Optimal. Leaf size=116 \[ \frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4 a-\text {$\#$1}^4 c+a^2+a b+a c\& ,\frac {\text {$\#$1}^4 \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+\text {$\#$1}^4 (-\log (x))-2 a \log \left (\sqrt [4]{a x^4-b}-\text {$\#$1} x\right )+2 a \log (x)}{-2 \text {$\#$1}^5+2 \text {$\#$1} a+\text {$\#$1} c}\& \right ] \]
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Rubi [B] time = 1.92, antiderivative size = 539, normalized size of antiderivative = 4.65, number of steps used = 10, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6728, 377, 212, 208, 205} \begin {gather*} \frac {a^{3/4} \left (\frac {2 b-c}{\sqrt {c^2-4 a b}}+1\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {c^2-4 a b}+2 b+c}}{\sqrt [4]{c-\sqrt {c^2-4 a b}} \sqrt [4]{a x^4-b}}\right )}{2 \left (c-\sqrt {c^2-4 a b}\right )^{3/4} \sqrt [4]{-\sqrt {c^2-4 a b}+2 b+c}}+\frac {a^{3/4} \left (1-\frac {2 b-c}{\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 \left (\sqrt {c^2-4 a b}+c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}+2 b+c}}+\frac {a^{3/4} \left (\frac {2 b-c}{\sqrt {c^2-4 a b}}+1\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {c^2-4 a b}+2 b+c}}{\sqrt [4]{c-\sqrt {c^2-4 a b}} \sqrt [4]{a x^4-b}}\right )}{2 \left (c-\sqrt {c^2-4 a b}\right )^{3/4} \sqrt [4]{-\sqrt {c^2-4 a b}+2 b+c}}+\frac {a^{3/4} \left (1-\frac {2 b-c}{\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 \left (\sqrt {c^2-4 a b}+c\right )^{3/4} \sqrt [4]{\sqrt {c^2-4 a b}+2 b+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 212
Rule 377
Rule 6728
Rubi steps
\begin {align*} \int \frac {b+a x^4}{\sqrt [4]{-b+a x^4} \left (b+c x^4+a x^8\right )} \, dx &=\int \left (\frac {a+\frac {a (2 b-c)}{\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 {a-\frac {a (2 b-c)}{\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 (a \left (1-\frac {2 b-c}{\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 (a \left (1+\frac {2 b-c}{\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 (a \left (1-\frac {2 b-c}{\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 (a \left (1+\frac {2 b-c}{\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 {\left (a \left (1+\frac {2 b-c}{\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 (a \left (1+\frac {2 b-c}{\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 (a \left (1-\frac {2 b-c}{\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 (a \left (1-\frac {2 b-c}{\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 {a^{3/4} \left (1+\frac {2 b-c}{\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 \left (c-\sqrt {-4 a b+c^2}\right )^{3/4} \sqrt [4]{2 b+c-\sqrt {-4 a b+c^2}}}+\frac {a^{3/4} \left (1-\frac {2 b-c}{\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 \left (c+\sqrt {-4 a b+c^2}\right )^{3/4} \sqrt [4]{2 b+c+\sqrt {-4 a b+c^2}}}+\frac {a^{3/4} \left (1+\frac {2 b-c}{\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 \left (c-\sqrt {-4 a b+c^2}\right )^{3/4} \sqrt [4]{2 b+c-\sqrt {-4 a b+c^2}}}+\frac {a^{3/4} \left (1-\frac {2 b-c}{\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 \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 = 1.47, size = 510, normalized size = 4.40 \begin {gather*} \frac {1}{2} a^{3/4} \left (\frac {\left (\frac {2 b-c}{\sqrt {c^2-4 a b}}+1\right ) \tan ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {c^2-4 a b}+2 b+c}}{\sqrt [4]{c-\sqrt {c^2-4 a b}} \sqrt [4]{a x^4-b}}\right )}{\left (c-\sqrt {c^2-4 a b}\right )^{3/4} \sqrt [4]{-\sqrt {c^2-4 a b}+2 b+c}}+\frac {\left (\frac {c-2 b}{\sqrt {c^2-4 a b}}+1\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 {\left (\frac {2 b-c}{\sqrt {c^2-4 a b}}+1\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{a} x \sqrt [4]{-\sqrt {c^2-4 a b}+2 b+c}}{\sqrt [4]{c-\sqrt {c^2-4 a b}} \sqrt [4]{a x^4-b}}\right )}{\left (c-\sqrt {c^2-4 a b}\right )^{3/4} \sqrt [4]{-\sqrt {c^2-4 a b}+2 b+c}}+\frac {\left (\frac {c-2 b}{\sqrt {c^2-4 a b}}+1\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}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.74, size = 117, normalized size = 1.01 \begin {gather*} \frac {1}{4} \text {RootSum}\left [a^2+a b+a c-2 a \text {$\#$1}^4-c \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 a \log (x)+2 a \log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-2 a \text {$\#$1}-c \text {$\#$1}+2 \text {$\#$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 {a 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 = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}+b}{\left (a \,x^{4}-b \right )^{\frac {1}{4}} \left (a \,x^{8}+c \,x^{4}+b \right )}\, dx\]
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 {a 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 {a\,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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