Optimal. Leaf size=183 \[ \frac{7 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{4}\right )}{15 \sqrt{x^4+x^2+1}}+\frac{1}{9} \left (x^4+x^2+1\right )^{3/2} x^3+\frac{1}{3} \left (x^4+x^2+1\right )^{3/2} x+\frac{2}{45} \left (6 x^2+7\right ) \sqrt{x^4+x^2+1} x+\frac{26 \sqrt{x^4+x^2+1} x}{45 \left (x^2+1\right )}-\frac{26 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{45 \sqrt{x^4+x^2+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0882292, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1206, 1679, 1176, 1197, 1103, 1195} \[ \frac{1}{9} \left (x^4+x^2+1\right )^{3/2} x^3+\frac{1}{3} \left (x^4+x^2+1\right )^{3/2} x+\frac{2}{45} \left (6 x^2+7\right ) \sqrt{x^4+x^2+1} x+\frac{26 \sqrt{x^4+x^2+1} x}{45 \left (x^2+1\right )}+\frac{7 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{15 \sqrt{x^4+x^2+1}}-\frac{26 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{45 \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1206
Rule 1679
Rule 1176
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \left (1+x^2\right )^3 \sqrt{1+x^2+x^4} \, dx &=\frac{1}{9} x^3 \left (1+x^2+x^4\right )^{3/2}+\frac{1}{9} \int \sqrt{1+x^2+x^4} \left (9+24 x^2+21 x^4\right ) \, dx\\ &=\frac{1}{3} x \left (1+x^2+x^4\right )^{3/2}+\frac{1}{9} x^3 \left (1+x^2+x^4\right )^{3/2}+\frac{1}{63} \int \left (42+84 x^2\right ) \sqrt{1+x^2+x^4} \, dx\\ &=\frac{2}{45} x \left (7+6 x^2\right ) \sqrt{1+x^2+x^4}+\frac{1}{3} x \left (1+x^2+x^4\right )^{3/2}+\frac{1}{9} x^3 \left (1+x^2+x^4\right )^{3/2}+\frac{1}{945} \int \frac{336+546 x^2}{\sqrt{1+x^2+x^4}} \, dx\\ &=\frac{2}{45} x \left (7+6 x^2\right ) \sqrt{1+x^2+x^4}+\frac{1}{3} x \left (1+x^2+x^4\right )^{3/2}+\frac{1}{9} x^3 \left (1+x^2+x^4\right )^{3/2}-\frac{26}{45} \int \frac{1-x^2}{\sqrt{1+x^2+x^4}} \, dx+\frac{14}{15} \int \frac{1}{\sqrt{1+x^2+x^4}} \, dx\\ &=\frac{26 x \sqrt{1+x^2+x^4}}{45 \left (1+x^2\right )}+\frac{2}{45} x \left (7+6 x^2\right ) \sqrt{1+x^2+x^4}+\frac{1}{3} x \left (1+x^2+x^4\right )^{3/2}+\frac{1}{9} x^3 \left (1+x^2+x^4\right )^{3/2}-\frac{26 \left (1+x^2\right ) \sqrt{\frac{1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{45 \sqrt{1+x^2+x^4}}+\frac{7 \left (1+x^2\right ) \sqrt{\frac{1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{15 \sqrt{1+x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.313023, size = 169, normalized size = 0.92 \[ \frac{2 (-1)^{5/6} \left (4 \sqrt{3}+9 i\right ) \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \text{EllipticF}\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )+x \left (5 x^{10}+25 x^8+57 x^6+81 x^4+61 x^2+29\right )+26 \sqrt [3]{-1} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )}{45 \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.174, size = 263, normalized size = 1.4 \begin{align*}{\frac{{x}^{7}}{9}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{4\,{x}^{5}}{9}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{32\,{x}^{3}}{45}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{29\,x}{45}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{32}{45\,\sqrt{-2+2\,i\sqrt{3}}}\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}-{\frac{104}{45\,\sqrt{-2+2\,i\sqrt{3}} \left ( i\sqrt{3}+1 \right ) }\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} \sqrt{x^{4} + x^{2} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]