Optimal. Leaf size=264 \[ -\frac {5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}-\frac {5 a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{9/4}}+\frac {5 a^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{9/4}}-\frac {5 a^2 \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt {2} b^{9/4}}+\frac {5 a^2 \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt {2} b^{9/4}} \]
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Rubi [A]
time = 0.11, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {327, 246, 217,
1179, 642, 1176, 631, 210} \begin {gather*} -\frac {5 a^2 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{9/4}}+\frac {5 a^2 \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt {2} b^{9/4}}-\frac {5 a^2 \log \left (-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1\right )}{128 \sqrt {2} b^{9/4}}+\frac {5 a^2 \log \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1\right )}{128 \sqrt {2} b^{9/4}}-\frac {5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 246
Rule 327
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt [4]{a-b x^4}} \, dx &=-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}+\frac {(5 a) \int \frac {x^4}{\sqrt [4]{a-b x^4}} \, dx}{8 b}\\ &=-\frac {5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}+\frac {\left (5 a^2\right ) \int \frac {1}{\sqrt [4]{a-b x^4}} \, dx}{32 b^2}\\ &=-\frac {5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}+\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{1+b x^4} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{32 b^2}\\ &=-\frac {5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}+\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1-\sqrt {b} x^2}{1+b x^4} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{64 b^2}+\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1+\sqrt {b} x^2}{1+b x^4} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{64 b^2}\\ &=-\frac {5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}+\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {b}}-\frac {\sqrt {2} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{128 b^{5/2}}+\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{\sqrt {b}}+\frac {\sqrt {2} x}{\sqrt [4]{b}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{128 b^{5/2}}-\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2}}{\sqrt [4]{b}}+2 x}{-\frac {1}{\sqrt {b}}-\frac {\sqrt {2} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt {2} b^{9/4}}-\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2}}{\sqrt [4]{b}}-2 x}{-\frac {1}{\sqrt {b}}+\frac {\sqrt {2} x}{\sqrt [4]{b}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt {2} b^{9/4}}\\ &=-\frac {5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}-\frac {5 a^2 \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt {2} b^{9/4}}+\frac {5 a^2 \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt {2} b^{9/4}}+\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{9/4}}-\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{9/4}}\\ &=-\frac {5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac {x^5 \left (a-b x^4\right )^{3/4}}{8 b}-\frac {5 a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{9/4}}+\frac {5 a^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{9/4}}-\frac {5 a^2 \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt {2} b^{9/4}}+\frac {5 a^2 \log \left (1+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{128 \sqrt {2} b^{9/4}}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 159, normalized size = 0.60 \begin {gather*} \frac {-4 \sqrt [4]{b} x \left (a-b x^4\right )^{3/4} \left (5 a+4 b x^4\right )-5 \sqrt {2} a^2 \tan ^{-1}\left (\frac {-\sqrt {b} x^2+\sqrt {a-b x^4}}{\sqrt {2} \sqrt [4]{b} x \sqrt [4]{a-b x^4}}\right )+5 \sqrt {2} a^2 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x \sqrt [4]{a-b x^4}}{\sqrt {b} x^2+\sqrt {a-b x^4}}\right )}{128 b^{9/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{8}}{\left (-b \,x^{4}+a \right )^{\frac {1}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 265, normalized size = 1.00 \begin {gather*} -\frac {5 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {b} + \frac {\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{\frac {1}{4}}}{x} + \frac {\sqrt {-b x^{4} + a}}{x^{2}}\right )}{b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} - \frac {\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{\frac {1}{4}}}{x} + \frac {\sqrt {-b x^{4} + a}}{x^{2}}\right )}{b^{\frac {1}{4}}}\right )} a^{2}}{256 \, b^{2}} - \frac {\frac {9 \, {\left (-b x^{4} + a\right )}^{\frac {3}{4}} a^{2} b}{x^{3}} + \frac {5 \, {\left (-b x^{4} + a\right )}^{\frac {7}{4}} a^{2}}{x^{7}}}{32 \, {\left (b^{4} - \frac {2 \, {\left (b x^{4} - a\right )} b^{3}}{x^{4}} + \frac {{\left (b x^{4} - a\right )}^{2} b^{2}}{x^{8}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 243, normalized size = 0.92 \begin {gather*} \frac {20 \, b^{2} \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {1}{4}} \arctan \left (-\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{6} b^{2} \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {1}{4}} - b^{2} x \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {1}{4}} \sqrt {-\frac {a^{8} b^{5} x^{2} \sqrt {-\frac {a^{8}}{b^{9}}} - \sqrt {-b x^{4} + a} a^{12}}{x^{2}}}}{a^{8} x}\right ) - 5 \, b^{2} \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {1}{4}} \log \left (\frac {125 \, {\left (b^{7} x \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {3}{4}} + {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{6}\right )}}{x}\right ) + 5 \, b^{2} \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {1}{4}} \log \left (-\frac {125 \, {\left (b^{7} x \left (-\frac {a^{8}}{b^{9}}\right )^{\frac {3}{4}} - {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{6}\right )}}{x}\right ) - 4 \, {\left (4 \, b x^{5} + 5 \, a x\right )} {\left (-b x^{4} + a\right )}^{\frac {3}{4}}}{128 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.52, size = 39, normalized size = 0.15 \begin {gather*} \frac {x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac {13}{4}\right )} \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 = -0.00 \begin {gather*} \int \frac {x^8}{{\left (a-b\,x^4\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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