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

Optimal. Leaf size=268 \[ \frac{92150}{429} \left (x^4+3 x^2+4\right )^{5/2} x+\frac{\left (131080 x^2+452001\right ) \left (x^4+3 x^2+4\right )^{3/2} x}{1287}+\frac{7 \left (174989 x^2+661429\right ) \sqrt{x^4+3 x^2+4} x}{2145}+\frac{12665086 \sqrt{x^4+3 x^2+4} x}{2145 \left (x^2+2\right )}+\frac{2383556 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{429 \sqrt{x^4+3 x^2+4}}-\frac{12665086 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{2145 \sqrt{x^4+3 x^2+4}}+\frac{125}{3} \left (x^4+3 x^2+4\right )^{5/2} x^5+\frac{2250}{13} \left (x^4+3 x^2+4\right )^{5/2} x^3 \]

[Out]

(12665086*x*Sqrt[4 + 3*x^2 + x^4])/(2145*(2 + x^2)) + (7*x*(661429 + 174989*x^2)
*Sqrt[4 + 3*x^2 + x^4])/2145 + (x*(452001 + 131080*x^2)*(4 + 3*x^2 + x^4)^(3/2))
/1287 + (92150*x*(4 + 3*x^2 + x^4)^(5/2))/429 + (2250*x^3*(4 + 3*x^2 + x^4)^(5/2
))/13 + (125*x^5*(4 + 3*x^2 + x^4)^(5/2))/3 - (12665086*Sqrt[2]*(2 + x^2)*Sqrt[(
4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(2145*Sqrt[4
+ 3*x^2 + x^4]) + (2383556*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]
*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(429*Sqrt[4 + 3*x^2 + x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.320155, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{92150}{429} \left (x^4+3 x^2+4\right )^{5/2} x+\frac{\left (131080 x^2+452001\right ) \left (x^4+3 x^2+4\right )^{3/2} x}{1287}+\frac{7 \left (174989 x^2+661429\right ) \sqrt{x^4+3 x^2+4} x}{2145}+\frac{12665086 \sqrt{x^4+3 x^2+4} x}{2145 \left (x^2+2\right )}+\frac{2383556 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{429 \sqrt{x^4+3 x^2+4}}-\frac{12665086 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{2145 \sqrt{x^4+3 x^2+4}}+\frac{125}{3} \left (x^4+3 x^2+4\right )^{5/2} x^5+\frac{2250}{13} \left (x^4+3 x^2+4\right )^{5/2} x^3 \]

Antiderivative was successfully verified.

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

[Out]

(12665086*x*Sqrt[4 + 3*x^2 + x^4])/(2145*(2 + x^2)) + (7*x*(661429 + 174989*x^2)
*Sqrt[4 + 3*x^2 + x^4])/2145 + (x*(452001 + 131080*x^2)*(4 + 3*x^2 + x^4)^(3/2))
/1287 + (92150*x*(4 + 3*x^2 + x^4)^(5/2))/429 + (2250*x^3*(4 + 3*x^2 + x^4)^(5/2
))/13 + (125*x^5*(4 + 3*x^2 + x^4)^(5/2))/3 - (12665086*Sqrt[2]*(2 + x^2)*Sqrt[(
4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(2145*Sqrt[4
+ 3*x^2 + x^4]) + (2383556*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]
*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(429*Sqrt[4 + 3*x^2 + x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 63.0751, size = 269, normalized size = 1. \[ \frac{125 x^{5} \left (x^{4} + 3 x^{2} + 4\right )^{\frac{5}{2}}}{3} + \frac{2250 x^{3} \left (x^{4} + 3 x^{2} + 4\right )^{\frac{5}{2}}}{13} + \frac{x \left (\frac{917560 x^{2}}{143} + \frac{287637}{13}\right ) \left (x^{4} + 3 x^{2} + 4\right )^{\frac{3}{2}}}{63} + \frac{x \left (\frac{25723383 x^{2}}{143} + \frac{97230063}{143}\right ) \sqrt{x^{4} + 3 x^{2} + 4}}{315} + \frac{92150 x \left (x^{4} + 3 x^{2} + 4\right )^{\frac{5}{2}}}{429} + \frac{25330172 x \sqrt{x^{4} + 3 x^{2} + 4}}{2145 \left (2 x^{2} + 4\right )} - \frac{12665086 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{2145 \sqrt{x^{4} + 3 x^{2} + 4}} + \frac{2383556 \sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{429 \sqrt{x^{4} + 3 x^{2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125*x**5*(x**4 + 3*x**2 + 4)**(5/2)/3 + 2250*x**3*(x**4 + 3*x**2 + 4)**(5/2)/13
+ x*(917560*x**2/143 + 287637/13)*(x**4 + 3*x**2 + 4)**(3/2)/63 + x*(25723383*x*
*2/143 + 97230063/143)*sqrt(x**4 + 3*x**2 + 4)/315 + 92150*x*(x**4 + 3*x**2 + 4)
**(5/2)/429 + 25330172*x*sqrt(x**4 + 3*x**2 + 4)/(2145*(2*x**2 + 4)) - 12665086*
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)/(2145*sqrt(x**4 + 3*x**2 + 4)) + 2383556*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)
/(429*sqrt(x**4 + 3*x**2 + 4))

_______________________________________________________________________________________

Mathematica [C]  time = 1.19101, size = 364, normalized size = 1.36 \[ \frac{21 \sqrt{2} \left (904649 \sqrt{7}-477617 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )-18997629 \sqrt{2} \left (\sqrt{7}+3 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )+2 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left (268125 x^{16}+3526875 x^{14}[phone] x^{12}[phone] x^{10}[phone] x^8+377574349 x^6+472235001 x^4+391419623 x^2+180184116\right )}{12870 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(180184116 + 391419623*x^2 + 472235001*x^4 + 37
7574349*x^6 + 212188905*x^8 + 83076275*x^10 + 21862875*x^12 + 3526875*x^14 + 268
125*x^16) - 18997629*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*A
rcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])] + 21*S
qrt[2]*(-477617*I + 904649*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sq
rt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sq
rt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I + Sqrt[7])])/(12870*Sqrt[(-
I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])

_______________________________________________________________________________________

Maple [C]  time = 0.049, size = 326, normalized size = 1.2 \[{\frac{356027\,{x}^{5}}{39}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{64070384\,{x}^{3}}{6435}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{15015343\,x}{2145}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{89363792}{2145\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{405282752}{2145\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{6863530\,{x}^{7}}{1287}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{841525\,{x}^{9}}{429}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{5500\,{x}^{11}}{13}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{125\,{x}^{13}}{3}\sqrt{{x}^{4}+3\,{x}^{2}+4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

356027/39*x^5*(x^4+3*x^2+4)^(1/2)[phone]/6435*x^3*(x^4+3*x^2+4)^(1/2)[phone]
/2145*x*(x^4+3*x^2+4)^(1/2)[phone]/2145/(-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)*Ellip
ticF(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2))-405282752/2145/(-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)))+6863530/1287*x^7*(x^4+3*x^2+4)^(1/2)+841525/429*x^9*(x^4+3*x^2+
4)^(1/2)+5500/13*x^11*(x^4+3*x^2+4)^(1/2)+125/3*x^13*(x^4+3*x^2+4)^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (625 \, x^{12} + 5375 \, x^{10} + 20350 \, x^{8} + 42910 \, x^{6} + 52381 \, x^{4} + 34643 \, x^{2} + 9604\right )} \sqrt{x^{4} + 3 \, x^{2} + 4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((625*x^12 + 5375*x^10 + 20350*x^8 + 42910*x^6 + 52381*x^4 + 34643*x^2 +
9604)*sqrt(x^4 + 3*x^2 + 4), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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