\(\int (7+5 x^2)^2 \sqrt {2+3 x^2+x^4} \, dx\) [287]

Optimal result

Integrand size = 24, antiderivative size = 168 \[ \int \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4} \, dx=\frac {31 x \left (2+x^2\right )}{\sqrt {2+3 x^2+x^4}}+\frac {1}{21} x \left (407+114 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {25}{7} x \left (2+3 x^2+x^4\right )^{3/2}-\frac {31 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {2+3 x^2+x^4}}+\frac {472 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{21 \sqrt {2+3 x^2+x^4}} \]

[Out]

25/7*x*(x^4+3*x^2+2)^(3/2)+31*x*(x^2+2)/(x^4+3*x^2+2)^(1/2)-31*(x^2+1)^(3/2)*(1/(x^2+1))^(1/2)*EllipticE(x/(x^
2+1)^(1/2),1/2*2^(1/2))*2^(1/2)*((x^2+2)/(x^2+1))^(1/2)/(x^4+3*x^2+2)^(1/2)+472/21*(x^2+1)^(3/2)*(1/(x^2+1))^(
1/2)*EllipticF(x/(x^2+1)^(1/2),1/2*2^(1/2))*2^(1/2)*((x^2+2)/(x^2+1))^(1/2)/(x^4+3*x^2+2)^(1/2)+1/21*x*(114*x^
2+407)*(x^4+3*x^2+2)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1220, 1190, 1203, 1113, 1149} \[ \int \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4} \, dx=\frac {472 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{21 \sqrt {x^4+3 x^2+2}}-\frac {31 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{\sqrt {x^4+3 x^2+2}}+\frac {25}{7} x \left (x^4+3 x^2+2\right )^{3/2}+\frac {1}{21} x \left (114 x^2+407\right ) \sqrt {x^4+3 x^2+2}+\frac {31 x \left (x^2+2\right )}{\sqrt {x^4+3 x^2+2}} \]

[In]

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

[Out]

(31*x*(2 + x^2))/Sqrt[2 + 3*x^2 + x^4] + (x*(407 + 114*x^2)*Sqrt[2 + 3*x^2 + x^4])/21 + (25*x*(2 + 3*x^2 + x^4
)^(3/2))/7 - (31*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/Sqrt[2 + 3*x^2 + x^4]
+ (472*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(21*Sqrt[2 + 3*x^2 + x^4])

Rule 1113

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

Rule 1149

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[x*((b +
q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4])), x] - Simp[Rt[(b + q)/(2*a), 2]*(2*a + (b + q)*x^2)*(Sqrt[(2*a + (
b - q)*x^2)/(2*a + (b + q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan[Rt[(b + q)/(2*a), 2]*x], 2*(q
/(b + q))], x] /; PosQ[(b + q)/a] &&  !(PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; Fre
eQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]

Rule 1190

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> 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] + Dist[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 1203

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[d, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[e, Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b +
q)/a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]

Rule 1220

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[e^q*x^(2*q - 3)*((
a + b*x^2 + c*x^4)^(p + 1)/(c*(4*p + 2*q + 1))), x] + Dist[1/(c*(4*p + 2*q + 1)), Int[(a + b*x^2 + c*x^4)^p*Ex
pandToSum[c*(4*p + 2*q + 1)*(d + e*x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - b*(2*p + 2*q - 1)*e^q*x^(2*q - 2) -
c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && IGtQ[q, 1]

Rubi steps \begin{align*} \text {integral}& = \frac {25}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac {1}{7} \int \left (293+190 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx \\ & = \frac {1}{21} x \left (407+114 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {25}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac {1}{105} \int \frac {4720+3255 x^2}{\sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {1}{21} x \left (407+114 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {25}{7} x \left (2+3 x^2+x^4\right )^{3/2}+31 \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {944}{21} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {31 x \left (2+x^2\right )}{\sqrt {2+3 x^2+x^4}}+\frac {1}{21} x \left (407+114 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {25}{7} x \left (2+3 x^2+x^4\right )^{3/2}-\frac {31 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{\sqrt {2+3 x^2+x^4}}+\frac {472 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{21 \sqrt {2+3 x^2+x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.69 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.68 \[ \int \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4} \, dx=\frac {1114 x+2349 x^3+1724 x^5+564 x^7+75 x^9-651 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-293 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{21 \sqrt {2+3 x^2+x^4}} \]

[In]

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

[Out]

(1114*x + 2349*x^3 + 1724*x^5 + 564*x^7 + 75*x^9 - (651*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/S
qrt[2]], 2] - (293*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(21*Sqrt[2 + 3*x^2 + x^4
])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.93 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.79

method	result	size
risch	\(\frac {x \left (75 x^{4}+339 x^{2}+557\right ) \sqrt {x^{4}+3 x^{2}+2}}{21}-\frac {472 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{21 \sqrt {x^{4}+3 x^{2}+2}}+\frac {31 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}\)	\(133\)
default	\(\frac {557 x \sqrt {x^{4}+3 x^{2}+2}}{21}-\frac {472 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{21 \sqrt {x^{4}+3 x^{2}+2}}+\frac {31 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}+\frac {25 x^{5} \sqrt {x^{4}+3 x^{2}+2}}{7}+\frac {113 x^{3} \sqrt {x^{4}+3 x^{2}+2}}{7}\)	\(155\)
elliptic	\(\frac {557 x \sqrt {x^{4}+3 x^{2}+2}}{21}-\frac {472 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{21 \sqrt {x^{4}+3 x^{2}+2}}+\frac {31 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{2 \sqrt {x^{4}+3 x^{2}+2}}+\frac {25 x^{5} \sqrt {x^{4}+3 x^{2}+2}}{7}+\frac {113 x^{3} \sqrt {x^{4}+3 x^{2}+2}}{7}\)	\(155\)

[In]

int((5*x^2+7)^2*(x^4+3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/21*x*(75*x^4+339*x^2+557)*(x^4+3*x^2+2)^(1/2)-472/21*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))+31/2*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*(Elli
pticF(1/2*I*2^(1/2)*x,2^(1/2))-EllipticE(1/2*I*2^(1/2)*x,2^(1/2)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.35 \[ \int \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4} \, dx=\frac {-651 i \, x E(\arcsin \left (\frac {i}{x}\right )\,|\,2) + 1595 i \, x F(\arcsin \left (\frac {i}{x}\right )\,|\,2) + {\left (75 \, x^{6} + 339 \, x^{4} + 557 \, x^{2} + 651\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}}{21 \, x} \]

[In]

integrate((5*x^2+7)^2*(x^4+3*x^2+2)^(1/2),x, algorithm="fricas")

[Out]

1/21*(-651*I*x*elliptic_e(arcsin(I/x), 2) + 1595*I*x*elliptic_f(arcsin(I/x), 2) + (75*x^6 + 339*x^4 + 557*x^2
+ 651)*sqrt(x^4 + 3*x^2 + 2))/x

Sympy [F]

\[ \int \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4} \, dx=\int \sqrt {\left (x^{2} + 1\right ) \left (x^{2} + 2\right )} \left (5 x^{2} + 7\right )^{2}\, dx \]

[In]

integrate((5*x**2+7)**2*(x**4+3*x**2+2)**(1/2),x)

[Out]

Integral(sqrt((x**2 + 1)*(x**2 + 2))*(5*x**2 + 7)**2, x)

Maxima [F]

\[ \int \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4} \, dx=\int { \sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{2} \,d x } \]

[In]

integrate((5*x^2+7)^2*(x^4+3*x^2+2)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4} \, dx=\int { \sqrt {x^{4} + 3 \, x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{2} \,d x } \]

[In]

integrate((5*x^2+7)^2*(x^4+3*x^2+2)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \left (7+5 x^2\right )^2 \sqrt {2+3 x^2+x^4} \, dx=\int {\left (5\,x^2+7\right )}^2\,\sqrt {x^4+3\,x^2+2} \,d x \]

[In]

int((5*x^2 + 7)^2*(3*x^2 + x^4 + 2)^(1/2),x)

[Out]

int((5*x^2 + 7)^2*(3*x^2 + x^4 + 2)^(1/2), x)