Optimal. Leaf size=150 \[ \frac {3 a^3 x \sqrt [4]{a+b x^4}}{112 b^3}-\frac {3 a^2 x^5 \sqrt [4]{a+b x^4}}{280 b^2}+\frac {a x^9 \sqrt [4]{a+b x^4}}{140 b}+\frac {1}{14} x^{13} \sqrt [4]{a+b x^4}+\frac {3 a^{7/2} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{112 b^{5/2} \left (a+b x^4\right )^{3/4}} \]
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Rubi [A]
time = 0.06, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {285, 327, 243,
342, 281, 237} \begin {gather*} \frac {3 a^{7/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-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} \end {gather*}
Antiderivative was successfully verified.
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Rule 237
Rule 243
Rule 281
Rule 285
Rule 327
Rule 342
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 ) \text {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 ) \text {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} \cot ^{-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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 7.15, size = 105, normalized size = 0.70 \begin {gather*} \frac {x \sqrt [4]{a+b x^4} \left (\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 )-15 a^3 \, _2F_1\left (-\frac {1}{4},\frac {1}{4};\frac {5}{4};-\frac {b x^4}{a}\right )\right )}{280 b^3 \sqrt [4]{1+\frac {b x^4}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{12} \left (b \,x^{4}+a \right )^{\frac {1}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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]
time = 0.08, size = 15, normalized size = 0.10 \begin {gather*} {\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{12}, x\right ) \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 = 0.73, size = 39, normalized size = 0.26 \begin {gather*} \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^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {17}{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 = -0.01 \begin {gather*} \int x^{12}\,{\left (b\,x^4+a\right )}^{1/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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