Optimal. Leaf size=314 \[ -\frac{\left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ),\frac{1}{8}\right )}{8624 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{555 \sqrt{x^4+3 x^2+4} x}{758912 \left (x^2+2\right )}+\frac{2775 \sqrt{x^4+3 x^2+4} x}{758912 \left (5 x^2+7\right )}+\frac{25 \sqrt{x^4+3 x^2+4} x}{1232 \left (5 x^2+7\right )^2}-\frac{3285 \sqrt{\frac{5}{77}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )}{3035648}+\frac{555 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{379456 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{18615 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{21249536 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.289723, antiderivative size = 314, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {1223, 1696, 1714, 1195, 1708, 1103, 1706} \[ -\frac{555 \sqrt{x^4+3 x^2+4} x}{758912 \left (x^2+2\right )}+\frac{2775 \sqrt{x^4+3 x^2+4} x}{758912 \left (5 x^2+7\right )}+\frac{25 \sqrt{x^4+3 x^2+4} x}{1232 \left (5 x^2+7\right )^2}-\frac{3285 \sqrt{\frac{5}{77}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{x^4+3 x^2+4}}\right )}{3035648}-\frac{\left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{8624 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{555 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{379456 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{18615 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{21249536 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1223
Rule 1696
Rule 1714
Rule 1195
Rule 1708
Rule 1103
Rule 1706
Rubi steps
\begin{align*} \int \frac{1}{\left (7+5 x^2\right )^3 \sqrt{4+3 x^2+x^4}} \, dx &=\frac{25 x \sqrt{4+3 x^2+x^4}}{1232 \left (7+5 x^2\right )^2}-\frac{\int \frac{-76-10 x^2-25 x^4}{\left (7+5 x^2\right )^2 \sqrt{4+3 x^2+x^4}} \, dx}{1232}\\ &=\frac{25 x \sqrt{4+3 x^2+x^4}}{1232 \left (7+5 x^2\right )^2}+\frac{2775 x \sqrt{4+3 x^2+x^4}}{758912 \left (7+5 x^2\right )}+\frac{\int \frac{-4412-4690 x^2-2775 x^4}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{758912}\\ &=\frac{25 x \sqrt{4+3 x^2+x^4}}{1232 \left (7+5 x^2\right )^2}+\frac{2775 x \sqrt{4+3 x^2+x^4}}{758912 \left (7+5 x^2\right )}+\frac{\int \frac{-60910-31775 x^2}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{3794560}+\frac{555 \int \frac{1-\frac{x^2}{2}}{\sqrt{4+3 x^2+x^4}} \, dx}{379456}\\ &=-\frac{555 x \sqrt{4+3 x^2+x^4}}{758912 \left (2+x^2\right )}+\frac{25 x \sqrt{4+3 x^2+x^4}}{1232 \left (7+5 x^2\right )^2}+\frac{2775 x \sqrt{4+3 x^2+x^4}}{758912 \left (7+5 x^2\right )}+\frac{555 \left (2+x^2\right ) \sqrt{\frac{4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{379456 \sqrt{2} \sqrt{4+3 x^2+x^4}}-\frac{\int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx}{4312}-\frac{5475 \int \frac{1+\frac{x^2}{2}}{\left (7+5 x^2\right ) \sqrt{4+3 x^2+x^4}} \, dx}{379456}\\ &=-\frac{555 x \sqrt{4+3 x^2+x^4}}{758912 \left (2+x^2\right )}+\frac{25 x \sqrt{4+3 x^2+x^4}}{1232 \left (7+5 x^2\right )^2}+\frac{2775 x \sqrt{4+3 x^2+x^4}}{758912 \left (7+5 x^2\right )}-\frac{3285 \sqrt{\frac{5}{77}} \tan ^{-1}\left (\frac{2 \sqrt{\frac{11}{35}} x}{\sqrt{4+3 x^2+x^4}}\right )}{3035648}+\frac{555 \left (2+x^2\right ) \sqrt{\frac{4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{379456 \sqrt{2} \sqrt{4+3 x^2+x^4}}-\frac{\left (2+x^2\right ) \sqrt{\frac{4+3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{8624 \sqrt{2} \sqrt{4+3 x^2+x^4}}-\frac{18615 \left (2+x^2\right ) \sqrt{\frac{4+3 x^2+x^4}{\left (2+x^2\right )^2}} \Pi \left (-\frac{9}{280};2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{21249536 \sqrt{2} \sqrt{4+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.89413, size = 308, normalized size = 0.98 \[ \frac{\frac{700 x \left (555 x^2+1393\right ) \left (x^4+3 x^2+4\right )}{\left (5 x^2+7\right )^2}+i \sqrt{6+2 i \sqrt{7}} \sqrt{1-\frac{2 i x^2}{\sqrt{7}-3 i}} \sqrt{1+\frac{2 i x^2}{\sqrt{7}+3 i}} \left (\left (-9401+3885 i \sqrt{7}\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{\sqrt{7}-3 i}} x\right ),\frac{-\sqrt{7}+3 i}{\sqrt{7}+3 i}\right )+3885 \left (3-i \sqrt{7}\right ) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+6570 \Pi \left (\frac{5}{14} \left (3+i \sqrt{7}\right );i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )\right )}{21249536 \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.025, size = 434, normalized size = 1.4 \begin{align*}{\frac{25\,x}{1232\, \left ( 5\,{x}^{2}+7 \right ) ^{2}}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{2775\,x}{3794560\,{x}^{2}+5312384}\sqrt{{x}^{4}+3\,{x}^{2}+4}}-{\frac{23}{27104\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{555}{23716\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{555}{23716\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{3285}{5312384\,\sqrt{-3/8+i/8\sqrt{7}}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}-{\frac{i}{8}}{x}^{2}\sqrt{7}}\sqrt{1+{\frac{3\,{x}^{2}}{8}}+{\frac{i}{8}}{x}^{2}\sqrt{7}}{\it EllipticPi} \left ( \sqrt{-{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7}}x,-{\frac{5}{-{\frac{21}{8}}+{\frac{7\,i}{8}}\sqrt{7}}},{\frac{\sqrt{-{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7}}}{\sqrt{-{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7}}}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}} \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 \frac{1}{\sqrt{x^{4} + 3 \, x^{2} + 4}{\left (5 \, x^{2} + 7\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 (\frac{\sqrt{x^{4} + 3 \, x^{2} + 4}}{125 \, x^{10} + 900 \, x^{8} + 2810 \, x^{6} + 4648 \, x^{4} + 3969 \, x^{2} + 1372}, 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 \frac{1}{\sqrt{\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )} \left (5 x^{2} + 7\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 \frac{1}{\sqrt{x^{4} + 3 \, x^{2} + 4}{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]