Optimal. Leaf size=99 \[ -\frac {3 b}{5 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{5 x^5}+\frac {3 b^{3/2} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{5 \sqrt {a} \sqrt [4]{a+b x^4}} \]
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Rubi [A]
time = 0.03, 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 = {283, 318, 287,
342, 281, 202} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 281
Rule 283
Rule 287
Rule 318
Rule 342
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 ) \text {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 ) \text {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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.02, size = 51, normalized size = 0.52 \begin {gather*} -\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 (1+\frac {b x^4}{a}\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, 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 \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 = 0.15 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{6}}, 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 = 0.61, size = 31, normalized size = 0.31 \begin {gather*} - \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}} \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 )}^{3/4}}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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