3.4.60 \(\int (4+3 x^2+x^4)^{3/2} \, dx\) [360]

3.4.60.1 Optimal result

Integrand size = 14, antiderivative size = 198 \[ \int \left (4+3 x^2+x^4\right )^{3/2} \, dx=\frac {138 x \sqrt {4+3 x^2+x^4}}{35 \left (2+x^2\right )}+\frac {1}{35} x \left (49+9 x^2\right ) \sqrt {4+3 x^2+x^4}+\frac {1}{7} x \left (4+3 x^2+x^4\right )^{3/2}-\frac {138 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{35 \sqrt {4+3 x^2+x^4}}+\frac {4 \sqrt {2} \left (2+x^2\right ) \sqrt {\frac {4+3 x^2+x^4}{\left (2+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {4+3 x^2+x^4}} \]

output

1/7*x*(x^4+3*x^2+4)^(3/2)+138/35*x*(x^4+3*x^2+4)^(1/2)/(x^2+2)+1/35*x*(9*x 
^2+49)*(x^4+3*x^2+4)^(1/2)-138/35*(x^2+2)*(cos(2*arctan(1/2*x*2^(1/2)))^2) 
^(1/2)/cos(2*arctan(1/2*x*2^(1/2)))*EllipticE(sin(2*arctan(1/2*x*2^(1/2))) 
,1/4*2^(1/2))*2^(1/2)*((x^4+3*x^2+4)/(x^2+2)^2)^(1/2)/(x^4+3*x^2+4)^(1/2)+ 
4*(x^2+2)*(cos(2*arctan(1/2*x*2^(1/2)))^2)^(1/2)/cos(2*arctan(1/2*x*2^(1/2 
)))*EllipticF(sin(2*arctan(1/2*x*2^(1/2))),1/4*2^(1/2))*((x^4+3*x^2+4)/(x^ 
2+2)^2)^(1/2)*2^(1/2)/(x^4+3*x^2+4)^(1/2)
 

3.4.60.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.78 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.73 \[ \int \left (4+3 x^2+x^4\right )^{3/2} \, dx=\frac {2 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} x \left (276+303 x^2+161 x^4+39 x^6+5 x^8\right )-69 \sqrt {2} \left (3 i+\sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right )|\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )+\sqrt {2} \left (-77 i+69 \sqrt {7}\right ) \sqrt {\frac {-3 i+\sqrt {7}-2 i x^2}{-3 i+\sqrt {7}}} \sqrt {\frac {3 i+\sqrt {7}+2 i x^2}{3 i+\sqrt {7}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {2 i}{-3 i+\sqrt {7}}} x\right ),\frac {3 i-\sqrt {7}}{3 i+\sqrt {7}}\right )}{70 \sqrt {-\frac {i}{-3 i+\sqrt {7}}} \sqrt {4+3 x^2+x^4}} \]

input

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

output

(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 + S 
qrt[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*Arc 
Sinh[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 + Sqr 
t[7])]*Sqrt[(3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSi 
nh[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])
 

3.4.60.3 Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1404, 1490, 27, 1511, 27, 1416, 1509}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1404

\(\displaystyle \frac {3}{7} \int \left (3 x^2+8\right ) \sqrt {x^4+3 x^2+4}dx+\frac {1}{7} x \left (x^4+3 x^2+4\right )^{3/2}\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {3}{7} \left (\frac {1}{15} \int \frac {2 \left (69 x^2+142\right )}{\sqrt {x^4+3 x^2+4}}dx+\frac {1}{15} x \sqrt {x^4+3 x^2+4} \left (9 x^2+49\right )\right )+\frac {1}{7} x \left (x^4+3 x^2+4\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{7} \left (\frac {2}{15} \int \frac {69 x^2+142}{\sqrt {x^4+3 x^2+4}}dx+\frac {1}{15} x \sqrt {x^4+3 x^2+4} \left (9 x^2+49\right )\right )+\frac {1}{7} x \left (x^4+3 x^2+4\right )^{3/2}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {3}{7} \left (\frac {2}{15} \left (280 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx-138 \int \frac {2-x^2}{2 \sqrt {x^4+3 x^2+4}}dx\right )+\frac {1}{15} x \sqrt {x^4+3 x^2+4} \left (9 x^2+49\right )\right )+\frac {1}{7} x \left (x^4+3 x^2+4\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{7} \left (\frac {2}{15} \left (280 \int \frac {1}{\sqrt {x^4+3 x^2+4}}dx-69 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx\right )+\frac {1}{15} x \sqrt {x^4+3 x^2+4} \left (9 x^2+49\right )\right )+\frac {1}{7} x \left (x^4+3 x^2+4\right )^{3/2}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {3}{7} \left (\frac {2}{15} \left (\frac {70 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-69 \int \frac {2-x^2}{\sqrt {x^4+3 x^2+4}}dx\right )+\frac {1}{15} x \sqrt {x^4+3 x^2+4} \left (9 x^2+49\right )\right )+\frac {1}{7} x \left (x^4+3 x^2+4\right )^{3/2}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {3}{7} \left (\frac {2}{15} \left (\frac {70 \sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right ),\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-69 \left (\frac {\sqrt {2} \left (x^2+2\right ) \sqrt {\frac {x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \arctan \left (\frac {x}{\sqrt {2}}\right )|\frac {1}{8}\right )}{\sqrt {x^4+3 x^2+4}}-\frac {x \sqrt {x^4+3 x^2+4}}{x^2+2}\right )\right )+\frac {1}{15} x \sqrt {x^4+3 x^2+4} \left (9 x^2+49\right )\right )+\frac {1}{7} x \left (x^4+3 x^2+4\right )^{3/2}\)

input

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

output

(x*(4 + 3*x^2 + x^4)^(3/2))/7 + (3*((x*(49 + 9*x^2)*Sqrt[4 + 3*x^2 + x^4]) 
/15 + (2*(-69*(-((x*Sqrt[4 + 3*x^2 + x^4])/(2 + x^2)) + (Sqrt[2]*(2 + x^2) 
*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/ 
Sqrt[4 + 3*x^2 + x^4]) + (70*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + 
 x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/Sqrt[4 + 3*x^2 + x^4]))/15)) 
/7
 

3.4.60.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1404

Int[((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*((a + b 
*x^2 + c*x^4)^p/(4*p + 1)), x] + Simp[2*(p/(4*p + 1))   Int[(2*a + b*x^2)*( 
a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a* 
c, 0] && GtQ[p, 0] && IntegerQ[2*p]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1490

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c 
*x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) 
 Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) 
- b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

3.4.60.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.19

method	result	size
risch	\(\frac {x \left (5 x^{4}+24 x^{2}+69\right ) \sqrt {x^{4}+3 x^{2}+4}}{35}+\frac {1136 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{35 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {4416 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{35 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}\)	\(236\)
default	\(\frac {x^{5} \sqrt {x^{4}+3 x^{2}+4}}{7}+\frac {24 x^{3} \sqrt {x^{4}+3 x^{2}+4}}{35}+\frac {69 x \sqrt {x^{4}+3 x^{2}+4}}{35}+\frac {1136 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{35 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {4416 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{35 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}\)	\(258\)
elliptic	\(\frac {x^{5} \sqrt {x^{4}+3 x^{2}+4}}{7}+\frac {24 x^{3} \sqrt {x^{4}+3 x^{2}+4}}{35}+\frac {69 x \sqrt {x^{4}+3 x^{2}+4}}{35}+\frac {1136 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )}{35 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}}-\frac {4416 \sqrt {1-\left (-\frac {3}{8}+\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{8}-\frac {i \sqrt {7}}{8}\right ) x^{2}}\, \left (F\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )-E\left (\frac {x \sqrt {-6+2 i \sqrt {7}}}{4}, \frac {\sqrt {2+6 i \sqrt {7}}}{4}\right )\right )}{35 \sqrt {-6+2 i \sqrt {7}}\, \sqrt {x^{4}+3 x^{2}+4}\, \left (3+i \sqrt {7}\right )}\)	\(258\)

input

int((x^4+3*x^2+4)^(3/2),x,method=_RETURNVERBOSE)
 

output

1/35*x*(5*x^4+24*x^2+69)*(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)/(3+I*7^(1/2))*(El 
lipticF(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)))
 

3.4.60.5 Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.65 \[ \int \left (4+3 x^2+x^4\right )^{3/2} \, dx=\frac {138 \, \sqrt {2} {\left (\sqrt {-7} x - 3 \, x\right )} \sqrt {\sqrt {-7} - 3} E(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-7} - 3}}{2 \, x}\right )\,|\,\frac {3}{8} \, \sqrt {-7} + \frac {1}{8}) - \sqrt {2} {\left (67 \, \sqrt {-7} x - 627 \, x\right )} \sqrt {\sqrt {-7} - 3} F(\arcsin \left (\frac {\sqrt {2} \sqrt {\sqrt {-7} - 3}}{2 \, x}\right )\,|\,\frac {3}{8} \, \sqrt {-7} + \frac {1}{8}) + 4 \, {\left (5 \, x^{6} + 24 \, x^{4} + 69 \, x^{2} + 138\right )} \sqrt {x^{4} + 3 \, x^{2} + 4}}{140 \, x} \]

input

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

output

1/140*(138*sqrt(2)*(sqrt(-7)*x - 3*x)*sqrt(sqrt(-7) - 3)*elliptic_e(arcsin 
(1/2*sqrt(2)*sqrt(sqrt(-7) - 3)/x), 3/8*sqrt(-7) + 1/8) - sqrt(2)*(67*sqrt 
(-7)*x - 627*x)*sqrt(sqrt(-7) - 3)*elliptic_f(arcsin(1/2*sqrt(2)*sqrt(sqrt 
(-7) - 3)/x), 3/8*sqrt(-7) + 1/8) + 4*(5*x^6 + 24*x^4 + 69*x^2 + 138)*sqrt 
(x^4 + 3*x^2 + 4))/x
 

3.4.60.6 Sympy [F]

\[ \int \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int \left (x^{4} + 3 x^{2} + 4\right )^{\frac {3}{2}}\, dx \]

input

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

output

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

3.4.60.7 Maxima [F]

\[ \int \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} \,d x } \]

input

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

output

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

3.4.60.8 Giac [F]

\[ \int \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int { {\left (x^{4} + 3 \, x^{2} + 4\right )}^{\frac {3}{2}} \,d x } \]

input

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

output

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

3.4.60.9 Mupad [F(-1)]

Timed out. \[ \int \left (4+3 x^2+x^4\right )^{3/2} \, dx=\int {\left (x^4+3\,x^2+4\right )}^{3/2} \,d x \]

input

int((3*x^2 + x^4 + 4)^(3/2),x)
 

output

int((3*x^2 + x^4 + 4)^(3/2), x)