Optimal. Leaf size=123 \[ \frac{45 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}+\frac{45 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}+\frac{9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}-\frac{45 a x \left (a+b x^4\right )^{3/4}}{32 b^3}-\frac{x^9}{b \sqrt [4]{a+b x^4}} \]
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Rubi [A] time = 0.0451936, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {288, 321, 240, 212, 206, 203} \[ \frac{45 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}+\frac{45 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}+\frac{9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}-\frac{45 a x \left (a+b x^4\right )^{3/4}}{32 b^3}-\frac{x^9}{b \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 288
Rule 321
Rule 240
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{x^{12}}{\left (a+b x^4\right )^{5/4}} \, dx &=-\frac{x^9}{b \sqrt [4]{a+b x^4}}+\frac{9 \int \frac{x^8}{\sqrt [4]{a+b x^4}} \, dx}{b}\\ &=-\frac{x^9}{b \sqrt [4]{a+b x^4}}+\frac{9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}-\frac{(45 a) \int \frac{x^4}{\sqrt [4]{a+b x^4}} \, dx}{8 b^2}\\ &=-\frac{x^9}{b \sqrt [4]{a+b x^4}}-\frac{45 a x \left (a+b x^4\right )^{3/4}}{32 b^3}+\frac{9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}+\frac{\left (45 a^2\right ) \int \frac{1}{\sqrt [4]{a+b x^4}} \, dx}{32 b^3}\\ &=-\frac{x^9}{b \sqrt [4]{a+b x^4}}-\frac{45 a x \left (a+b x^4\right )^{3/4}}{32 b^3}+\frac{9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}+\frac{\left (45 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{32 b^3}\\ &=-\frac{x^9}{b \sqrt [4]{a+b x^4}}-\frac{45 a x \left (a+b x^4\right )^{3/4}}{32 b^3}+\frac{9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}+\frac{\left (45 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{64 b^3}+\frac{\left (45 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{b} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{64 b^3}\\ &=-\frac{x^9}{b \sqrt [4]{a+b x^4}}-\frac{45 a x \left (a+b x^4\right )^{3/4}}{32 b^3}+\frac{9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}+\frac{45 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}+\frac{45 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}\\ \end{align*}
Mathematica [C] time = 0.0235217, size = 67, normalized size = 0.54 \[ \frac{x^5 \left (9 a \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{5}{4},\frac{5}{4};\frac{9}{4};-\frac{b x^4}{a}\right )-9 a+4 b x^4\right )}{32 b^2 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{{x}^{12} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71986, size = 647, normalized size = 5.26 \begin{align*} \frac{180 \,{\left (b^{4} x^{4} + a b^{3}\right )} \left (\frac{a^{8}}{b^{13}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6} b^{3} \left (\frac{a^{8}}{b^{13}}\right )^{\frac{1}{4}} - b^{3} x \sqrt{\frac{a^{8} b^{7} x^{2} \sqrt{\frac{a^{8}}{b^{13}}} + \sqrt{b x^{4} + a} a^{12}}{x^{2}}} \left (\frac{a^{8}}{b^{13}}\right )^{\frac{1}{4}}}{a^{8} x}\right ) + 45 \,{\left (b^{4} x^{4} + a b^{3}\right )} \left (\frac{a^{8}}{b^{13}}\right )^{\frac{1}{4}} \log \left (\frac{91125 \,{\left (b^{10} x \left (\frac{a^{8}}{b^{13}}\right )^{\frac{3}{4}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6}\right )}}{x}\right ) - 45 \,{\left (b^{4} x^{4} + a b^{3}\right )} \left (\frac{a^{8}}{b^{13}}\right )^{\frac{1}{4}} \log \left (-\frac{91125 \,{\left (b^{10} x \left (\frac{a^{8}}{b^{13}}\right )^{\frac{3}{4}} -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6}\right )}}{x}\right ) + 4 \,{\left (4 \, b^{2} x^{9} - 9 \, a b x^{5} - 45 \, a^{2} x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{128 \,{\left (b^{4} x^{4} + a b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.21406, size = 37, normalized size = 0.3 \begin{align*} \frac{x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} \Gamma \left (\frac{17}{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{x^{12}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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