3.298 \(\int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=198 \[ \frac{25}{11} x \left (x^4+3 x^2+2\right )^{5/2}+\frac{1}{693} x \left (2240 x^2+7281\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac{x \left (10643 x^2+36783\right ) \sqrt{x^4+3 x^2+2}}{1155}+\frac{742 x \left (x^2+2\right )}{15 \sqrt{x^4+3 x^2+2}}+\frac{13879 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{385 \sqrt{x^4+3 x^2+2}}-\frac{742 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{15 \sqrt{x^4+3 x^2+2}} \]

[Out]

(742*x*(2 + x^2))/(15*Sqrt[2 + 3*x^2 + x^4]) + (x*(36783 + 10643*x^2)*Sqrt[2 + 3
*x^2 + x^4])/1155 + (x*(7281 + 2240*x^2)*(2 + 3*x^2 + x^4)^(3/2))/693 + (25*x*(2
+ 3*x^2 + x^4)^(5/2))/11 - (742*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*Ell
ipticE[ArcTan[x], 1/2])/(15*Sqrt[2 + 3*x^2 + x^4]) + (13879*Sqrt[2]*(1 + x^2)*Sq
rt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(385*Sqrt[2 + 3*x^2 + x^4])

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Rubi [A]  time = 0.195864, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{25}{11} x \left (x^4+3 x^2+2\right )^{5/2}+\frac{1}{693} x \left (2240 x^2+7281\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac{x \left (10643 x^2+36783\right ) \sqrt{x^4+3 x^2+2}}{1155}+\frac{742 x \left (x^2+2\right )}{15 \sqrt{x^4+3 x^2+2}}+\frac{13879 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{385 \sqrt{x^4+3 x^2+2}}-\frac{742 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{15 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[In]  Int[(7 + 5*x^2)^2*(2 + 3*x^2 + x^4)^(3/2),x]

[Out]

(742*x*(2 + x^2))/(15*Sqrt[2 + 3*x^2 + x^4]) + (x*(36783 + 10643*x^2)*Sqrt[2 + 3
*x^2 + x^4])/1155 + (x*(7281 + 2240*x^2)*(2 + 3*x^2 + x^4)^(3/2))/693 + (25*x*(2
+ 3*x^2 + x^4)^(5/2))/11 - (742*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*Ell
ipticE[ArcTan[x], 1/2])/(15*Sqrt[2 + 3*x^2 + x^4]) + (13879*Sqrt[2]*(1 + x^2)*Sq
rt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(385*Sqrt[2 + 3*x^2 + x^4])

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Rubi in Sympy [A]  time = 33.077, size = 189, normalized size = 0.95 \[ \frac{371 x \left (2 x^{2} + 4\right )}{15 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{x \left (\frac{2240 x^{2}}{11} + \frac{7281}{11}\right ) \left (x^{4} + 3 x^{2} + 2\right )^{\frac{3}{2}}}{63} + \frac{x \left (\frac{31929 x^{2}}{11} + \frac{110349}{11}\right ) \sqrt{x^{4} + 3 x^{2} + 2}}{315} + \frac{25 x \left (x^{4} + 3 x^{2} + 2\right )^{\frac{5}{2}}}{11} - \frac{371 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{30 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{13879 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{1540 \sqrt{x^{4} + 3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**2+7)**2*(x**4+3*x**2+2)**(3/2),x)

[Out]

371*x*(2*x**2 + 4)/(15*sqrt(x**4 + 3*x**2 + 2)) + x*(2240*x**2/11 + 7281/11)*(x*
*4 + 3*x**2 + 2)**(3/2)/63 + x*(31929*x**2/11 + 110349/11)*sqrt(x**4 + 3*x**2 +
2)/315 + 25*x*(x**4 + 3*x**2 + 2)**(5/2)/11 - 371*sqrt((2*x**2 + 4)/(x**2 + 1))*
(4*x**2 + 4)*elliptic_e(atan(x), 1/2)/(30*sqrt(x**4 + 3*x**2 + 2)) + 13879*sqrt(
(2*x**2 + 4)/(x**2 + 1))*(4*x**2 + 4)*elliptic_f(atan(x), 1/2)/(1540*sqrt(x**4 +
3*x**2 + 2))

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Mathematica [C]  time = 0.0832228, size = 124, normalized size = 0.63 \[ \frac{7875 x^{13}+82075 x^{11}+363480 x^9+892084 x^7+1333551 x^5+1160065 x^3-78420 i \sqrt{x^2+1} \sqrt{x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-171402 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+429318 x}{3465 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(7 + 5*x^2)^2*(2 + 3*x^2 + x^4)^(3/2),x]

[Out]

(429318*x + 1160065*x^3 + 1333551*x^5 + 892084*x^7 + 363480*x^9 + 82075*x^11 + 7
875*x^13 - (171402*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]]
, 2] - (78420*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])
/(3465*Sqrt[2 + 3*x^2 + x^4])

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Maple [C]  time = 0.011, size = 189, normalized size = 1. \[{\frac{11492\,{x}^{5}}{231}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{258044\,{x}^{3}}{3465}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{23851\,x}{385}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{13879\,i}{385}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{371\,i}{15}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{\frac{1670\,{x}^{7}}{99}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{25\,{x}^{9}}{11}\sqrt{{x}^{4}+3\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2+7)^2*(x^4+3*x^2+2)^(3/2),x)

[Out]

11492/231*x^5*(x^4+3*x^2+2)^(1/2)+258044/3465*x^3*(x^4+3*x^2+2)^(1/2)+23851/385*
x*(x^4+3*x^2+2)^(1/2)-13879/385*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x
^2+2)^(1/2)*EllipticF(1/2*I*2^(1/2)*x,2^(1/2))+371/15*I*2^(1/2)*(2*x^2+4)^(1/2)*
(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*(EllipticF(1/2*I*2^(1/2)*x,2^(1/2))-EllipticE(
1/2*I*2^(1/2)*x,2^(1/2)))+1670/99*x^7*(x^4+3*x^2+2)^(1/2)+25/11*x^9*(x^4+3*x^2+2
)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 3*x^2 + 2)^(3/2)*(5*x^2 + 7)^2,x, algorithm="maxima")

[Out]

integrate((x^4 + 3*x^2 + 2)^(3/2)*(5*x^2 + 7)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (25 \, x^{8} + 145 \, x^{6} + 309 \, x^{4} + 287 \, x^{2} + 98\right )} \sqrt{x^{4} + 3 \, x^{2} + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 3*x^2 + 2)^(3/2)*(5*x^2 + 7)^2,x, algorithm="fricas")

[Out]

integral((25*x^8 + 145*x^6 + 309*x^4 + 287*x^2 + 98)*sqrt(x^4 + 3*x^2 + 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac{3}{2}} \left (5 x^{2} + 7\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2+7)**2*(x**4+3*x**2+2)**(3/2),x)

[Out]

Integral(((x**2 + 1)*(x**2 + 2))**(3/2)*(5*x**2 + 7)**2, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 3*x^2 + 2)^(3/2)*(5*x^2 + 7)^2,x, algorithm="giac")

[Out]

integrate((x^4 + 3*x^2 + 2)^(3/2)*(5*x^2 + 7)^2, x)