Optimal. Leaf size=153 \[ \frac {8 b^3}{39 a^3 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}+\frac {10 b \left (a+b x^4\right )^{3/4}}{117 a^2 x^9}-\frac {4 b^2 \left (a+b x^4\right )^{3/4}}{39 a^3 x^5}-\frac {8 b^{7/2} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{39 a^{7/2} \sqrt [4]{a+b x^4}} \]
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Rubi [A]
time = 0.05, antiderivative size = 153, 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 = {331, 318, 287,
342, 281, 202} \begin {gather*} -\frac {8 b^{7/2} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{39 a^{7/2} \sqrt [4]{a+b x^4}}+\frac {8 b^3}{39 a^3 x \sqrt [4]{a+b x^4}}-\frac {4 b^2 \left (a+b x^4\right )^{3/4}}{39 a^3 x^5}+\frac {10 b \left (a+b x^4\right )^{3/4}}{117 a^2 x^9}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 281
Rule 287
Rule 318
Rule 331
Rule 342
Rubi steps
\begin {align*} \int \frac {1}{x^{14} \sqrt [4]{a+b x^4}} \, dx &=-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}-\frac {(10 b) \int \frac {1}{x^{10} \sqrt [4]{a+b x^4}} \, dx}{13 a}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}+\frac {10 b \left (a+b x^4\right )^{3/4}}{117 a^2 x^9}+\frac {\left (20 b^2\right ) \int \frac {1}{x^6 \sqrt [4]{a+b x^4}} \, dx}{39 a^2}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}+\frac {10 b \left (a+b x^4\right )^{3/4}}{117 a^2 x^9}-\frac {4 b^2 \left (a+b x^4\right )^{3/4}}{39 a^3 x^5}-\frac {\left (8 b^3\right ) \int \frac {1}{x^2 \sqrt [4]{a+b x^4}} \, dx}{39 a^3}\\ &=\frac {8 b^3}{39 a^3 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}+\frac {10 b \left (a+b x^4\right )^{3/4}}{117 a^2 x^9}-\frac {4 b^2 \left (a+b x^4\right )^{3/4}}{39 a^3 x^5}+\frac {\left (8 b^4\right ) \int \frac {x^2}{\left (a+b x^4\right )^{5/4}} \, dx}{39 a^3}\\ &=\frac {8 b^3}{39 a^3 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}+\frac {10 b \left (a+b x^4\right )^{3/4}}{117 a^2 x^9}-\frac {4 b^2 \left (a+b x^4\right )^{3/4}}{39 a^3 x^5}+\frac {\left (8 b^3 \sqrt [4]{1+\frac {a}{b x^4}} x\right ) \int \frac {1}{\left (1+\frac {a}{b x^4}\right )^{5/4} x^3} \, dx}{39 a^3 \sqrt [4]{a+b x^4}}\\ &=\frac {8 b^3}{39 a^3 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}+\frac {10 b \left (a+b x^4\right )^{3/4}}{117 a^2 x^9}-\frac {4 b^2 \left (a+b x^4\right )^{3/4}}{39 a^3 x^5}-\frac {\left (8 b^3 \sqrt [4]{1+\frac {a}{b x^4}} x\right ) \text {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{39 a^3 \sqrt [4]{a+b x^4}}\\ &=\frac {8 b^3}{39 a^3 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}+\frac {10 b \left (a+b x^4\right )^{3/4}}{117 a^2 x^9}-\frac {4 b^2 \left (a+b x^4\right )^{3/4}}{39 a^3 x^5}-\frac {\left (4 b^3 \sqrt [4]{1+\frac {a}{b x^4}} x\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x^2}\right )}{39 a^3 \sqrt [4]{a+b x^4}}\\ &=\frac {8 b^3}{39 a^3 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{13 a x^{13}}+\frac {10 b \left (a+b x^4\right )^{3/4}}{117 a^2 x^9}-\frac {4 b^2 \left (a+b x^4\right )^{3/4}}{39 a^3 x^5}-\frac {8 b^{7/2} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{39 a^{7/2} \sqrt [4]{a+b x^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 = 10.01, size = 51, normalized size = 0.33 \begin {gather*} -\frac {\sqrt [4]{1+\frac {b x^4}{a}} \, _2F_1\left (-\frac {13}{4},\frac {1}{4};-\frac {9}{4};-\frac {b x^4}{a}\right )}{13 x^{13} \sqrt [4]{a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{14} \left (b \,x^{4}+a \right )^{\frac {1}{4}}}\, dx\]
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 = 25, normalized size = 0.16 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{b x^{18} + a x^{14}}, 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 = 1.00, size = 44, normalized size = 0.29 \begin {gather*} \frac {\Gamma \left (- \frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {13}{4}, \frac {1}{4} \\ - \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} x^{13} \Gamma \left (- \frac {9}{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 \frac {1}{x^{14}\,{\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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