Optimal. Leaf size=78 \[ \frac{3 b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}+\frac{3 b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}-\frac{\sqrt [4]{a+b x^4}}{4 a x^4} \]
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Rubi [A] time = 0.0450485, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {266, 51, 63, 212, 206, 203} \[ \frac{3 b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}+\frac{3 b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}-\frac{\sqrt [4]{a+b x^4}}{4 a x^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (a+b x^4\right )^{3/4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{3/4}} \, dx,x,x^4\right )\\ &=-\frac{\sqrt [4]{a+b x^4}}{4 a x^4}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/4}} \, dx,x,x^4\right )}{16 a}\\ &=-\frac{\sqrt [4]{a+b x^4}}{4 a x^4}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )}{4 a}\\ &=-\frac{\sqrt [4]{a+b x^4}}{4 a x^4}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{8 a^{3/2}}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )}{8 a^{3/2}}\\ &=-\frac{\sqrt [4]{a+b x^4}}{4 a x^4}+\frac{3 b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}+\frac{3 b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}\\ \end{align*}
Mathematica [C] time = 0.0067605, size = 34, normalized size = 0.44 \[ \frac{b \sqrt [4]{a+b x^4} \, _2F_1\left (\frac{1}{4},2;\frac{5}{4};\frac{b x^4}{a}+1\right )}{a^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.032, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \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(-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 = 1.6547, size = 462, normalized size = 5.92 \begin{align*} -\frac{12 \, a x^{4} \left (\frac{b^{4}}{a^{7}}\right )^{\frac{1}{4}} \arctan \left (-\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{5} b \left (\frac{b^{4}}{a^{7}}\right )^{\frac{3}{4}} - \sqrt{a^{4} \sqrt{\frac{b^{4}}{a^{7}}} + \sqrt{b x^{4} + a} b^{2}} a^{5} \left (\frac{b^{4}}{a^{7}}\right )^{\frac{3}{4}}}{b^{4}}\right ) - 3 \, a x^{4} \left (\frac{b^{4}}{a^{7}}\right )^{\frac{1}{4}} \log \left (3 \, a^{2} \left (\frac{b^{4}}{a^{7}}\right )^{\frac{1}{4}} + 3 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b\right ) + 3 \, a x^{4} \left (\frac{b^{4}}{a^{7}}\right )^{\frac{1}{4}} \log \left (-3 \, a^{2} \left (\frac{b^{4}}{a^{7}}\right )^{\frac{1}{4}} + 3 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b\right ) + 4 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{16 \, a x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.80853, size = 39, normalized size = 0.5 \begin{align*} - \frac{\Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 b^{\frac{3}{4}} x^{7} \Gamma \left (\frac{11}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15133, size = 282, normalized size = 3.62 \begin{align*} \frac{1}{32} \, b{\left (\frac{6 \, \sqrt{2} \left (-a\right )^{\frac{1}{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^{2}} + \frac{6 \, \sqrt{2} \left (-a\right )^{\frac{1}{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^{2}} + \frac{3 \, \sqrt{2} \left (-a\right )^{\frac{1}{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^{2}} - \frac{3 \, \sqrt{2} \left (-a\right )^{\frac{1}{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^{2}} - \frac{8 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{a b x^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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