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

Optimal. Leaf size=207 \[ \frac{1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+4\right )^{3/2}+\frac{1}{105} x \left (289 x^2+1029\right ) \sqrt{x^4+3 x^2+4}+\frac{2798 x \sqrt{x^4+3 x^2+4}}{105 \left (x^2+2\right )}+\frac{74 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{3 \sqrt{x^4+3 x^2+4}}-\frac{2798 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{105 \sqrt{x^4+3 x^2+4}} \]

[Out]

(2798*x*Sqrt[4 + 3*x^2 + x^4])/(105*(2 + x^2)) + (x*(1029 + 289*x^2)*Sqrt[4 + 3*
x^2 + x^4])/105 + (x*(108 + 35*x^2)*(4 + 3*x^2 + x^4)^(3/2))/63 - (2798*Sqrt[2]*
(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8
])/(105*Sqrt[4 + 3*x^2 + x^4]) + (74*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2
+ x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(3*Sqrt[4 + 3*x^2 + x^4])

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Rubi [A]  time = 0.169919, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+4\right )^{3/2}+\frac{1}{105} x \left (289 x^2+1029\right ) \sqrt{x^4+3 x^2+4}+\frac{2798 x \sqrt{x^4+3 x^2+4}}{105 \left (x^2+2\right )}+\frac{74 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{3 \sqrt{x^4+3 x^2+4}}-\frac{2798 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{105 \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(2798*x*Sqrt[4 + 3*x^2 + x^4])/(105*(2 + x^2)) + (x*(1029 + 289*x^2)*Sqrt[4 + 3*
x^2 + x^4])/105 + (x*(108 + 35*x^2)*(4 + 3*x^2 + x^4)^(3/2))/63 - (2798*Sqrt[2]*
(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8
])/(105*Sqrt[4 + 3*x^2 + x^4]) + (74*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2
+ x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(3*Sqrt[4 + 3*x^2 + x^4])

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Rubi in Sympy [A]  time = 31.7825, size = 202, normalized size = 0.98 \[ \frac{x \left (35 x^{2} + 108\right ) \left (x^{4} + 3 x^{2} + 4\right )^{\frac{3}{2}}}{63} + \frac{x \left (867 x^{2} + 3087\right ) \sqrt{x^{4} + 3 x^{2} + 4}}{315} + \frac{5596 x \sqrt{x^{4} + 3 x^{2} + 4}}{105 \left (2 x^{2} + 4\right )} - \frac{2798 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{105 \sqrt{x^{4} + 3 x^{2} + 4}} + \frac{74 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{3 \sqrt{x^{4} + 3 x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(35*x**2 + 108)*(x**4 + 3*x**2 + 4)**(3/2)/63 + x*(867*x**2 + 3087)*sqrt(x**4
+ 3*x**2 + 4)/315 + 5596*x*sqrt(x**4 + 3*x**2 + 4)/(105*(2*x**2 + 4)) - 2798*sqr
t(2)*sqrt((x**4 + 3*x**2 + 4)/(x**2/2 + 1)**2)*(x**2/2 + 1)*elliptic_e(2*atan(sq
rt(2)*x/2), 1/8)/(105*sqrt(x**4 + 3*x**2 + 4)) + 74*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)/(3*sqrt(
x**4 + 3*x**2 + 4))

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Mathematica [C]  time = 1.04044, size = 349, normalized size = 1.69 \[ \frac{3 \sqrt{2} \left (1399 \sqrt{7}-567 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-4197 \sqrt{2} \left (\sqrt{7}+3 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+2 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left (175 x^{10}+1590 x^8+7082 x^6+19068 x^4+28489 x^2+20988\right )}{630 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(20988 + 28489*x^2 + 19068*x^4 + 7082*x^6 + 159
0*x^8 + 175*x^10) - 4197*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 - Sqrt[7])/(3*I + Sqrt[7])] +
3*Sqrt[2]*(-567*I + 1399*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])])/(630*Sqrt[(-I)/(
-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])

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Maple [C]  time = 0.01, size = 275, normalized size = 1.3 \[{\frac{71\,{x}^{5}}{21}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{3187\,{x}^{3}}{315}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{583\,x}{35}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{6352}{35\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{89536}{105\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{5\,{x}^{7}}{9}\sqrt{{x}^{4}+3\,{x}^{2}+4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

71/21*x^5*(x^4+3*x^2+4)^(1/2)+3187/315*x^3*(x^4+3*x^2+4)^(1/2)+583/35*x*(x^4+3*x
^2+4)^(1/2)+6352/35/(-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))-89536/105/(-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)))+5/9*x^7*(x
^4+3*x^2+4)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (5 \, x^{6} + 22 \, x^{4} + 41 \, x^{2} + 28\right )} \sqrt{x^{4} + 3 \, x^{2} + 4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((5*x^6 + 22*x^4 + 41*x^2 + 28)*sqrt(x^4 + 3*x^2 + 4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2+7)*(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), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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