Optimal. Leaf size=150 \[ -\frac {4 b^3}{65 a^2 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{13 x^{13}}-\frac {b \left (a+b x^4\right )^{3/4}}{39 a x^9}+\frac {2 b^2 \left (a+b x^4\right )^{3/4}}{65 a^2 x^5}+\frac {4 b^{7/2} \sqrt [4]{1+\frac {a}{b x^4}} x E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{65 a^{5/2} \sqrt [4]{a+b x^4}} \]
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Rubi [A]
time = 0.05, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {283, 331, 318,
287, 342, 281, 202} \begin {gather*} \frac {4 b^{7/2} 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 )}{65 a^{5/2} \sqrt [4]{a+b x^4}}-\frac {4 b^3}{65 a^2 x \sqrt [4]{a+b x^4}}+\frac {2 b^2 \left (a+b x^4\right )^{3/4}}{65 a^2 x^5}-\frac {\left (a+b x^4\right )^{3/4}}{13 x^{13}}-\frac {b \left (a+b x^4\right )^{3/4}}{39 a x^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 281
Rule 283
Rule 287
Rule 318
Rule 331
Rule 342
Rubi steps
\begin {align*} \int \frac {\left (a+b x^4\right )^{3/4}}{x^{14}} \, dx &=-\frac {\left (a+b x^4\right )^{3/4}}{13 x^{13}}+\frac {1}{13} (3 b) \int \frac {1}{x^{10} \sqrt [4]{a+b x^4}} \, dx\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{13 x^{13}}-\frac {b \left (a+b x^4\right )^{3/4}}{39 a x^9}-\frac {\left (2 b^2\right ) \int \frac {1}{x^6 \sqrt [4]{a+b x^4}} \, dx}{13 a}\\ &=-\frac {\left (a+b x^4\right )^{3/4}}{13 x^{13}}-\frac {b \left (a+b x^4\right )^{3/4}}{39 a x^9}+\frac {2 b^2 \left (a+b x^4\right )^{3/4}}{65 a^2 x^5}+\frac {\left (4 b^3\right ) \int \frac {1}{x^2 \sqrt [4]{a+b x^4}} \, dx}{65 a^2}\\ &=-\frac {4 b^3}{65 a^2 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{13 x^{13}}-\frac {b \left (a+b x^4\right )^{3/4}}{39 a x^9}+\frac {2 b^2 \left (a+b x^4\right )^{3/4}}{65 a^2 x^5}-\frac {\left (4 b^4\right ) \int \frac {x^2}{\left (a+b x^4\right )^{5/4}} \, dx}{65 a^2}\\ &=-\frac {4 b^3}{65 a^2 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{13 x^{13}}-\frac {b \left (a+b x^4\right )^{3/4}}{39 a x^9}+\frac {2 b^2 \left (a+b x^4\right )^{3/4}}{65 a^2 x^5}-\frac {\left (4 b^3 \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}{65 a^2 \sqrt [4]{a+b x^4}}\\ &=-\frac {4 b^3}{65 a^2 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{13 x^{13}}-\frac {b \left (a+b x^4\right )^{3/4}}{39 a x^9}+\frac {2 b^2 \left (a+b x^4\right )^{3/4}}{65 a^2 x^5}+\frac {\left (4 b^3 \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 )}{65 a^2 \sqrt [4]{a+b x^4}}\\ &=-\frac {4 b^3}{65 a^2 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{13 x^{13}}-\frac {b \left (a+b x^4\right )^{3/4}}{39 a x^9}+\frac {2 b^2 \left (a+b x^4\right )^{3/4}}{65 a^2 x^5}+\frac {\left (2 b^3 \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 )}{65 a^2 \sqrt [4]{a+b x^4}}\\ &=-\frac {4 b^3}{65 a^2 x \sqrt [4]{a+b x^4}}-\frac {\left (a+b x^4\right )^{3/4}}{13 x^{13}}-\frac {b \left (a+b x^4\right )^{3/4}}{39 a x^9}+\frac {2 b^2 \left (a+b x^4\right )^{3/4}}{65 a^2 x^5}+\frac {4 b^{7/2} \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 )}{65 a^{5/2} \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 = 10.01, size = 51, normalized size = 0.34 \begin {gather*} -\frac {\left (a+b x^4\right )^{3/4} \, _2F_1\left (-\frac {13}{4},-\frac {3}{4};-\frac {9}{4};-\frac {b x^4}{a}\right )}{13 x^{13} \left (1+\frac {b x^4}{a}\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{4}+a \right )^{\frac {3}{4}}}{x^{14}}\, dx\]
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 = 0.10 \begin {gather*} {\rm integral}\left (\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{x^{14}}, 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 = 1.18, size = 46, normalized size = 0.31 \begin {gather*} \frac {a^{\frac {3}{4}} \Gamma \left (- \frac {13}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {13}{4}, - \frac {3}{4} \\ - \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{13} \Gamma \left (- \frac {9}{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 {{\left (b\,x^4+a\right )}^{3/4}}{x^{14}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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