Optimal. Leaf size=179 \[ \frac{1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac{1}{105} x \left (149 x^2+519\right ) \sqrt{x^4+3 x^2+2}+\frac{116 x \left (x^2+2\right )}{15 \sqrt{x^4+3 x^2+2}}+\frac{197 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{35 \sqrt{x^4+3 x^2+2}}-\frac{116 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{15 \sqrt{x^4+3 x^2+2}} \]
[Out]
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Rubi [A] time = 0.150632, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac{1}{105} x \left (149 x^2+519\right ) \sqrt{x^4+3 x^2+2}+\frac{116 x \left (x^2+2\right )}{15 \sqrt{x^4+3 x^2+2}}+\frac{197 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{35 \sqrt{x^4+3 x^2+2}}-\frac{116 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{15 \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
[In] Int[(7 + 5*x^2)*(2 + 3*x^2 + x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 23.3048, size = 163, normalized size = 0.91 \[ \frac{58 x \left (2 x^{2} + 4\right )}{15 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{x \left (35 x^{2} + 108\right ) \left (x^{4} + 3 x^{2} + 2\right )^{\frac{3}{2}}}{63} + \frac{x \left (447 x^{2} + 1557\right ) \sqrt{x^{4} + 3 x^{2} + 2}}{315} - \frac{29 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{15 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{197 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{140 \sqrt{x^{4} + 3 x^{2} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5*x**2+7)*(x**4+3*x**2+2)**(3/2),x)
[Out]
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Mathematica [C] time = 0.0748744, size = 119, normalized size = 0.66 \[ \frac{175 x^{11}+1590 x^9+5962 x^7+12018 x^5+12745 x^3-1110 i \sqrt{x^2+1} \sqrt{x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-2436 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+5274 x}{315 \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
[In] Integrate[(7 + 5*x^2)*(2 + 3*x^2 + x^4)^(3/2),x]
[Out]
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Maple [C] time = 0.008, size = 172, normalized size = 1. \[{\frac{71\,{x}^{5}}{21}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{2417\,{x}^{3}}{315}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{293\,x}{35}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{197\,i}{35}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{58\,i}{15}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{\frac{5\,{x}^{7}}{9}\sqrt{{x}^{4}+3\,{x}^{2}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5*x^2+7)*(x^4+3*x^2+2)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 3*x^2 + 2)^(3/2)*(5*x^2 + 7),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (5 \, x^{6} + 22 \, x^{4} + 31 \, x^{2} + 14\right )} \sqrt{x^{4} + 3 \, x^{2} + 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 3*x^2 + 2)^(3/2)*(5*x^2 + 7),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac{3}{2}} \left (5 x^{2} + 7\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x**2+7)*(x**4+3*x**2+2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^4 + 3*x^2 + 2)^(3/2)*(5*x^2 + 7),x, algorithm="giac")
[Out]