Optimal. Leaf size=136 \[ -\frac{a \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.117721, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {1876, 211, 1165, 628, 1162, 617, 204, 1248, 635, 203, 260} \[ -\frac{a \log \left (3 x^2-6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (3 x^2+6^{3/4} x+\sqrt{6}\right )}{8 \sqrt [4]{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}+\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}+\frac{1}{12} d \log \left (3 x^4+2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1876
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rule 1248
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{a+b x+d x^3}{2+3 x^4} \, dx &=\int \left (\frac{a}{2+3 x^4}+\frac{x \left (b+d x^2\right )}{2+3 x^4}\right ) \, dx\\ &=a \int \frac{1}{2+3 x^4} \, dx+\int \frac{x \left (b+d x^2\right )}{2+3 x^4} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{b+d x}{2+3 x^2} \, dx,x,x^2\right )+\frac{a \int \frac{\sqrt{2}-\sqrt{3} x^2}{2+3 x^4} \, dx}{2 \sqrt{2}}+\frac{a \int \frac{\sqrt{2}+\sqrt{3} x^2}{2+3 x^4} \, dx}{2 \sqrt{2}}\\ &=\frac{a \int \frac{1}{\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx}{4 \sqrt{6}}+\frac{a \int \frac{1}{\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx}{4 \sqrt{6}}-\frac{a \int \frac{\frac{2^{3/4}}{\sqrt [4]{3}}+2 x}{-\sqrt{\frac{2}{3}}-\frac{2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{8 \sqrt [4]{6}}-\frac{a \int \frac{\frac{2^{3/4}}{\sqrt [4]{3}}-2 x}{-\sqrt{\frac{2}{3}}+\frac{2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{8 \sqrt [4]{6}}+\frac{1}{2} b \operatorname{Subst}\left (\int \frac{1}{2+3 x^2} \, dx,x,x^2\right )+\frac{1}{2} d \operatorname{Subst}\left (\int \frac{x}{2+3 x^2} \, dx,x,x^2\right )\\ &=\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}-\frac{a \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac{1}{12} d \log \left (2+3 x^4\right )+\frac{a \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}\\ &=\frac{b \tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}}-\frac{a \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac{a \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}-\frac{a \log \left (\sqrt{6}-6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac{a \log \left (\sqrt{6}+6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac{1}{12} d \log \left (2+3 x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0584346, size = 128, normalized size = 0.94 \[ \frac{1}{48} \left (-2 \sqrt{6} \left (\sqrt [4]{6} a+2 b\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \sqrt{6} \left (\sqrt [4]{6} a-2 b\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )-6^{3/4} a \log \left (\sqrt{6} x^2-2 \sqrt [4]{6} x+2\right )+6^{3/4} a \log \left (\sqrt{6} x^2+2 \sqrt [4]{6} x+2\right )+4 d \log \left (3 x^4+2\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.041, size = 140, normalized size = 1. \begin{align*}{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}-1 \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{48}\ln \left ({ \left ({x}^{2}+{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) \left ({x}^{2}-{\frac{\sqrt{3}\sqrt [4]{6}x\sqrt{2}}{3}}+{\frac{\sqrt{6}}{3}} \right ) ^{-1}} \right ) }+{\frac{a\sqrt{3}\sqrt [4]{6}\sqrt{2}}{24}\arctan \left ({\frac{\sqrt{2}\sqrt{3}{6}^{{\frac{3}{4}}}x}{6}}+1 \right ) }+{\frac{b\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) }+{\frac{d\ln \left ( 3\,{x}^{4}+2 \right ) }{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.45488, size = 231, normalized size = 1.7 \begin{align*} \frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d + 3 \, a\right )} \log \left (\sqrt{3} x^{2} + 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{144} \cdot 3^{\frac{3}{4}} 2^{\frac{3}{4}}{\left (2 \cdot 3^{\frac{1}{4}} 2^{\frac{1}{4}} d - 3 \, a\right )} \log \left (\sqrt{3} x^{2} - 3^{\frac{1}{4}} 2^{\frac{3}{4}} x + \sqrt{2}\right ) + \frac{1}{24} \, \sqrt{3}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} a - 2 \, \sqrt{2} b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x + 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) + \frac{1}{24} \, \sqrt{3}{\left (3^{\frac{1}{4}} 2^{\frac{3}{4}} a + 2 \, \sqrt{2} b\right )} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} 2^{\frac{1}{4}}{\left (2 \, \sqrt{3} x - 3^{\frac{1}{4}} 2^{\frac{3}{4}}\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.18933, size = 199, normalized size = 1.46 \begin{align*} \operatorname{RootSum}{\left (165888 t^{4} - 55296 t^{3} d + t^{2} \left (3456 b^{2} + 6912 d^{2}\right ) + t \left (- 864 a^{2} b - 576 b^{2} d - 384 d^{3}\right ) + 27 a^{4} + 72 a^{2} b d + 18 b^{4} + 24 b^{2} d^{2} + 8 d^{4}, \left ( t \mapsto t \log{\left (x + \frac{27648 t^{3} b^{2} + 1728 t^{2} a^{2} b - 6912 t^{2} b^{2} d + 216 t a^{4} - 288 t a^{2} b d + 288 t b^{4} + 576 t b^{2} d^{2} - 18 a^{4} d - 90 a^{2} b^{3} + 12 a^{2} b d^{2} - 24 b^{4} d - 16 b^{2} d^{3}}{27 a^{5} - 72 a b^{4}} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.10922, size = 169, normalized size = 1.24 \begin{align*} \frac{1}{24} \,{\left (6^{\frac{3}{4}} a - 2 \, \sqrt{6} b\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{24} \,{\left (6^{\frac{3}{4}} a + 2 \, \sqrt{6} b\right )} \arctan \left (\frac{3}{4} \, \sqrt{2} \left (\frac{2}{3}\right )^{\frac{3}{4}}{\left (2 \, x - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}}\right )}\right ) + \frac{1}{48} \,{\left (6^{\frac{3}{4}} a + 4 \, d\right )} \log \left (x^{2} + \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) - \frac{1}{48} \,{\left (6^{\frac{3}{4}} a - 4 \, d\right )} \log \left (x^{2} - \sqrt{2} \left (\frac{2}{3}\right )^{\frac{1}{4}} x + \sqrt{\frac{2}{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]