Optimal. Leaf size=152 \[ \frac {8 b^{9/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{231 a^{7/2} \left (a+b x^4\right )^{3/4}}-\frac {4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}-\frac {\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac {b \sqrt [4]{a+b x^4}}{165 a x^{11}} \]
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Rubi [A] time = 0.08, antiderivative size = 152, 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 = {277, 325, 237, 335, 275, 231} \[ -\frac {4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}+\frac {8 b^{9/2} x^3 \left (\frac {a}{b x^4}+1\right )^{3/4} F\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{231 a^{7/2} \left (a+b x^4\right )^{3/4}}-\frac {b \sqrt [4]{a+b x^4}}{165 a x^{11}}-\frac {\sqrt [4]{a+b x^4}}{15 x^{15}} \]
Antiderivative was successfully verified.
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Rule 231
Rule 237
Rule 275
Rule 277
Rule 325
Rule 335
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{a+b x^4}}{x^{16}} \, dx &=-\frac {\sqrt [4]{a+b x^4}}{15 x^{15}}+\frac {1}{15} b \int \frac {1}{x^{12} \left (a+b x^4\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac {b \sqrt [4]{a+b x^4}}{165 a x^{11}}-\frac {\left (2 b^2\right ) \int \frac {1}{x^8 \left (a+b x^4\right )^{3/4}} \, dx}{33 a}\\ &=-\frac {\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac {b \sqrt [4]{a+b x^4}}{165 a x^{11}}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}+\frac {\left (4 b^3\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}}{15 x^{15}}-\frac {b \sqrt [4]{a+b x^4}}{165 a x^{11}}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}-\frac {4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}-\frac {\left (8 b^4\right ) \int \frac {1}{\left (a+b x^4\right )^{3/4}} \, dx}{231 a^3}\\ &=-\frac {\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac {b \sqrt [4]{a+b x^4}}{165 a x^{11}}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}-\frac {4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}-\frac {\left (8 b^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}{231 a^3 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac {b \sqrt [4]{a+b x^4}}{165 a x^{11}}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}-\frac {4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}+\frac {\left (8 b^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 )}{231 a^3 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac {b \sqrt [4]{a+b x^4}}{165 a x^{11}}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}-\frac {4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}+\frac {\left (4 b^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 )}{231 a^3 \left (a+b x^4\right )^{3/4}}\\ &=-\frac {\sqrt [4]{a+b x^4}}{15 x^{15}}-\frac {b \sqrt [4]{a+b x^4}}{165 a x^{11}}+\frac {2 b^2 \sqrt [4]{a+b x^4}}{231 a^2 x^7}-\frac {4 b^3 \sqrt [4]{a+b x^4}}{231 a^3 x^3}+\frac {8 b^{9/2} \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 )}{231 a^{7/2} \left (a+b x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 51, normalized size = 0.34 \[ -\frac {\sqrt [4]{a+b x^4} \, _2F_1\left (-\frac {15}{4},-\frac {1}{4};-\frac {11}{4};-\frac {b x^4}{a}\right )}{15 x^{15} \sqrt [4]{\frac {b x^4}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x^{16}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x^{16}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{4}+a \right )^{\frac {1}{4}}}{x^{16}}\, dx \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {1}{4}}}{x^{16}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^4+a\right )}^{1/4}}{x^{16}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.38, size = 46, normalized size = 0.30 \[ \frac {\sqrt [4]{a} \Gamma \left (- \frac {15}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {15}{4}, - \frac {1}{4} \\ - \frac {11}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{15} \Gamma \left (- \frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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