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

Optimal. Leaf size=226 \[ \frac{25}{11} x \left (x^4+3 x^2+4\right )^{5/2}+\frac{1}{693} x \left (2240 x^2+6831\right ) \left (x^4+3 x^2+4\right )^{3/2}+\frac{x \left (18253 x^2+64533\right ) \sqrt{x^4+3 x^2+4}}{1155}+\frac{175346 x \sqrt{x^4+3 x^2+4}}{1155 \left (x^2+2\right )}+\frac{4628 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{33 \sqrt{x^4+3 x^2+4}}-\frac{175346 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{1155 \sqrt{x^4+3 x^2+4}} \]

[Out]

(175346*x*Sqrt[4 + 3*x^2 + x^4])/(1155*(2 + x^2)) + (x*(64533 + 18253*x^2)*Sqrt[
4 + 3*x^2 + x^4])/1155 + (x*(6831 + 2240*x^2)*(4 + 3*x^2 + x^4)^(3/2))/693 + (25
*x*(4 + 3*x^2 + x^4)^(5/2))/11 - (175346*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4
)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(1155*Sqrt[4 + 3*x^2 + x^4])
+ (4628*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTa
n[x/Sqrt[2]], 1/8])/(33*Sqrt[4 + 3*x^2 + x^4])

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Rubi [A]  time = 0.222728, antiderivative size = 226, 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 \[ \frac{25}{11} x \left (x^4+3 x^2+4\right )^{5/2}+\frac{1}{693} x \left (2240 x^2+6831\right ) \left (x^4+3 x^2+4\right )^{3/2}+\frac{x \left (18253 x^2+64533\right ) \sqrt{x^4+3 x^2+4}}{1155}+\frac{175346 x \sqrt{x^4+3 x^2+4}}{1155 \left (x^2+2\right )}+\frac{4628 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{33 \sqrt{x^4+3 x^2+4}}-\frac{175346 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{1155 \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(175346*x*Sqrt[4 + 3*x^2 + x^4])/(1155*(2 + x^2)) + (x*(64533 + 18253*x^2)*Sqrt[
4 + 3*x^2 + x^4])/1155 + (x*(6831 + 2240*x^2)*(4 + 3*x^2 + x^4)^(3/2))/693 + (25
*x*(4 + 3*x^2 + x^4)^(5/2))/11 - (175346*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4
)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(1155*Sqrt[4 + 3*x^2 + x^4])
+ (4628*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTa
n[x/Sqrt[2]], 1/8])/(33*Sqrt[4 + 3*x^2 + x^4])

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Rubi in Sympy [A]  time = 41.6745, size = 226, normalized size = 1. \[ \frac{x \left (\frac{2240 x^{2}}{11} + 621\right ) \left (x^{4} + 3 x^{2} + 4\right )^{\frac{3}{2}}}{63} + \frac{x \left (\frac{54759 x^{2}}{11} + \frac{193599}{11}\right ) \sqrt{x^{4} + 3 x^{2} + 4}}{315} + \frac{25 x \left (x^{4} + 3 x^{2} + 4\right )^{\frac{5}{2}}}{11} + \frac{350692 x \sqrt{x^{4} + 3 x^{2} + 4}}{1155 \left (2 x^{2} + 4\right )} - \frac{175346 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{1155 \sqrt{x^{4} + 3 x^{2} + 4}} + \frac{4628 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{33 \sqrt{x^{4} + 3 x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(2240*x**2/11 + 621)*(x**4 + 3*x**2 + 4)**(3/2)/63 + x*(54759*x**2/11 + 193599
/11)*sqrt(x**4 + 3*x**2 + 4)/315 + 25*x*(x**4 + 3*x**2 + 4)**(5/2)/11 + 350692*x
*sqrt(x**4 + 3*x**2 + 4)/(1155*(2*x**2 + 4)) - 175346*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)/(1155*
sqrt(x**4 + 3*x**2 + 4)) + 4628*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)/(33*sqrt(x**4 + 3*x**2 + 4))

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Mathematica [C]  time = 0.911577, size = 354, normalized size = 1.57 \[ \frac{3 \sqrt{2} \left (87673 \sqrt{7}-34209 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-263019 \sqrt{2} \left (\sqrt{7}+3 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+2 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left (7875 x^{12}+82075 x^{10}+408480 x^8+1229714 x^6+2435811 x^4+2932753 x^2+1824876\right )}{6930 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(1824876 + 2932753*x^2 + 2435811*x^4 + 1229714*
x^6 + 408480*x^8 + 82075*x^10 + 7875*x^12) - 263019*Sqrt[2]*(3*I + 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])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - S
qrt[7])/(3*I + Sqrt[7])] + 3*Sqrt[2]*(-34209*I + 87673*Sqrt[7])*Sqrt[(-3*I + Sqr
t[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])])/(6930*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])

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Maple [C]  time = 0.013, size = 292, normalized size = 1.3 \[{\frac{1222\,{x}^{5}}{21}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{391024\,{x}^{3}}{3465}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{50691\,x}{385}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{396304}{385\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{5611072}{1155\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{1670\,{x}^{7}}{99}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{25\,{x}^{9}}{11}\sqrt{{x}^{4}+3\,{x}^{2}+4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1222/21*x^5*(x^4+3*x^2+4)^(1/2)+391024/3465*x^3*(x^4+3*x^2+4)^(1/2)+50691/385*x*
(x^4+3*x^2+4)^(1/2)+396304/385/(-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))-5611072/1155/(-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))-EllipticE(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2
)))+1670/99*x^7*(x^4+3*x^2+4)^(1/2)+25/11*x^9*(x^4+3*x^2+4)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (25 \, x^{8} + 145 \, x^{6} + 359 \, x^{4} + 427 \, x^{2} + 196\right )} \sqrt{x^{4} + 3 \, x^{2} + 4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((25*x^8 + 145*x^6 + 359*x^4 + 427*x^2 + 196)*sqrt(x^4 + 3*x^2 + 4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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