Optimal. Leaf size=99 \[ \frac {3 b^{3/2} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {a} \sqrt [4]{a+b x^4}}-\frac {3 b}{5 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5} \]
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Rubi [A] time = 0.05, antiderivative size = 99, 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 = {277, 312, 281, 335, 275, 196} \[ \frac {3 b^{3/2} x \sqrt [4]{\frac {a}{b x^4}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {a} \sqrt [4]{a+b x^4}}-\frac {3 b}{5 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 196
Rule 275
Rule 277
Rule 281
Rule 312
Rule 335
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{3/4}}{x^6} \, dx &=-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5}+\frac {1}{5} (3 b) \int \frac {1}{x^2 \sqrt [4]{a+b x^4}} \, dx\\ &=-\frac {3 b}{5 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5}-\frac {1}{5} \left (3 b^2\right ) \int \frac {x^2}{\left (a+b x^4\right )^{5/4}} \, dx\\ &=-\frac {3 b}{5 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5}-\frac {\left (3 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 \sqrt [4]{a+b x^4}}\\ &=-\frac {3 b}{5 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5}+\frac {\left (3 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 \sqrt [4]{a+b x^4}}\\ &=-\frac {3 b}{5 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5}+\frac {\left (3 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 )}{10 \sqrt [4]{a+b x^4}}\\ &=-\frac {3 b}{5 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5}+\frac {3 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 \sqrt {a} \sqrt [4]{a+b x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 51, normalized size = 0.52 \[ -\frac {\left (a+b x^4\right )^{3/4} \, _2F_1\left (-\frac {5}{4},-\frac {3}{4};-\frac {1}{4};-\frac {b x^4}{a}\right )}{5 x^5 \left (\frac {b x^4}{a}+1\right )^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{6}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{6}}\,{d x} \]
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 \[ \int \frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}}}{x^{6}}\, dx \]
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 \[ \int \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{6}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^4+a\right )}^{3/4}}{x^6} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.58, size = 31, normalized size = 0.31 \[ - \frac {b^{\frac {3}{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{4}}} \right )}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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