 a b+c^2}\right )^{3/4} \sqrt [4]{-2 b+c+\sqrt {-4 a b+c^2}}}\\ \end {align*}
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Mathematica [B] time = 0.87, size = 594, normalized size = 3.00 \begin {gather*} \frac {\frac {3 \left (\frac {2 a b-c^2}{\sqrt {c^2-4 a b}}+c\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}{\sqrt [4]{\sqrt {c^2-4 a b}-c} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {c^2-4 a b}-c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}+\frac {3 \left (2 a b-c \left (\sqrt {c^2-4 a b}+c\right )\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}{\sqrt [4]{\sqrt {c^2-4 a b}+c} \sqrt [4]{a x^4-b}}\right )}{\sqrt {c^2-4 a b} \left (\sqrt {c^2-4 a b}+c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}+\frac {3 \left (\frac {2 a b-c^2}{\sqrt {c^2-4 a b}}+c\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}{\sqrt [4]{\sqrt {c^2-4 a b}-c} \sqrt [4]{a x^4-b}}\right )}{\left (\sqrt {c^2-4 a b}-c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}+2 b-c}}+\frac {3 \left (2 a b-c \left (\sqrt {c^2-4 a b}+c\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}{\sqrt [4]{\sqrt {c^2-4 a b}+c} \sqrt [4]{a x^4-b}}\right )}{\sqrt {c^2-4 a b} \left (\sqrt {c^2-4 a b}+c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}-2 b+c}}+2 \tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )+2 \tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{a x^4-b}}\right )}{2 \sqrt [4]{a}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 197, normalized size = 0.99 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b+a x^4}}\right )}{\sqrt [4]{a}}+\frac {3}{4} \text {RootSum}\left [a^2+a b-a c-2 a \text {$\#$1}^4+c \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {a \log (x)-c \log (x)-a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+c \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-2 a \text {$\#$1}+c \text {$\#$1}+2 \text {$\#$1}^5}\&\right ] \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 \, a x^{8} + c x^{4} - b}{{\left (a x^{8} - c 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 = 0.00, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{8}+c \,x^{4}-b}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (a \,x^{8}-c \,x^{4}+b \right )}\, dx\]
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 \, a x^{8} + c x^{4} - b}{{\left (a x^{8} - c 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\,a\,x^8+c\,x^4-b}{{\left (a\,x^4-b\right )}^{1/4}\,\left (a\,x^8-c\,x^4+b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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