Optimal. Leaf size=131 \[ -\frac {\sqrt [4]{a+b x^4}}{11 a x^{11}}+\frac {10 b \sqrt [4]{a+b x^4}}{77 a^2 x^7}-\frac {20 b^2 \sqrt [4]{a+b x^4}}{77 a^3 x^3}+\frac {40 b^{7/2} \left (1+\frac {a}{b x^4}\right )^{3/4} x^3 F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{77 a^{7/2} \left (a+b x^4\right )^{3/4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {331, 243, 342,
281, 237} \begin {gather*} \frac {40 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^{7/2} \left (a+b x^4\right )^{3/4}}-\frac {20 b^2 \sqrt [4]{a+b x^4}}{77 a^3 x^3}+\frac {10 b \sqrt [4]{a+b x^4}}{77 a^2 x^7}-\frac {\sqrt [4]{a+b x^4}}{11 a x^{11}} \end {gather*}
Antiderivative was successfully verified.
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Rule 237
Rule 243
Rule 281
Rule 331
Rule 342
Rubi steps
\begin {align*} \int \frac {1}{x^{12} \left (a+b x^4\right )^{3/4}} \, dx &=-\frac {\sqrt [4]{a+b x^4}}{11 a x^{11}}-\frac {(10 b) \int \frac {1}{x^8 \left (a+b x^4\right )^{3/4}} \, dx}{11 a}\\ &=-\frac {\sqrt [4]{a+b x^4}}{11 a x^{11}}+\frac {10 b \sqrt [4]{a+b x^4}}{77 a^2 x^7}+\frac {\left (60 b^2\right ) \int \frac {1}{x^4 \left (a+b x^4\right )^{3/4}} \, dx}{77 a^2}\\ &=-\frac {\sqrt [4]{a+b x^4}}{11 a x^{11}}+\frac {10 b \sqrt [4]{a+b x^4}}{77 a^2 x^7}-\frac {20 b^2 \sqrt [4]{a+b x^4}}{77 a^3 x^3}-\frac {\left (40 b^3\right ) \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{77 a^3}\\ &=-\frac {\sqrt [4]{a+b x^4}}{11 a x^{11}}+\frac {10 b \sqrt [4]{a+b x^4}}{77 a^2 x^7}-\frac {20 b^2 \sqrt [4]{a+b x^4}}{77 a^3 x^3}-\frac {\left (40 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^3 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{11 a x^{11}}+\frac {10 b \sqrt [4]{a+b x^4}}{77 a^2 x^7}-\frac {20 b^2 \sqrt [4]{a+b x^4}}{77 a^3 x^3}+\frac {\left (40 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^3 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{11 a x^{11}}+\frac {10 b \sqrt [4]{a+b x^4}}{77 a^2 x^7}-\frac {20 b^2 \sqrt [4]{a+b x^4}}{77 a^3 x^3}+\frac {\left (20 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^3 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{11 a x^{11}}+\frac {10 b \sqrt [4]{a+b x^4}}{77 a^2 x^7}-\frac {20 b^2 \sqrt [4]{a+b x^4}}{77 a^3 x^3}+\frac {40 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^{7/2} \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 = 0.39 \begin {gather*} -\frac {\left (1+\frac {b x^4}{a}\right )^{3/4} \, _2F_1\left (-\frac {11}{4},\frac {3}{4};-\frac {7}{4};-\frac {b x^4}{a}\right )}{11 x^{11} \left (a+b x^4\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{12} \left (b \,x^{4}+a \right )^{\frac {3}{4}}}\, dx\]
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 = 0.19 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{b x^{16} + a 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 = 0.94, size = 44, normalized size = 0.34 \begin {gather*} \frac {\Gamma \left (- \frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{4}, \frac {3}{4} \\ - \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{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 {1}{x^{12}\,{\left (b\,x^4+a\right )}^{3/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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