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

Optimal. Leaf size=207 \[ \frac{24 x \left (x^2+2\right )}{125 \sqrt{x^4+3 x^2+2}}+\frac{1}{75} x \left (3 x^2+11\right ) \sqrt{x^4+3 x^2+2}+\frac{56 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{375 \sqrt{x^4+3 x^2+2}}-\frac{24 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{125 \sqrt{x^4+3 x^2+2}}-\frac{9 \sqrt{2} \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{875 \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]

[Out]

(24*x*(2 + x^2))/(125*Sqrt[2 + 3*x^2 + x^4]) + (x*(11 + 3*x^2)*Sqrt[2 + 3*x^2 +
x^4])/75 - (24*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x],
1/2])/(125*Sqrt[2 + 3*x^2 + x^4]) + (56*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^
2)]*EllipticF[ArcTan[x], 1/2])/(375*Sqrt[2 + 3*x^2 + x^4]) - (9*Sqrt[2]*(2 + x^2
)*EllipticPi[2/7, ArcTan[x], 1/2])/(875*Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2
+ x^4])

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Rubi [A]  time = 0.488555, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{24 x \left (x^2+2\right )}{125 \sqrt{x^4+3 x^2+2}}+\frac{1}{75} x \left (3 x^2+11\right ) \sqrt{x^4+3 x^2+2}+\frac{56 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{375 \sqrt{x^4+3 x^2+2}}-\frac{24 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{125 \sqrt{x^4+3 x^2+2}}-\frac{9 \sqrt{2} \left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{875 \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(24*x*(2 + x^2))/(125*Sqrt[2 + 3*x^2 + x^4]) + (x*(11 + 3*x^2)*Sqrt[2 + 3*x^2 +
x^4])/75 - (24*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x],
1/2])/(125*Sqrt[2 + 3*x^2 + x^4]) + (56*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^
2)]*EllipticF[ArcTan[x], 1/2])/(375*Sqrt[2 + 3*x^2 + x^4]) - (9*Sqrt[2]*(2 + x^2
)*EllipticPi[2/7, ArcTan[x], 1/2])/(875*Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2
+ x^4])

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Rubi in Sympy [A]  time = 63.7172, size = 214, normalized size = 1.03 \[ \frac{x^{3} \sqrt{x^{4} + 3 x^{2} + 2}}{25} + \frac{12 x \left (2 x^{2} + 4\right )}{125 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{11 x \sqrt{x^{4} + 3 x^{2} + 2}}{75} - \frac{6 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{125 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{47 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{1500 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{48 \sqrt{2} \sqrt{x^{4} + 3 x^{2} + 2} \Pi \left (- \frac{3}{7}; \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -1\right )}{875 \sqrt{\frac{2 x^{2} + 2}{x^{2} + 2}} \left (2 x^{2} + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*sqrt(x**4 + 3*x**2 + 2)/25 + 12*x*(2*x**2 + 4)/(125*sqrt(x**4 + 3*x**2 + 2)
) + 11*x*sqrt(x**4 + 3*x**2 + 2)/75 - 6*sqrt((2*x**2 + 4)/(x**2 + 1))*(4*x**2 +
4)*elliptic_e(atan(x), 1/2)/(125*sqrt(x**4 + 3*x**2 + 2)) + 47*sqrt((2*x**2 + 4)
/(x**2 + 1))*(4*x**2 + 4)*elliptic_f(atan(x), 1/2)/(1500*sqrt(x**4 + 3*x**2 + 2)
) + 48*sqrt(2)*sqrt(x**4 + 3*x**2 + 2)*elliptic_pi(-3/7, atan(sqrt(2)*x/2), -1)/
(875*sqrt((2*x**2 + 2)/(x**2 + 2))*(2*x**2 + 4))

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Mathematica [C]  time = 0.132934, size = 148, normalized size = 0.71 \[ \frac{525 x^7+3500 x^5+6825 x^3-1022 i \sqrt{x^2+1} \sqrt{x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-2520 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-108 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left (\frac{10}{7};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+3850 x}{13125 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(3850*x + 6825*x^3 + 3500*x^5 + 525*x^7 - (2520*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*E
llipticE[I*ArcSinh[x/Sqrt[2]], 2] - (1022*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*Ellipti
cF[I*ArcSinh[x/Sqrt[2]], 2] - (108*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticPi[10/
7, I*ArcSinh[x/Sqrt[2]], 2])/(13125*Sqrt[2 + 3*x^2 + x^4])

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Maple [C]  time = 0.021, size = 170, normalized size = 0.8 \[{\frac{{x}^{3}}{25}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{11\,x}{75}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{73\,i}{1875}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{{\frac{12\,i}{125}}\sqrt{2}{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{{\frac{36\,i}{4375}}\sqrt{2}\sqrt{1+{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{i}{2}}\sqrt{2}x,{\frac{10}{7}},\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25*x^3*(x^4+3*x^2+2)^(1/2)+11/75*x*(x^4+3*x^2+2)^(1/2)-73/1875*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))-
12/125*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticE(1/2
*I*2^(1/2)*x,2^(1/2))-36/4375*I*2^(1/2)*(1+1/2*x^2)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x
^2+2)^(1/2)*EllipticPi(1/2*I*2^(1/2)*x,10/7,2^(1/2))

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

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]

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

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

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]

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

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

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)

[Out]

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

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

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]

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