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

Optimal. Leaf size=198 \[ \frac{1}{7} x \left (x^4+3 x^2+4\right )^{3/2}+\frac{1}{35} x \left (9 x^2+49\right ) \sqrt{x^4+3 x^2+4}+\frac{138 x \sqrt{x^4+3 x^2+4}}{35 \left (x^2+2\right )}+\frac{4 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{\sqrt{x^4+3 x^2+4}}-\frac{138 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{35 \sqrt{x^4+3 x^2+4}} \]

[Out]

(138*x*Sqrt[4 + 3*x^2 + x^4])/(35*(2 + x^2)) + (x*(49 + 9*x^2)*Sqrt[4 + 3*x^2 +
x^4])/35 + (x*(4 + 3*x^2 + x^4)^(3/2))/7 - (138*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^
2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(35*Sqrt[4 + 3*x^2 +
x^4]) + (4*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*Arc
Tan[x/Sqrt[2]], 1/8])/Sqrt[4 + 3*x^2 + x^4]

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Rubi [A]  time = 0.157542, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{1}{7} x \left (x^4+3 x^2+4\right )^{3/2}+\frac{1}{35} x \left (9 x^2+49\right ) \sqrt{x^4+3 x^2+4}+\frac{138 x \sqrt{x^4+3 x^2+4}}{35 \left (x^2+2\right )}+\frac{4 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{\sqrt{x^4+3 x^2+4}}-\frac{138 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{35 \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

[In]  Int[(4 + 3*x^2 + x^4)^(3/2),x]

[Out]

(138*x*Sqrt[4 + 3*x^2 + x^4])/(35*(2 + x^2)) + (x*(49 + 9*x^2)*Sqrt[4 + 3*x^2 +
x^4])/35 + (x*(4 + 3*x^2 + x^4)^(3/2))/7 - (138*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^
2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(35*Sqrt[4 + 3*x^2 +
x^4]) + (4*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*Arc
Tan[x/Sqrt[2]], 1/8])/Sqrt[4 + 3*x^2 + x^4]

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Rubi in Sympy [A]  time = 36.5299, size = 194, normalized size = 0.98 \[ \frac{x \left (9 x^{2} + 49\right ) \sqrt{x^{4} + 3 x^{2} + 4}}{35} + \frac{x \left (x^{4} + 3 x^{2} + 4\right )^{\frac{3}{2}}}{7} + \frac{276 x \sqrt{x^{4} + 3 x^{2} + 4}}{35 \left (2 x^{2} + 4\right )} - \frac{138 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{35 \sqrt{x^{4} + 3 x^{2} + 4}} + \frac{4 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{\sqrt{x^{4} + 3 x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(9*x**2 + 49)*sqrt(x**4 + 3*x**2 + 4)/35 + x*(x**4 + 3*x**2 + 4)**(3/2)/7 + 27
6*x*sqrt(x**4 + 3*x**2 + 4)/(35*(2*x**2 + 4)) - 138*sqrt(2)*sqrt((x**4 + 3*x**2
+ 4)/(x**2/2 + 1)**2)*(x**2/2 + 1)*elliptic_e(2*atan(sqrt(2)*x/2), 1/8)/(35*sqrt
(x**4 + 3*x**2 + 4)) + 4*sqrt(2)*sqrt((x**4 + 3*x**2 + 4)/(x**2/2 + 1)**2)*(x**2
/2 + 1)*elliptic_f(2*atan(sqrt(2)*x/2), 1/8)/sqrt(x**4 + 3*x**2 + 4)

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Mathematica [C]  time = 0.921861, size = 343, normalized size = 1.73 \[ \frac{\sqrt{2} \left (69 \sqrt{7}-77 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-69 \sqrt{2} \left (\sqrt{7}+3 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+2 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left (5 x^8+39 x^6+161 x^4+303 x^2+276\right )}{70 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(276 + 303*x^2 + 161*x^4 + 39*x^6 + 5*x^8) - 69
*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqr
t[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(
-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + Sqrt[2]*(-77*I + 69*Sqrt
[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sqrt[7] + (
2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x],
(3*I - Sqrt[7])/(3*I + Sqrt[7])])/(70*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^
2 + x^4])

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Maple [C]  time = 0.005, size = 258, normalized size = 1.3 \[{\frac{{x}^{5}}{7}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{24\,{x}^{3}}{35}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{69\,x}{35}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{1136}{35\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{4416}{35\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+3*x^2+4)^(3/2),x)

[Out]

1/7*x^5*(x^4+3*x^2+4)^(1/2)+24/35*x^3*(x^4+3*x^2+4)^(1/2)+69/35*x*(x^4+3*x^2+4)^
(1/2)+1136/35/(-6+2*I*7^(1/2))^(1/2)*(1-(-3/8+1/8*I*7^(1/2))*x^2)^(1/2)*(1-(-3/8
-1/8*I*7^(1/2))*x^2)^(1/2)/(x^4+3*x^2+4)^(1/2)*EllipticF(1/4*x*(-6+2*I*7^(1/2))^
(1/2),1/4*(2+6*I*7^(1/2))^(1/2))-4416/35/(-6+2*I*7^(1/2))^(1/2)*(1-(-3/8+1/8*I*7
^(1/2))*x^2)^(1/2)*(1-(-3/8-1/8*I*7^(1/2))*x^2)^(1/2)/(x^4+3*x^2+4)^(1/2)/(I*7^(
1/2)+3)*(EllipticF(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2))-Ellip
ticE(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x^4 + 3*x^2 + 4)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac{3}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((x^4 + 3*x^2 + 4)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x**4 + 3*x**2 + 4)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x^4 + 3*x^2 + 4)^(3/2), x)