Optimal. Leaf size=263 \[ -\frac {3 a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{7/4}}+\frac {3 a^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt {2} b^{7/4}}+\frac {3 a^2 \log \left (-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1\right )}{128 \sqrt {2} b^{7/4}}-\frac {3 a^2 \log \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+1\right )}{128 \sqrt {2} b^{7/4}}+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}-\frac {a x^3 \sqrt [4]{a-b x^4}}{32 b} \]
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Rubi [A] time = 0.16, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {279, 321, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac {3 a^2 \log \left (\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt {2} b^{7/4}}-\frac {3 a^2 \log \left (\frac {\sqrt {b} x^2}{\sqrt {a-b x^4}}+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{128 \sqrt {2} b^{7/4}}-\frac {3 a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{7/4}}+\frac {3 a^2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt {2} b^{7/4}}+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}-\frac {a x^3 \sqrt [4]{a-b x^4}}{32 b} \]
Antiderivative was successfully verified.
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Rule 204
Rule 279
Rule 297
Rule 321
Rule 331
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int x^6 \sqrt [4]{a-b x^4} \, dx &=\frac {1}{8} x^7 \sqrt [4]{a-b x^4}+\frac {1}{8} a \int \frac {x^6}{\left (a-b x^4\right )^{3/4}} \, dx\\ &=-\frac {a x^3 \sqrt [4]{a-b x^4}}{32 b}+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}+\frac {\left (3 a^2\right ) \int \frac {x^2}{\left (a-b x^4\right )^{3/4}} \, dx}{32 b}\\ &=-\frac {a x^3 \sqrt [4]{a-b x^4}}{32 b}+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+b x^4} \, dx,x,\frac {x}{\sqrt [4]{a-b x^4}}\right )}{32 b}\\ &=-\frac {a x^3 \sqrt [4]{a-b x^4}}{32 b}+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}-\frac {\left (3 a^2\right ) \operatorname {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^{3/2}}+\frac {\left (3 a^2\right ) \operatorname {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^{3/2}}\\ &=-\frac {a x^3 \sqrt [4]{a-b x^4}}{32 b}+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}+\frac {\left (3 a^2\right ) \operatorname {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^2}+\frac {\left (3 a^2\right ) \operatorname {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^2}+\frac {\left (3 a^2\right ) \operatorname {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^{7/4}}+\frac {\left (3 a^2\right ) \operatorname {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^{7/4}}\\ &=-\frac {a x^3 \sqrt [4]{a-b x^4}}{32 b}+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}+\frac {3 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^{7/4}}-\frac {3 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^{7/4}}+\frac {\left (3 a^2\right ) \operatorname {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^{7/4}}-\frac {\left (3 a^2\right ) \operatorname {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^{7/4}}\\ &=-\frac {a x^3 \sqrt [4]{a-b x^4}}{32 b}+\frac {1}{8} x^7 \sqrt [4]{a-b x^4}-\frac {3 a^2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{7/4}}+\frac {3 a^2 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt {2} b^{7/4}}+\frac {3 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^{7/4}}-\frac {3 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^{7/4}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 66, normalized size = 0.25 \[ \frac {x^3 \sqrt [4]{a-b x^4} \left (\frac {a \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {b x^4}{a}\right )}{\sqrt [4]{1-\frac {b x^4}{a}}}-a+b x^4\right )}{8 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 234, normalized size = 0.89 \[ -\frac {12 \, \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b \arctan \left (-\frac {{\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{2} \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {3}{4}} b^{5} - \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {3}{4}} b^{5} x \sqrt {\frac {\sqrt {-\frac {a^{8}}{b^{7}}} b^{4} x^{2} + \sqrt {-b x^{4} + a} a^{4}}{x^{2}}}}{a^{8} x}\right ) + 3 \, \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b \log \left (\frac {3 \, {\left (\left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b^{2} x + {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{2}\right )}}{x}\right ) - 3 \, \left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b \log \left (-\frac {3 \, {\left (\left (-\frac {a^{8}}{b^{7}}\right )^{\frac {1}{4}} b^{2} x - {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{2}\right )}}{x}\right ) - 4 \, {\left (4 \, b x^{7} - a x^{3}\right )} {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{128 \, b} \]
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 \[ \int {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \left (-b \,x^{4}+a \right )^{\frac {1}{4}} x^{6}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.09, size = 272, normalized size = 1.03 \[ \frac {\frac {3 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}} a^{2} b}{x} - \frac {{\left (-b x^{4} + a\right )}^{\frac {5}{4}} a^{2}}{x^{5}}}{32 \, {\left (b^{3} - \frac {2 \, {\left (b x^{4} - a\right )} b^{2}}{x^{4}} + \frac {{\left (b x^{4} - a\right )}^{2} b}{x^{8}}\right )}} - \frac {3 \, {\left (\frac {2 \, \sqrt {2} a^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} + \frac {2 \, \sqrt {2} a^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - \frac {2 \, {\left (-b x^{4} + a\right )}^{\frac {1}{4}}}{x}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {3}{4}}} + \frac {\sqrt {2} a^{2} \log \left (\sqrt {b} + \frac {\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{\frac {1}{4}}}{x} + \frac {\sqrt {-b x^{4} + a}}{x^{2}}\right )}{b^{\frac {3}{4}}} - \frac {\sqrt {2} a^{2} \log \left (\sqrt {b} - \frac {\sqrt {2} {\left (-b x^{4} + a\right )}^{\frac {1}{4}} b^{\frac {1}{4}}}{x} + \frac {\sqrt {-b x^{4} + a}}{x^{2}}\right )}{b^{\frac {3}{4}}}\right )}}{256 \, b} \]
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 \[ \int x^6\,{\left (a-b\,x^4\right )}^{1/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.08, size = 41, normalized size = 0.16 \[ \frac {\sqrt [4]{a} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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