\(\int (7+5 x^2) (2+3 x^2+x^4)^{3/2} \, dx\) [295]

Optimal result

Integrand size = 22, antiderivative size = 179 \[ \int \left (7+5 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {116 x \left (2+x^2\right )}{15 \sqrt {2+3 x^2+x^4}}+\frac {1}{105} x \left (519+149 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {1}{63} x \left (108+35 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}-\frac {116 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{15 \sqrt {2+3 x^2+x^4}}+\frac {197 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{35 \sqrt {2+3 x^2+x^4}} \]

[Out]

1/63*x*(35*x^2+108)*(x^4+3*x^2+2)^(3/2)+116/15*x*(x^2+2)/(x^4+3*x^2+2)^(1/2)-116/15*(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)+197/35*(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)+1/105*x*(149*x^2+519)*(x^4+3*x^2+2)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1190, 1203, 1113, 1149} \[ \int \left (7+5 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {197 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{35 \sqrt {x^4+3 x^2+2}}-\frac {116 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{15 \sqrt {x^4+3 x^2+2}}+\frac {1}{63} x \left (35 x^2+108\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac {1}{105} x \left (149 x^2+519\right ) \sqrt {x^4+3 x^2+2}+\frac {116 x \left (x^2+2\right )}{15 \sqrt {x^4+3 x^2+2}} \]

[In]

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

[Out]

(116*x*(2 + x^2))/(15*Sqrt[2 + 3*x^2 + x^4]) + (x*(519 + 149*x^2)*Sqrt[2 + 3*x^2 + x^4])/105 + (x*(108 + 35*x^
2)*(2 + 3*x^2 + x^4)^(3/2))/63 - (116*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(
15*Sqrt[2 + 3*x^2 + x^4]) + (197*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(35*Sq
rt[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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{63} x \left (108+35 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {1}{21} \int \left (222+149 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx \\ & = \frac {1}{105} x \left (519+149 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {1}{63} x \left (108+35 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {1}{315} \int \frac {3546+2436 x^2}{\sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {1}{105} x \left (519+149 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {1}{63} x \left (108+35 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac {116}{15} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {394}{35} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {116 x \left (2+x^2\right )}{15 \sqrt {2+3 x^2+x^4}}+\frac {1}{105} x \left (519+149 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {1}{63} x \left (108+35 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}-\frac {116 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{15 \sqrt {2+3 x^2+x^4}}+\frac {197 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{35 \sqrt {2+3 x^2+x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.82 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.66 \[ \int \left (7+5 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {5274 x+12745 x^3+12018 x^5+5962 x^7+1590 x^9+175 x^{11}-2436 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-1110 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{315 \sqrt {2+3 x^2+x^4}} \]

[In]

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

[Out]

(5274*x + 12745*x^3 + 12018*x^5 + 5962*x^7 + 1590*x^9 + 175*x^11 - (2436*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*Ellipt
icE[I*ArcSinh[x/Sqrt[2]], 2] - (1110*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(315*S
qrt[2 + 3*x^2 + x^4])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.17 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.77

method	result	size
risch	\(\frac {x \left (175 x^{6}+1065 x^{4}+2417 x^{2}+2637\right ) \sqrt {x^{4}+3 x^{2}+2}}{315}-\frac {197 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{35 \sqrt {x^{4}+3 x^{2}+2}}+\frac {58 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{15 \sqrt {x^{4}+3 x^{2}+2}}\)	\(138\)
default	\(\frac {71 x^{5} \sqrt {x^{4}+3 x^{2}+2}}{21}+\frac {2417 x^{3} \sqrt {x^{4}+3 x^{2}+2}}{315}+\frac {293 x \sqrt {x^{4}+3 x^{2}+2}}{35}-\frac {197 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{35 \sqrt {x^{4}+3 x^{2}+2}}+\frac {58 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{15 \sqrt {x^{4}+3 x^{2}+2}}+\frac {5 x^{7} \sqrt {x^{4}+3 x^{2}+2}}{9}\)	\(172\)
elliptic	\(\frac {71 x^{5} \sqrt {x^{4}+3 x^{2}+2}}{21}+\frac {2417 x^{3} \sqrt {x^{4}+3 x^{2}+2}}{315}+\frac {293 x \sqrt {x^{4}+3 x^{2}+2}}{35}-\frac {197 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{35 \sqrt {x^{4}+3 x^{2}+2}}+\frac {58 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{15 \sqrt {x^{4}+3 x^{2}+2}}+\frac {5 x^{7} \sqrt {x^{4}+3 x^{2}+2}}{9}\)	\(172\)

[In]

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

[Out]

1/315*x*(175*x^6+1065*x^4+2417*x^2+2637)*(x^4+3*x^2+2)^(1/2)-197/35*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))+58/15*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))-EllipticE(1/2*I*2^(1/2)*x,2^(1/2)))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.35 \[ \int \left (7+5 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {-2436 i \, x E(\arcsin \left (\frac {i}{x}\right )\,|\,2) + 5982 i \, x F(\arcsin \left (\frac {i}{x}\right )\,|\,2) + {\left (175 \, x^{8} + 1065 \, x^{6} + 2417 \, x^{4} + 2637 \, x^{2} + 2436\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}}{315 \, x} \]

[In]

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

[Out]

1/315*(-2436*I*x*elliptic_e(arcsin(I/x), 2) + 5982*I*x*elliptic_f(arcsin(I/x), 2) + (175*x^8 + 1065*x^6 + 2417
*x^4 + 2637*x^2 + 2436)*sqrt(x^4 + 3*x^2 + 2))/x

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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