Optimal. Leaf size=156 \[ -\frac {3 a^{7/2} x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{112 b^{5/2} \left (a-b x^4\right )^{3/4}}-\frac {3 a^3 x \sqrt [4]{a-b x^4}}{112 b^3}-\frac {3 a^2 x^5 \sqrt [4]{a-b x^4}}{280 b^2}+\frac {1}{14} x^{13} \sqrt [4]{a-b x^4}-\frac {a x^9 \sqrt [4]{a-b x^4}}{140 b} \]
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Rubi [A] time = 0.08, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {279, 321, 237, 335, 275, 232} \[ -\frac {3 a^2 x^5 \sqrt [4]{a-b x^4}}{280 b^2}-\frac {3 a^3 x \sqrt [4]{a-b x^4}}{112 b^3}-\frac {3 a^{7/2} x^3 \left (1-\frac {a}{b x^4}\right )^{3/4} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{112 b^{5/2} \left (a-b x^4\right )^{3/4}}+\frac {1}{14} x^{13} \sqrt [4]{a-b x^4}-\frac {a x^9 \sqrt [4]{a-b x^4}}{140 b} \]
Antiderivative was successfully verified.
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Rule 232
Rule 237
Rule 275
Rule 279
Rule 321
Rule 335
Rubi steps
\begin {align*} \int x^{12} \sqrt [4]{a-b x^4} \, dx &=\frac {1}{14} x^{13} \sqrt [4]{a-b x^4}+\frac {1}{14} a \int \frac {x^{12}}{\left (a-b x^4\right )^{3/4}} \, dx\\ &=-\frac {a x^9 \sqrt [4]{a-b x^4}}{140 b}+\frac {1}{14} x^{13} \sqrt [4]{a-b x^4}+\frac {\left (9 a^2\right ) \int \frac {x^8}{\left (a-b x^4\right )^{3/4}} \, dx}{140 b}\\ &=-\frac {3 a^2 x^5 \sqrt [4]{a-b x^4}}{280 b^2}-\frac {a x^9 \sqrt [4]{a-b x^4}}{140 b}+\frac {1}{14} x^{13} \sqrt [4]{a-b x^4}+\frac {\left (3 a^3\right ) \int \frac {x^4}{\left (a-b x^4\right )^{3/4}} \, dx}{56 b^2}\\ &=-\frac {3 a^3 x \sqrt [4]{a-b x^4}}{112 b^3}-\frac {3 a^2 x^5 \sqrt [4]{a-b x^4}}{280 b^2}-\frac {a x^9 \sqrt [4]{a-b x^4}}{140 b}+\frac {1}{14} x^{13} \sqrt [4]{a-b x^4}+\frac {\left (3 a^4\right ) \int \frac {1}{\left (a-b x^4\right )^{3/4}} \, dx}{112 b^3}\\ &=-\frac {3 a^3 x \sqrt [4]{a-b x^4}}{112 b^3}-\frac {3 a^2 x^5 \sqrt [4]{a-b x^4}}{280 b^2}-\frac {a x^9 \sqrt [4]{a-b x^4}}{140 b}+\frac {1}{14} x^{13} \sqrt [4]{a-b x^4}+\frac {\left (3 a^4 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \int \frac {1}{\left (1-\frac {a}{b x^4}\right )^{3/4} x^3} \, dx}{112 b^3 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {3 a^3 x \sqrt [4]{a-b x^4}}{112 b^3}-\frac {3 a^2 x^5 \sqrt [4]{a-b x^4}}{280 b^2}-\frac {a x^9 \sqrt [4]{a-b x^4}}{140 b}+\frac {1}{14} x^{13} \sqrt [4]{a-b x^4}-\frac {\left (3 a^4 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1-\frac {a x^4}{b}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )}{112 b^3 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {3 a^3 x \sqrt [4]{a-b x^4}}{112 b^3}-\frac {3 a^2 x^5 \sqrt [4]{a-b x^4}}{280 b^2}-\frac {a x^9 \sqrt [4]{a-b x^4}}{140 b}+\frac {1}{14} x^{13} \sqrt [4]{a-b x^4}-\frac {\left (3 a^4 \left (1-\frac {a}{b x^4}\right )^{3/4} x^3\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {a x^2}{b}\right )^{3/4}} \, dx,x,\frac {1}{x^2}\right )}{224 b^3 \left (a-b x^4\right )^{3/4}}\\ &=-\frac {3 a^3 x \sqrt [4]{a-b x^4}}{112 b^3}-\frac {3 a^2 x^5 \sqrt [4]{a-b x^4}}{280 b^2}-\frac {a x^9 \sqrt [4]{a-b x^4}}{140 b}+\frac {1}{14} x^{13} \sqrt [4]{a-b x^4}-\frac {3 a^{7/2} \left (1-\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{112 b^{5/2} \left (a-b x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 108, normalized size = 0.69 \[ \frac {x \sqrt [4]{a-b x^4} \left (15 a^3 \, _2F_1\left (-\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {b x^4}{a}\right )-\sqrt [4]{1-\frac {b x^4}{a}} \left (15 a^3+3 a^2 b x^4+2 a b^2 x^8-20 b^3 x^{12}\right )\right )}{280 b^3 \sqrt [4]{1-\frac {b x^4}{a}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{12}, x\right ) \]
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 \[ \int {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{12}\,{d x} \]
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 \[ \int \left (-b \,x^{4}+a \right )^{\frac {1}{4}} x^{12}\, dx \]
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 \[ \int {\left (-b x^{4} + a\right )}^{\frac {1}{4}} x^{12}\,{d x} \]
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 \[ \int x^{12}\,{\left (a-b\,x^4\right )}^{1/4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.25, size = 41, normalized size = 0.26 \[ \frac {\sqrt [4]{a} x^{13} \Gamma \left (\frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {13}{4} \\ \frac {17}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {17}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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