Optimal. Leaf size=128 \[ -\frac {\sqrt [4]{a+b x^4}}{11 x^{11}}-\frac {b \sqrt [4]{a+b x^4}}{77 a x^7}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{77 a^2 x^3}-\frac {4 b^{7/2} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{77 a^{5/2} \left (a+b x^4\right )^{3/4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {283, 331, 243,
342, 281, 237} \begin {gather*} -\frac {4 b^{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 )}{77 a^{5/2} \left (a+b x^4\right )^{3/4}}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{77 a^2 x^3}-\frac {\sqrt [4]{a+b x^4}}{11 x^{11}}-\frac {b \sqrt [4]{a+b x^4}}{77 a x^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 237
Rule 243
Rule 281
Rule 283
Rule 331
Rule 342
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a+b x^4}}{x^{12}} \, dx &=-\frac {\sqrt [4]{a+b x^4}}{11 x^{11}}+\frac {1}{11} b \int \frac {1}{x^8 \left (a+b x^4\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{a+b x^4}}{11 x^{11}}-\frac {b \sqrt [4]{a+b x^4}}{77 a x^7}-\frac {\left (6 b^2\right ) \int \frac {1}{x^4 \left (a+b x^4\right )^{3/4}} \, dx}{77 a}\\ &=-\frac {\sqrt [4]{a+b x^4}}{11 x^{11}}-\frac {b \sqrt [4]{a+b x^4}}{77 a x^7}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{77 a^2 x^3}+\frac {\left (4 b^3\right ) \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{77 a^2}\\ &=-\frac {\sqrt [4]{a+b x^4}}{11 x^{11}}-\frac {b \sqrt [4]{a+b x^4}}{77 a x^7}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{77 a^2 x^3}+\frac {\left (4 b^3 \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}{77 a^2 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{11 x^{11}}-\frac {b \sqrt [4]{a+b x^4}}{77 a x^7}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{77 a^2 x^3}-\frac {\left (4 b^3 \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 )}{77 a^2 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{11 x^{11}}-\frac {b \sqrt [4]{a+b x^4}}{77 a x^7}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{77 a^2 x^3}-\frac {\left (2 b^3 \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 )}{77 a^2 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{11 x^{11}}-\frac {b \sqrt [4]{a+b x^4}}{77 a x^7}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{77 a^2 x^3}-\frac {4 b^{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 )}{77 a^{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 = 10.01, size = 51, normalized size = 0.40 \begin {gather*} -\frac {\sqrt [4]{a+b x^4} \, _2F_1\left (-\frac {11}{4},-\frac {1}{4};-\frac {7}{4};-\frac {b x^4}{a}\right )}{11 x^{11} \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 \[\int \frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{x^{12}}\, dx\]
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 \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 = 0.07, size = 15, normalized size = 0.12 \begin {gather*} {\rm integral}\left (\frac {{\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 = 0.82, size = 46, normalized size = 0.36 \begin {gather*} \frac {\sqrt [4]{a} \Gamma \left (- \frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{4}, - \frac {1}{4} \\ - \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{11} \Gamma \left (- \frac {7}{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 {{\left (b\,x^4+a\right )}^{1/4}}{x^{12}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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