Optimal. Leaf size=460 \[ \frac {\left (-1-\sqrt [4]{-1}\right ) \tan ^{-1}\left (\frac {(-1)^{7/8} \sqrt {2+\sqrt {2}} x \sqrt [8]{a^2-2 b} \sqrt [4]{a x^4+b}}{(-1)^{3/4} x^2 \sqrt [4]{a^2-2 b}+\sqrt {a x^4+b}}\right )}{8 \sqrt [8]{a^2-2 b}}+\frac {\left (\sqrt {2}+i \sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \tan ^{-1}\left (\frac {(-1)^{7/8} \left (\sqrt {2}-2\right ) x \sqrt [8]{a^2-2 b} \sqrt [4]{a x^4+b}}{(-1)^{3/4} \sqrt {2-\sqrt {2}} x^2 \sqrt [4]{a^2-2 b}+\sqrt {2-\sqrt {2}} \sqrt {a x^4+b}}\right )}{16 \sqrt [8]{a^2-2 b}}-\frac {i \left (\sqrt {2 \left (3-2 \sqrt {2}\right )}-i \sqrt {2}\right ) \tanh ^{-1}\left (\frac {(-1)^{7/8} x^2 \sqrt [4]{a^2-2 b}-\sqrt [8]{-1} \sqrt {a x^4+b}}{\sqrt {2-\sqrt {2}} x \sqrt [8]{a^2-2 b} \sqrt [4]{a x^4+b}}\right )}{16 \sqrt [8]{a^2-2 b}}+\frac {\left (-1-\sqrt [4]{-1}\right ) \tanh ^{-1}\left (\frac {(-1)^{7/8} x^2 \sqrt [4]{a^2-2 b}-\sqrt [8]{-1} \sqrt {a x^4+b}}{\sqrt {2+\sqrt {2}} x \sqrt [8]{a^2-2 b} \sqrt [4]{a x^4+b}}\right )}{8 \sqrt [8]{a^2-2 b}} \]
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Rubi [A] time = 0.59, antiderivative size = 457, normalized size of antiderivative = 0.99, number of steps used = 19, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1428, 408, 240, 212, 206, 203, 377, 208, 205} \begin {gather*} -\frac {\left (-a \sqrt {a^2-2 b}+a^2-2 b\right )^{3/4} \tan ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2-2 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-2 b}} \sqrt [4]{a x^4+b}}\right )}{4 \left (a-\sqrt {a^2-2 b}\right )^{3/4} \sqrt {a^2-2 b}}+\frac {\left (a \sqrt {a^2-2 b}+a^2-2 b\right )^{3/4} \tan ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2-2 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-2 b}+a} \sqrt [4]{a x^4+b}}\right )}{4 \left (\sqrt {a^2-2 b}+a\right )^{3/4} \sqrt {a^2-2 b}}-\frac {\left (-a \sqrt {a^2-2 b}+a^2-2 b\right )^{3/4} \tanh ^{-1}\left (\frac {x \sqrt [4]{-a \sqrt {a^2-2 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-2 b}} \sqrt [4]{a x^4+b}}\right )}{4 \left (a-\sqrt {a^2-2 b}\right )^{3/4} \sqrt {a^2-2 b}}+\frac {\left (a \sqrt {a^2-2 b}+a^2-2 b\right )^{3/4} \tanh ^{-1}\left (\frac {x \sqrt [4]{a \sqrt {a^2-2 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-2 b}+a} \sqrt [4]{a x^4+b}}\right )}{4 \left (\sqrt {a^2-2 b}+a\right )^{3/4} \sqrt {a^2-2 b}} \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 408
Rule 1428
Rubi steps
\begin {align*} \int \frac {\left (b+a x^4\right )^{3/4}}{b+2 a x^4+2 x^8} \, dx &=\frac {2 \int \frac {\left (b+a x^4\right )^{3/4}}{2 a-2 \sqrt {a^2-2 b}+4 x^4} \, dx}{\sqrt {a^2-2 b}}-\frac {2 \int \frac {\left (b+a x^4\right )^{3/4}}{2 a+2 \sqrt {a^2-2 b}+4 x^4} \, dx}{\sqrt {a^2-2 b}}\\ &=\frac {\left (a \left (a+\sqrt {a^2-2 b}\right )-2 b\right ) \int \frac {1}{\left (2 a+2 \sqrt {a^2-2 b}+4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2-2 b}}-\frac {\left (a^2-a \sqrt {a^2-2 b}-2 b\right ) \int \frac {1}{\left (2 a-2 \sqrt {a^2-2 b}+4 x^4\right ) \sqrt [4]{b+a x^4}} \, dx}{\sqrt {a^2-2 b}}\\ &=\frac {\left (a \left (a+\sqrt {a^2-2 b}\right )-2 b\right ) \operatorname {Subst}\left (\int \frac {1}{2 a+2 \sqrt {a^2-2 b}-\left (a \left (2 a+2 \sqrt {a^2-2 b}\right )-4 b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2-2 b}}-\frac {\left (a^2-a \sqrt {a^2-2 b}-2 b\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-2 \sqrt {a^2-2 b}-\left (a \left (2 a-2 \sqrt {a^2-2 b}\right )-4 b\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{\sqrt {a^2-2 b}}\\ &=\frac {\left (a \left (a+\sqrt {a^2-2 b}\right )-2 b\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-2 b}}-\sqrt {a^2+a \sqrt {a^2-2 b}-2 b} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a+\sqrt {a^2-2 b}} \sqrt {a^2-2 b}}+\frac {\left (a \left (a+\sqrt {a^2-2 b}\right )-2 b\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\sqrt {a^2-2 b}}+\sqrt {a^2+a \sqrt {a^2-2 b}-2 b} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a+\sqrt {a^2-2 b}} \sqrt {a^2-2 b}}-\frac {\left (a^2-a \sqrt {a^2-2 b}-2 b\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-2 b}}-\sqrt {a^2-a \sqrt {a^2-2 b}-2 b} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a-\sqrt {a^2-2 b}} \sqrt {a^2-2 b}}-\frac {\left (a^2-a \sqrt {a^2-2 b}-2 b\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-\sqrt {a^2-2 b}}+\sqrt {a^2-a \sqrt {a^2-2 b}-2 b} x^2} \, dx,x,\frac {x}{\sqrt [4]{b+a x^4}}\right )}{4 \sqrt {a-\sqrt {a^2-2 b}} \sqrt {a^2-2 b}}\\ &=-\frac {\left (a^2-a \sqrt {a^2-2 b}-2 b\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-2 b}-2 b} x}{\sqrt [4]{a-\sqrt {a^2-2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \left (a-\sqrt {a^2-2 b}\right )^{3/4} \sqrt {a^2-2 b}}+\frac {\left (a^2+a \sqrt {a^2-2 b}-2 b\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-2 b}-2 b} x}{\sqrt [4]{a+\sqrt {a^2-2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \left (a+\sqrt {a^2-2 b}\right )^{3/4} \sqrt {a^2-2 b}}-\frac {\left (a^2-a \sqrt {a^2-2 b}-2 b\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2-a \sqrt {a^2-2 b}-2 b} x}{\sqrt [4]{a-\sqrt {a^2-2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \left (a-\sqrt {a^2-2 b}\right )^{3/4} \sqrt {a^2-2 b}}+\frac {\left (a^2+a \sqrt {a^2-2 b}-2 b\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{a^2+a \sqrt {a^2-2 b}-2 b} x}{\sqrt [4]{a+\sqrt {a^2-2 b}} \sqrt [4]{b+a x^4}}\right )}{4 \left (a+\sqrt {a^2-2 b}\right )^{3/4} \sqrt {a^2-2 b}}\\ \end {align*}
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Mathematica [F] time = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b+a x^4\right )^{3/4}}{b+2 a x^4+2 x^8} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 37.48, size = 510, normalized size = 1.11 \begin {gather*} \frac {\left (\sqrt {2}-i \sqrt {2 \left (3+2 \sqrt {2}\right )}\right ) \tan ^{-1}\left (\frac {\left ((-1+i)-(1+i) (-1)^{3/4}\right ) x^2 \sqrt [4]{a^2-2 b}+(1+i) (-1)^{3/4} \sqrt {a x^4+b}+(1+i) \sqrt {a x^4+b}}{2 x \sqrt [8]{a^2-2 b} \sqrt [4]{a x^4+b}}\right )}{16 \sqrt [8]{a^2-2 b}}-\frac {i \left (\sqrt {2 \left (3-2 \sqrt {2}\right )}-i \sqrt {2}\right ) \tan ^{-1}\left (\frac {2 x \sqrt [8]{a^2-2 b} \sqrt [4]{a x^4+b}}{\left (\sqrt {2}+(1-i)\right ) x^2 \sqrt [4]{a^2-2 b}-\sqrt {2} \sqrt {a x^4+b}-(1+i) \sqrt {a x^4+b}}\right )}{16 \sqrt [8]{a^2-2 b}}+\frac {\left (-(-1)^{3/4}-i\right ) \tanh ^{-1}\left (\frac {\left ((-2+2 i)-(2+2 i) (-1)^{3/4}\right ) x^2 \sqrt [4]{a^2-2 b}-(2+2 i) (-1)^{3/4} \sqrt {a x^4+b}-(2+2 i) \sqrt {a x^4+b}}{4 x \sqrt [8]{a^2-2 b} \sqrt [4]{a x^4+b}}\right )}{8 \sqrt [8]{a^2-2 b}}+\frac {\left (\sqrt {2}+i \sqrt {2 \left (3-2 \sqrt {2}\right )}\right ) \tanh ^{-1}\left (\frac {\left (\sqrt {2}+(1-i)\right ) x^2 \sqrt [4]{a^2-2 b}+\sqrt {2} \sqrt {a x^4+b}+(1+i) \sqrt {a x^4+b}}{2 x \sqrt [8]{a^2-2 b} \sqrt [4]{a x^4+b}}\right )}{16 \sqrt [8]{a^2-2 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 {{\left (a x^{4} + b\right )}^{\frac {3}{4}}}{2 \, x^{8} + 2 \, a x^{4} + b}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.53, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a \,x^{4}+b \right )^{\frac {3}{4}}}{2 x^{8}+2 a \,x^{4}+b}\, 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 {{\left (a x^{4} + b\right )}^{\frac {3}{4}}}{2 \, x^{8} + 2 \, a x^{4} + b}\,{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 {{\left (a\,x^4+b\right )}^{3/4}}{2\,x^8+2\,a\,x^4+b} \,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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