Optimal. Leaf size=150 \[ -\frac{4 b^3}{65 a^2 x \sqrt [4]{a+b x^4}}+\frac{2 b^2 \left (a+b x^4\right )^{3/4}}{65 a^2 x^5}+\frac{4 b^{7/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{65 a^{5/2} \sqrt [4]{a+b x^4}}-\frac{b \left (a+b x^4\right )^{3/4}}{39 a x^9}-\frac{\left (a+b x^4\right )^{3/4}}{13 x^{13}} \]
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Rubi [A] time = 0.0711166, antiderivative size = 150, normalized size of antiderivative = 1., 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 = {277, 325, 312, 281, 335, 275, 196} \[ -\frac{4 b^3}{65 a^2 x \sqrt [4]{a+b x^4}}+\frac{2 b^2 \left (a+b x^4\right )^{3/4}}{65 a^2 x^5}+\frac{4 b^{7/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{65 a^{5/2} \sqrt [4]{a+b x^4}}-\frac{b \left (a+b x^4\right )^{3/4}}{39 a x^9}-\frac{\left (a+b x^4\right )^{3/4}}{13 x^{13}} \]
Antiderivative was successfully verified.
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Rule 277
Rule 325
Rule 312
Rule 281
Rule 335
Rule 275
Rule 196
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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [C] time = 0.0095917, size = 51, normalized size = 0.34 \[ -\frac{\left (a+b x^4\right )^{3/4} \, _2F_1\left (-\frac{13}{4},-\frac{3}{4};-\frac{9}{4};-\frac{b x^4}{a}\right )}{13 x^{13} \left (\frac{b x^4}{a}+1\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{14}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{14}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{14}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 6.38297, size = 46, normalized size = 0.31 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{14}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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