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

Optimal. Leaf size=219 \[ \frac{3825}{143} \left (x^4+3 x^2+2\right )^{5/2} x+\frac{\left (65345 x^2+208212\right ) \left (x^4+3 x^2+2\right )^{3/2} x}{3003}+\frac{\left (297911 x^2+1032541\right ) \sqrt{x^4+3 x^2+2} x}{5005}+\frac{20884 \left (x^2+2\right ) x}{65 \sqrt{x^4+3 x^2+2}}+\frac{1171349 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5005 \sqrt{x^4+3 x^2+2}}-\frac{20884 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{65 \sqrt{x^4+3 x^2+2}}+\frac{125}{13} \left (x^4+3 x^2+2\right )^{5/2} x^3 \]

[Out]

(20884*x*(2 + x^2))/(65*Sqrt[2 + 3*x^2 + x^4]) + (x*(1032541 + 297911*x^2)*Sqrt[
2 + 3*x^2 + x^4])/5005 + (x*(208212 + 65345*x^2)*(2 + 3*x^2 + x^4)^(3/2))/3003 +
(3825*x*(2 + 3*x^2 + x^4)^(5/2))/143 + (125*x^3*(2 + 3*x^2 + x^4)^(5/2))/13 - (
20884*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(65
*Sqrt[2 + 3*x^2 + x^4]) + (1171349*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*E
llipticF[ArcTan[x], 1/2])/(5005*Sqrt[2 + 3*x^2 + x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.241824, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{3825}{143} \left (x^4+3 x^2+2\right )^{5/2} x+\frac{\left (65345 x^2+208212\right ) \left (x^4+3 x^2+2\right )^{3/2} x}{3003}+\frac{\left (297911 x^2+1032541\right ) \sqrt{x^4+3 x^2+2} x}{5005}+\frac{20884 \left (x^2+2\right ) x}{65 \sqrt{x^4+3 x^2+2}}+\frac{1171349 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5005 \sqrt{x^4+3 x^2+2}}-\frac{20884 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{65 \sqrt{x^4+3 x^2+2}}+\frac{125}{13} \left (x^4+3 x^2+2\right )^{5/2} x^3 \]

Antiderivative was successfully verified.

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

[Out]

(20884*x*(2 + x^2))/(65*Sqrt[2 + 3*x^2 + x^4]) + (x*(1032541 + 297911*x^2)*Sqrt[
2 + 3*x^2 + x^4])/5005 + (x*(208212 + 65345*x^2)*(2 + 3*x^2 + x^4)^(3/2))/3003 +
(3825*x*(2 + 3*x^2 + x^4)^(5/2))/143 + (125*x^3*(2 + 3*x^2 + x^4)^(5/2))/13 - (
20884*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(65
*Sqrt[2 + 3*x^2 + x^4]) + (1171349*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*E
llipticF[ArcTan[x], 1/2])/(5005*Sqrt[2 + 3*x^2 + x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 42.2697, size = 209, normalized size = 0.95 \[ \frac{125 x^{3} \left (x^{4} + 3 x^{2} + 2\right )^{\frac{5}{2}}}{13} + \frac{10442 x \left (2 x^{2} + 4\right )}{65 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{x \left (\frac{196035 x^{2}}{143} + \frac{624636}{143}\right ) \left (x^{4} + 3 x^{2} + 2\right )^{\frac{3}{2}}}{63} + \frac{x \left (\frac{2681199 x^{2}}{143} + \frac{9292869}{143}\right ) \sqrt{x^{4} + 3 x^{2} + 2}}{315} + \frac{3825 x \left (x^{4} + 3 x^{2} + 2\right )^{\frac{5}{2}}}{143} - \frac{5221 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{65 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{1171349 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{20020 \sqrt{x^{4} + 3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125*x**3*(x**4 + 3*x**2 + 2)**(5/2)/13 + 10442*x*(2*x**2 + 4)/(65*sqrt(x**4 + 3*
x**2 + 2)) + x*(196035*x**2/143 + 624636/143)*(x**4 + 3*x**2 + 2)**(3/2)/63 + x*
(2681199*x**2/143 + 9292869/143)*sqrt(x**4 + 3*x**2 + 2)/315 + 3825*x*(x**4 + 3*
x**2 + 2)**(5/2)/143 - 5221*sqrt((2*x**2 + 4)/(x**2 + 1))*(4*x**2 + 4)*elliptic_
e(atan(x), 1/2)/(65*sqrt(x**4 + 3*x**2 + 2)) + 1171349*sqrt((2*x**2 + 4)/(x**2 +
1))*(4*x**2 + 4)*elliptic_f(atan(x), 1/2)/(20020*sqrt(x**4 + 3*x**2 + 2))

_______________________________________________________________________________________

Mathematica [C]  time = 0.087309, size = 129, normalized size = 0.59 \[ \frac{144375 x^{15}+1701000 x^{13}+8705725 x^{11}[phone] x^9+46218643 x^7+54938052 x^5+40493455 x^3-2203890 i \sqrt{x^2+1} \sqrt{x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-4824204 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )[phone] x}{15015 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(13572486*x + 40493455*x^3 + 54938052*x^5 + 46218643*x^7 + 25350660*x^9 + 870572
5*x^11 + 1701000*x^13 + 144375*x^15 - (4824204*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*El
lipticE[I*ArcSinh[x/Sqrt[2]], 2] - (2203890*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*Ellip
ticF[I*ArcSinh[x/Sqrt[2]], 2])/(15015*Sqrt[2 + 3*x^2 + x^4])

_______________________________________________________________________________________

Maple [C]  time = 0.026, size = 206, normalized size = 0.9 \[{\frac{598324\,{x}^{5}}{1001}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{10067363\,{x}^{3}}{15015}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{2262081\,x}{5005}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{1171349\,i}{5005}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{10442\,i}{65}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{\frac{131810\,{x}^{7}}{429}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{12075\,{x}^{9}}{143}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{125\,{x}^{11}}{13}\sqrt{{x}^{4}+3\,{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

598324/1001*x^5*(x^4+3*x^2+2)^(1/2)[phone]/15015*x^3*(x^4+3*x^2+2)^(1/2)+22620
81/5005*x*(x^4+3*x^2+2)^(1/2)-1171349/5005*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))+10442/65*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)))+131810/429*x^7*(x^4+3*x^2+2)^(1/2)+12075/
143*x^9*(x^4+3*x^2+2)^(1/2)+125/13*x^11*(x^4+3*x^2+2)^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (125 \, x^{10} + 900 \, x^{8} + 2560 \, x^{6} + 3598 \, x^{4} + 2499 \, x^{2} + 686\right )} \sqrt{x^{4} + 3 \, x^{2} + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((125*x^10 + 900*x^8 + 2560*x^6 + 3598*x^4 + 2499*x^2 + 686)*sqrt(x^4 +
3*x^2 + 2), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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