Optimal. Leaf size=75 \[ -\frac{\left (a+b x^4\right )^{3/4}}{4 x^4}+\frac{3 b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0467913, antiderivative size = 75, 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, 47, 63, 298, 203, 206} \[ -\frac{\left (a+b x^4\right )^{3/4}}{4 x^4}+\frac{3 b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 47
Rule 63
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (a+b x^4\right )^{3/4}}{x^5} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{(a+b x)^{3/4}}{x^2} \, dx,x,x^4\right )\\ &=-\frac{\left (a+b x^4\right )^{3/4}}{4 x^4}+\frac{1}{16} (3 b) \operatorname{Subst}\left (\int \frac{1}{x \sqrt [4]{a+b x}} \, dx,x,x^4\right )\\ &=-\frac{\left (a+b x^4\right )^{3/4}}{4 x^4}+\frac{3}{4} \operatorname{Subst}\left (\int \frac{x^2}{-\frac{a}{b}+\frac{x^4}{b}} \, dx,x,\sqrt [4]{a+b x^4}\right )\\ &=-\frac{\left (a+b x^4\right )^{3/4}}{4 x^4}-\frac{1}{8} (3 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )+\frac{1}{8} (3 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+x^2} \, dx,x,\sqrt [4]{a+b x^4}\right )\\ &=-\frac{\left (a+b x^4\right )^{3/4}}{4 x^4}+\frac{3 b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 \sqrt [4]{a}}\\ \end{align*}
Mathematica [C] time = 0.0081006, size = 37, normalized size = 0.49 \[ \frac{b \left (a+b x^4\right )^{7/4} \, _2F_1\left (\frac{7}{4},2;\frac{11}{4};\frac{b x^4}{a}+1\right )}{7 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.031, 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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.9191, size = 425, normalized size = 5.67 \begin{align*} -\frac{12 \, \left (\frac{b^{4}}{a}\right )^{\frac{1}{4}} x^{4} \arctan \left (-\frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (\frac{b^{4}}{a}\right )^{\frac{1}{4}} b^{3} - \sqrt{\sqrt{b x^{4} + a} b^{6} + \sqrt{\frac{b^{4}}{a}} a b^{4}} \left (\frac{b^{4}}{a}\right )^{\frac{1}{4}}}{b^{4}}\right ) + 3 \, \left (\frac{b^{4}}{a}\right )^{\frac{1}{4}} x^{4} \log \left (27 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3} + 27 \, \left (\frac{b^{4}}{a}\right )^{\frac{3}{4}} a\right ) - 3 \, \left (\frac{b^{4}}{a}\right )^{\frac{1}{4}} x^{4} \log \left (27 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3} - 27 \, \left (\frac{b^{4}}{a}\right )^{\frac{3}{4}} a\right ) + 4 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{16 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 1.67734, size = 39, normalized size = 0.52 \begin{align*} - \frac{b^{\frac{3}{4}} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 x \Gamma \left (\frac{5}{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.13955, size = 278, normalized size = 3.71 \begin{align*} -\frac{1}{32} \,{\left (\frac{6 \, \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} + \frac{6 \, \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} - \frac{3 \, \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} + \frac{3 \, \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} + \frac{8 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{b x^{4}}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]