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

Optimal. Leaf size=193 \[ \frac{275}{7} \left (x^4+3 x^2+2\right )^{3/2} x+\frac{1}{21} \left (757 x^2+2608\right ) \sqrt{x^4+3 x^2+2} x+\frac{577 \left (x^2+2\right ) x}{3 \sqrt{x^4+3 x^2+2}}+\frac{2945 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{21 \sqrt{x^4+3 x^2+2}}-\frac{577 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{x^4+3 x^2+2}}+\frac{125}{9} \left (x^4+3 x^2+2\right )^{3/2} x^3 \]

[Out]

(577*x*(2 + x^2))/(3*Sqrt[2 + 3*x^2 + x^4]) + (x*(2608 + 757*x^2)*Sqrt[2 + 3*x^2
+ x^4])/21 + (275*x*(2 + 3*x^2 + x^4)^(3/2))/7 + (125*x^3*(2 + 3*x^2 + x^4)^(3/
2))/9 - (577*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/
2])/(3*Sqrt[2 + 3*x^2 + x^4]) + (2945*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)
]*EllipticF[ArcTan[x], 1/2])/(21*Sqrt[2 + 3*x^2 + x^4])

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Rubi [A]  time = 0.207848, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{275}{7} \left (x^4+3 x^2+2\right )^{3/2} x+\frac{1}{21} \left (757 x^2+2608\right ) \sqrt{x^4+3 x^2+2} x+\frac{577 \left (x^2+2\right ) x}{3 \sqrt{x^4+3 x^2+2}}+\frac{2945 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{21 \sqrt{x^4+3 x^2+2}}-\frac{577 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{3 \sqrt{x^4+3 x^2+2}}+\frac{125}{9} \left (x^4+3 x^2+2\right )^{3/2} x^3 \]

Antiderivative was successfully verified.

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

[Out]

(577*x*(2 + x^2))/(3*Sqrt[2 + 3*x^2 + x^4]) + (x*(2608 + 757*x^2)*Sqrt[2 + 3*x^2
+ x^4])/21 + (275*x*(2 + 3*x^2 + x^4)^(3/2))/7 + (125*x^3*(2 + 3*x^2 + x^4)^(3/
2))/9 - (577*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/
2])/(3*Sqrt[2 + 3*x^2 + x^4]) + (2945*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)
]*EllipticF[ArcTan[x], 1/2])/(21*Sqrt[2 + 3*x^2 + x^4])

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Rubi in Sympy [A]  time = 36.0919, size = 182, normalized size = 0.94 \[ \frac{125 x^{3} \left (x^{4} + 3 x^{2} + 2\right )^{\frac{3}{2}}}{9} + \frac{577 x \left (2 x^{2} + 4\right )}{6 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{x \left (\frac{3785 x^{2}}{7} + \frac{13040}{7}\right ) \sqrt{x^{4} + 3 x^{2} + 2}}{15} + \frac{275 x \left (x^{4} + 3 x^{2} + 2\right )^{\frac{3}{2}}}{7} - \frac{577 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{12 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{2945 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{84 \sqrt{x^{4} + 3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**2+7)**3*(x**4+3*x**2+2)**(1/2),x)

[Out]

125*x**3*(x**4 + 3*x**2 + 2)**(3/2)/9 + 577*x*(2*x**2 + 4)/(6*sqrt(x**4 + 3*x**2
+ 2)) + x*(3785*x**2/7 + 13040/7)*sqrt(x**4 + 3*x**2 + 2)/15 + 275*x*(x**4 + 3*
x**2 + 2)**(3/2)/7 - 577*sqrt((2*x**2 + 4)/(x**2 + 1))*(4*x**2 + 4)*elliptic_e(a
tan(x), 1/2)/(12*sqrt(x**4 + 3*x**2 + 2)) + 2945*sqrt((2*x**2 + 4)/(x**2 + 1))*(
4*x**2 + 4)*elliptic_f(atan(x), 1/2)/(84*sqrt(x**4 + 3*x**2 + 2))

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Mathematica [C]  time = 0.0936792, size = 119, normalized size = 0.62 \[ \frac{875 x^{11}+7725 x^9+28496 x^7+57312 x^5+61214 x^3-5553 i \sqrt{x^2+1} \sqrt{x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-12117 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+25548 x}{63 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(25548*x + 61214*x^3 + 57312*x^5 + 28496*x^7 + 7725*x^9 + 875*x^11 - (12117*I)*S
qrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (5553*I)*Sqrt[1
+ x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(63*Sqrt[2 + 3*x^2 + x^
4])

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Maple [C]  time = 0.029, size = 172, normalized size = 0.9 \[{\frac{4258\,x}{21}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{2945\,i}{21}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{577\,i}{6}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{\frac{11446\,{x}^{3}}{63}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{1700\,{x}^{5}}{21}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{125\,{x}^{7}}{9}\sqrt{{x}^{4}+3\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2+7)^3*(x^4+3*x^2+2)^(1/2),x)

[Out]

4258/21*x*(x^4+3*x^2+2)^(1/2)-2945/21*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))+577/6*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))-Ellip
ticE(1/2*I*2^(1/2)*x,2^(1/2)))+11446/63*x^3*(x^4+3*x^2+2)^(1/2)+1700/21*x^5*(x^4
+3*x^2+2)^(1/2)+125/9*x^7*(x^4+3*x^2+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343\right )} \sqrt{x^{4} + 3 \, x^{2} + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((125*x^6 + 525*x^4 + 735*x^2 + 343)*sqrt(x^4 + 3*x^2 + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2+7)**3*(x**4+3*x**2+2)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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