Optimal. Leaf size=116 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {2} x \sqrt [4]{-b x^2+a x^6}}{-x^2+\sqrt {-b x^2+a x^6}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {-b x^2+a x^6}}{\sqrt {2}}}{x \sqrt [4]{-b x^2+a x^6}}\right )}{\sqrt {2}} \]
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Rubi [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 1.43, antiderivative size = 447, normalized size of antiderivative = 3.85, number of steps
used = 20, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2081, 6847,
6860, 252, 251, 1452, 441, 440, 525, 524} \begin {gather*} -\frac {2 x \sqrt [4]{1-\frac {a x^4}{b}} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {2 a^2 x^4}{2 a b-\sqrt {4 a b+1}+1},\frac {a x^4}{b}\right )}{\sqrt [4]{a x^6-b x^2}}-\frac {2 x \sqrt [4]{1-\frac {a x^4}{b}} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {2 a^2 x^4}{2 a b+\sqrt {4 a b+1}+1},\frac {a x^4}{b}\right )}{\sqrt [4]{a x^6-b x^2}}+\frac {2 a x^3 \left (1-\sqrt {4 a b+1}\right ) \sqrt [4]{1-\frac {a x^4}{b}} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {2 a^2 x^4}{2 a b-\sqrt {4 a b+1}+1},\frac {a x^4}{b}\right )}{5 \left (2 a b-\sqrt {4 a b+1}+1\right ) \sqrt [4]{a x^6-b x^2}}+\frac {2 a x^3 \left (\sqrt {4 a b+1}+1\right ) \sqrt [4]{1-\frac {a x^4}{b}} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {2 a^2 x^4}{2 a b+\sqrt {4 a b+1}+1},\frac {a x^4}{b}\right )}{5 \left (2 a b+\sqrt {4 a b+1}+1\right ) \sqrt [4]{a x^6-b x^2}}+\frac {2 x \sqrt [4]{1-\frac {a x^4}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};\frac {a x^4}{b}\right )}{\sqrt [4]{a x^6-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 252
Rule 440
Rule 441
Rule 524
Rule 525
Rule 1452
Rule 2081
Rule 6847
Rule 6860
Rubi steps
\begin {align*} \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx &=\frac {\left (\sqrt {x} \sqrt [4]{-b+a x^4}\right ) \int \frac {b+a x^4}{\sqrt {x} \sqrt [4]{-b+a x^4} \left (-b+x^2+a x^4\right )} \, dx}{\sqrt [4]{-b x^2+a x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {b+a x^8}{\sqrt [4]{-b+a x^8} \left (-b+x^4+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt [4]{-b+a x^8}}+\frac {2 b-x^4}{\sqrt [4]{-b+a x^8} \left (-b+x^4+a x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {2 b-x^4}{\sqrt [4]{-b+a x^8} \left (-b+x^4+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}\\ &=\frac {\left (2 \sqrt {x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \left (\frac {-1+\sqrt {1+4 a b}}{\left (1-\sqrt {1+4 a b}+2 a x^4\right ) \sqrt [4]{-b+a x^8}}+\frac {-1-\sqrt {1+4 a b}}{\left (1+\sqrt {1+4 a b}+2 a x^4\right ) \sqrt [4]{-b+a x^8}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {\left (2 \sqrt {x} \sqrt [4]{1-\frac {a x^4}{b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1-\frac {a x^8}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}\\ &=\frac {2 x \sqrt [4]{1-\frac {a x^4}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};\frac {a x^4}{b}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {\left (2 \left (-1-\sqrt {1+4 a b}\right ) \sqrt {x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\sqrt {1+4 a b}+2 a x^4\right ) \sqrt [4]{-b+a x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {\left (2 \left (-1+\sqrt {1+4 a b}\right ) \sqrt {x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (1-\sqrt {1+4 a b}+2 a x^4\right ) \sqrt [4]{-b+a x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}\\ &=\frac {2 x \sqrt [4]{1-\frac {a x^4}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};\frac {a x^4}{b}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {\left (2 \left (-1-\sqrt {1+4 a b}\right ) \sqrt {x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \left (\frac {1+\sqrt {1+4 a b}}{2 \sqrt [4]{-b+a x^8} \left (1+2 a b+\sqrt {1+4 a b}-2 a^2 x^8\right )}+\frac {a x^4}{\sqrt [4]{-b+a x^8} \left (-1-2 a b-\sqrt {1+4 a b}+2 a^2 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {\left (2 \left (-1+\sqrt {1+4 a b}\right ) \sqrt {x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \left (\frac {-1+\sqrt {1+4 a b}}{2 \sqrt [4]{-b+a x^8} \left (-1-2 a b+\sqrt {1+4 a b}+2 a^2 x^8\right )}+\frac {a x^4}{\sqrt [4]{-b+a x^8} \left (-1-2 a b+\sqrt {1+4 a b}+2 a^2 x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}\\ &=\frac {2 x \sqrt [4]{1-\frac {a x^4}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};\frac {a x^4}{b}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {\left (2 a \left (-1-\sqrt {1+4 a b}\right ) \sqrt {x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt [4]{-b+a x^8} \left (-1-2 a b-\sqrt {1+4 a b}+2 a^2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {\left (2 a \left (-1+\sqrt {1+4 a b}\right ) \sqrt {x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt [4]{-b+a x^8} \left (-1-2 a b+\sqrt {1+4 a b}+2 a^2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {\left (\left (-1+\sqrt {1+4 a b}\right )^2 \sqrt {x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^8} \left (-1-2 a b+\sqrt {1+4 a b}+2 a^2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {\left (\left (-1-\sqrt {1+4 a b}\right ) \left (1+\sqrt {1+4 a b}\right ) \sqrt {x} \sqrt [4]{-b+a x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-b+a x^8} \left (1+2 a b+\sqrt {1+4 a b}-2 a^2 x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}\\ &=\frac {2 x \sqrt [4]{1-\frac {a x^4}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};\frac {a x^4}{b}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {\left (2 a \left (-1-\sqrt {1+4 a b}\right ) \sqrt {x} \sqrt [4]{1-\frac {a x^4}{b}}\right ) \text {Subst}\left (\int \frac {x^4}{\left (-1-2 a b-\sqrt {1+4 a b}+2 a^2 x^8\right ) \sqrt [4]{1-\frac {a x^8}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {\left (2 a \left (-1+\sqrt {1+4 a b}\right ) \sqrt {x} \sqrt [4]{1-\frac {a x^4}{b}}\right ) \text {Subst}\left (\int \frac {x^4}{\left (-1-2 a b+\sqrt {1+4 a b}+2 a^2 x^8\right ) \sqrt [4]{1-\frac {a x^8}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {\left (\left (-1+\sqrt {1+4 a b}\right )^2 \sqrt {x} \sqrt [4]{1-\frac {a x^4}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-2 a b+\sqrt {1+4 a b}+2 a^2 x^8\right ) \sqrt [4]{1-\frac {a x^8}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {\left (\left (-1-\sqrt {1+4 a b}\right ) \left (1+\sqrt {1+4 a b}\right ) \sqrt {x} \sqrt [4]{1-\frac {a x^4}{b}}\right ) \text {Subst}\left (\int \frac {1}{\left (1+2 a b+\sqrt {1+4 a b}-2 a^2 x^8\right ) \sqrt [4]{1-\frac {a x^8}{b}}} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{-b x^2+a x^6}}\\ &=-\frac {2 x \sqrt [4]{1-\frac {a x^4}{b}} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {2 a^2 x^4}{1+2 a b-\sqrt {1+4 a b}},\frac {a x^4}{b}\right )}{\sqrt [4]{-b x^2+a x^6}}-\frac {2 x \sqrt [4]{1-\frac {a x^4}{b}} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};\frac {2 a^2 x^4}{1+2 a b+\sqrt {1+4 a b}},\frac {a x^4}{b}\right )}{\sqrt [4]{-b x^2+a x^6}}+\frac {2 a \left (1-\sqrt {1+4 a b}\right ) x^3 \sqrt [4]{1-\frac {a x^4}{b}} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {2 a^2 x^4}{1+2 a b-\sqrt {1+4 a b}},\frac {a x^4}{b}\right )}{5 \left (1+2 a b-\sqrt {1+4 a b}\right ) \sqrt [4]{-b x^2+a x^6}}+\frac {2 a \left (1+\sqrt {1+4 a b}\right ) x^3 \sqrt [4]{1-\frac {a x^4}{b}} F_1\left (\frac {5}{8};1,\frac {1}{4};\frac {13}{8};\frac {2 a^2 x^4}{1+2 a b+\sqrt {1+4 a b}},\frac {a x^4}{b}\right )}{5 \left (1+2 a b+\sqrt {1+4 a b}\right ) \sqrt [4]{-b x^2+a x^6}}+\frac {2 x \sqrt [4]{1-\frac {a x^4}{b}} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};\frac {a x^4}{b}\right )}{\sqrt [4]{-b x^2+a x^6}}\\ \end {align*}
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Mathematica [F]
time = 21.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b+a x^4}{\left (-b+x^2+a x^4\right ) \sqrt [4]{-b x^2+a x^6}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{4}+b}{\left (a \,x^{4}+x^{2}-b \right ) \left (a \,x^{6}-b \,x^{2}\right )^{\frac {1}{4}}}\, dx\]
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 [F(-1)] Timed out
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x^{4} + b}{\sqrt [4]{x^{2} \left (a x^{4} - b\right )} \left (a x^{4} - b + x^{2}\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 {a\,x^4+b}{{\left (a\,x^6-b\,x^2\right )}^{1/4}\,\left (a\,x^4+x^2-b\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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