Optimal. Leaf size=97 \[ -\frac{5 b}{4 a^2 \sqrt [4]{a+b x^4}}-\frac{5 b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{9/4}}+\frac{5 b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{9/4}}-\frac{1}{4 a x^4 \sqrt [4]{a+b x^4}} \]
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Rubi [A] time = 0.059758, antiderivative size = 96, normalized size of antiderivative = 0.99, 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 = {266, 51, 63, 298, 203, 206} \[ -\frac{5 \left (a+b x^4\right )^{3/4}}{4 a^2 x^4}-\frac{5 b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{9/4}}+\frac{5 b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{9/4}}+\frac{1}{a x^4 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (a+b x^4\right )^{5/4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{5/4}} \, dx,x,x^4\right )\\ &=\frac{1}{a x^4 \sqrt [4]{a+b x^4}}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt [4]{a+b x}} \, dx,x,x^4\right )}{4 a}\\ &=\frac{1}{a x^4 \sqrt [4]{a+b x^4}}-\frac{5 \left (a+b x^4\right )^{3/4}}{4 a^2 x^4}-\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt [4]{a+b x}} \, dx,x,x^4\right )}{16 a^2}\\ &=\frac{1}{a x^4 \sqrt [4]{a+b x^4}}-\frac{5 \left (a+b x^4\right )^{3/4}}{4 a^2 x^4}-\frac{5 \operatorname{Subst}\left (\int \frac{x^2}{-\frac{a}{b}+\frac{x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )}{4 a^2}\\ &=\frac{1}{a x^4 \sqrt [4]{a+b x^4}}-\frac{5 \left (a+b x^4\right )^{3/4}}{4 a^2 x^4}+\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{8 a^2}-\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{8 a^2}\\ &=\frac{1}{a x^4 \sqrt [4]{a+b x^4}}-\frac{5 \left (a+b x^4\right )^{3/4}}{4 a^2 x^4}-\frac{5 b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{9/4}}+\frac{5 b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{9/4}}\\ \end{align*}
Mathematica [C] time = 0.0075967, size = 35, normalized size = 0.36 \[ -\frac{b \, _2F_1\left (-\frac{1}{4},2;\frac{3}{4};\frac{b x^4}{a}+1\right )}{a^2 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \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 = 1.6331, size = 587, normalized size = 6.05 \begin{align*} \frac{20 \,{\left (a^{2} b x^{8} + a^{3} x^{4}\right )} \left (\frac{b^{4}}{a^{9}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} b^{3} \left (\frac{b^{4}}{a^{9}}\right )^{\frac{1}{4}} - \sqrt{a^{5} b^{4} \sqrt{\frac{b^{4}}{a^{9}}} + \sqrt{b x^{4} + a} b^{6}} a^{2} \left (\frac{b^{4}}{a^{9}}\right )^{\frac{1}{4}}}{b^{4}}\right ) + 5 \,{\left (a^{2} b x^{8} + a^{3} x^{4}\right )} \left (\frac{b^{4}}{a^{9}}\right )^{\frac{1}{4}} \log \left (125 \, a^{7} \left (\frac{b^{4}}{a^{9}}\right )^{\frac{3}{4}} + 125 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3}\right ) - 5 \,{\left (a^{2} b x^{8} + a^{3} x^{4}\right )} \left (\frac{b^{4}}{a^{9}}\right )^{\frac{1}{4}} \log \left (-125 \, a^{7} \left (\frac{b^{4}}{a^{9}}\right )^{\frac{3}{4}} + 125 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3}\right ) - 4 \,{\left (5 \, b x^{4} + a\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{16 \,{\left (a^{2} b x^{8} + a^{3} x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.83758, size = 39, normalized size = 0.4 \begin{align*} - \frac{\Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 b^{\frac{5}{4}} x^{9} \Gamma \left (\frac{13}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14392, size = 305, normalized size = 3.14 \begin{align*} \frac{1}{32} \, b{\left (\frac{10 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{3}} + \frac{10 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{3}} - \frac{5 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \log \left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{3}} + \frac{5 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \log \left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{3}} - \frac{8 \,{\left (5 \, b x^{4} + a\right )}}{{\left ({\left (b x^{4} + a\right )}^{\frac{5}{4}} -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a\right )} a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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