Optimal. Leaf size=105 \[ -\frac {2 b^{3/2} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a+b x^4}}+\frac {2 b}{5 a x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^5} \]
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Rubi [A] time = 0.05, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {325, 312, 281, 335, 275, 196} \[ -\frac {2 b^{3/2} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{5 a^{3/2} \sqrt [4]{a+b x^4}}+\frac {2 b}{5 a x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^5} \]
Antiderivative was successfully verified.
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Rule 196
Rule 275
Rule 281
Rule 312
Rule 325
Rule 335
Rubi steps
\begin {align*} \int \frac {1}{x^6 \sqrt [4]{a+b x^4}} \, dx &=-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^5}-\frac {(2 b) \int \frac {1}{x^2 \sqrt [4]{a+b x^4}} \, dx}{5 a}\\ &=\frac {2 b}{5 a x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^5}+\frac {\left (2 b^2\right ) \int \frac {x^2}{\left (a+b x^4\right )^{5/4}} \, dx}{5 a}\\ &=\frac {2 b}{5 a x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^5}+\frac {\left (2 b \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}{5 a \sqrt [4]{a+b x^4}}\\ &=\frac {2 b}{5 a x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^5}-\frac {\left (2 b \sqrt [4]{1+\frac {a}{b x^4}} x\right ) \operatorname {Subst}\left (\int \frac {x}{\left (1+\frac {a x^4}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{5 a \sqrt [4]{a+b x^4}}\\ &=\frac {2 b}{5 a x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^5}-\frac {\left (b \sqrt [4]{1+\frac {a}{b x^4}} x\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x^2}\right )}{5 a \sqrt [4]{a+b x^4}}\\ &=\frac {2 b}{5 a x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 a x^5}-\frac {2 b^{3/2} \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 )}{5 a^{3/2} \sqrt [4]{a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 51, normalized size = 0.49 \[ -\frac {\sqrt [4]{\frac {b x^4}{a}+1} \, _2F_1\left (-\frac {5}{4},\frac {1}{4};-\frac {1}{4};-\frac {b x^4}{a}\right )}{5 x^5 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{b x^{10} + a x^{6}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{4}+a \right )^{\frac {1}{4}} x^{6}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} x^{6}}\,{d x} \]
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 \[ \int \frac {1}{x^6\,{\left (b\,x^4+a\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.73, size = 29, normalized size = 0.28 \[ - \frac {{{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{6 \sqrt [4]{b} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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