Optimal. Leaf size=187 \[ \frac{13 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ),\frac{1}{8}\right )}{2 \sqrt{2} \sqrt{x^4+3 x^2+4}}+25 \sqrt{x^4+3 x^2+4} x^3-\frac{15 \sqrt{x^4+3 x^2+4} x}{x^2+2}+75 \sqrt{x^4+3 x^2+4} x+\frac{15 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{\sqrt{x^4+3 x^2+4}} \]
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Rubi [A] time = 0.0927102, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1206, 1679, 1197, 1103, 1195} \[ 25 \sqrt{x^4+3 x^2+4} x^3-\frac{15 \sqrt{x^4+3 x^2+4} x}{x^2+2}+75 \sqrt{x^4+3 x^2+4} x+\frac{13 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{2 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{15 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{\sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Rule 1206
Rule 1679
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \frac{\left (7+5 x^2\right )^3}{\sqrt{4+3 x^2+x^4}} \, dx &=25 x^3 \sqrt{4+3 x^2+x^4}+\frac{1}{5} \int \frac{1715+2175 x^2+1125 x^4}{\sqrt{4+3 x^2+x^4}} \, dx\\ &=75 x \sqrt{4+3 x^2+x^4}+25 x^3 \sqrt{4+3 x^2+x^4}+\frac{1}{15} \int \frac{645-225 x^2}{\sqrt{4+3 x^2+x^4}} \, dx\\ &=75 x \sqrt{4+3 x^2+x^4}+25 x^3 \sqrt{4+3 x^2+x^4}+13 \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx+30 \int \frac{1-\frac{x^2}{2}}{\sqrt{4+3 x^2+x^4}} \, dx\\ &=75 x \sqrt{4+3 x^2+x^4}+25 x^3 \sqrt{4+3 x^2+x^4}-\frac{15 x \sqrt{4+3 x^2+x^4}}{2+x^2}+\frac{15 \sqrt{2} \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 )}{\sqrt{4+3 x^2+x^4}}+\frac{13 \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 )}{2 \sqrt{2} \sqrt{4+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.483693, size = 337, normalized size = 1.8 \[ \frac{-\sqrt{2} \left (15 \sqrt{7}+131 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{\sqrt{7}-3 i}} x\right ),\frac{-\sqrt{7}+3 i}{\sqrt{7}+3 i}\right )+100 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left (x^6+6 x^4+13 x^2+12\right )+15 \sqrt{2} \left (\sqrt{7}+3 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )}{4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.029, size = 241, normalized size = 1.3 \begin{align*} 25\,{x}^{3}\sqrt{{x}^{4}+3\,{x}^{2}+4}+75\,x\sqrt{{x}^{4}+3\,{x}^{2}+4}+172\,{\frac{\sqrt{1- \left ( -3/8+i/8\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -3/8-i/8\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ( 1/4\,x\sqrt{-6+2\,i\sqrt{7}},1/4\,\sqrt{2+6\,i\sqrt{7}} \right ) }{\sqrt{-6+2\,i\sqrt{7}}\sqrt{{x}^{4}+3\,{x}^{2}+4}}}+480\,{\frac{\sqrt{1- \left ( -3/8+i/8\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -3/8-i/8\sqrt{7} \right ){x}^{2}} \left ({\it EllipticF} \left ( 1/4\,x\sqrt{-6+2\,i\sqrt{7}},1/4\,\sqrt{2+6\,i\sqrt{7}} \right ) -{\it EllipticE} \left ( 1/4\,x\sqrt{-6+2\,i\sqrt{7}},1/4\,\sqrt{2+6\,i\sqrt{7}} \right ) \right ) }{\sqrt{-6+2\,i\sqrt{7}}\sqrt{{x}^{4}+3\,{x}^{2}+4} \left ( i\sqrt{7}+3 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x^{2} + 7\right )}^{3}}{\sqrt{x^{4} + 3 \, x^{2} + 4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343}{\sqrt{x^{4} + 3 \, x^{2} + 4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (5 x^{2} + 7\right )^{3}}{\sqrt{\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x^{2} + 7\right )}^{3}}{\sqrt{x^{4} + 3 \, x^{2} + 4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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