\(\int (7+5 x^2)^3 \sqrt {2+3 x^2+x^4} \, dx\) [286]

Optimal result

Integrand size = 24, antiderivative size = 193 \[ \int \left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4} \, dx=\frac {577 x \left (2+x^2\right )}{3 \sqrt {2+3 x^2+x^4}}+\frac {1}{21} x \left (2608+757 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {275}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac {125}{9} x^3 \left (2+3 x^2+x^4\right )^{3/2}-\frac {577 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{3 \sqrt {2+3 x^2+x^4}}+\frac {2945 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{21 \sqrt {2+3 x^2+x^4}} \]

[Out]

275/7*x*(x^4+3*x^2+2)^(3/2)+125/9*x^3*(x^4+3*x^2+2)^(3/2)+577/3*x*(x^2+2)/(x^4+3*x^2+2)^(1/2)-577/3*(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)+2945/21*(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*(757*x^2+2608)*(x^4+3*x^2+2)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1220, 1693, 1190, 1203, 1113, 1149} \[ \int \left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4} \, dx=\frac {2945 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{21 \sqrt {x^4+3 x^2+2}}-\frac {577 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{3 \sqrt {x^4+3 x^2+2}}+\frac {275}{7} \left (x^4+3 x^2+2\right )^{3/2} x+\frac {1}{21} \left (757 x^2+2608\right ) \sqrt {x^4+3 x^2+2} x+\frac {577 \left (x^2+2\right ) x}{3 \sqrt {x^4+3 x^2+2}}+\frac {125}{9} \left (x^4+3 x^2+2\right )^{3/2} x^3 \]

[In]

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

[Out]

(577*x*(2 + x^2))/(3*Sqrt[2 + 3*x^2 + x^4]) + (x*(2608 + 757*x^2)*Sqrt[2 + 3*x^2 + x^4])/21 + (275*x*(2 + 3*x^
2 + x^4)^(3/2))/7 + (125*x^3*(2 + 3*x^2 + x^4)^(3/2))/9 - (577*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*Ell
ipticE[ArcTan[x], 1/2])/(3*Sqrt[2 + 3*x^2 + x^4]) + (2945*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*Elliptic
F[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]

Rule 1693

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{q = Expon[Pq, x^2], e = Coeff[Pq, x^2,
Expon[Pq, x^2]]}, Simp[e*x^(2*q - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(2*q + 4*p + 1))), x] + Dist[1/(c*(2*q +
4*p + 1)), Int[(a + b*x^2 + c*x^4)^p*ExpandToSum[c*(2*q + 4*p + 1)*Pq - a*e*(2*q - 3)*x^(2*q - 4) - b*e*(2*q
+ 2*p - 1)*x^(2*q - 2) - c*e*(2*q + 4*p + 1)*x^(2*q), x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2]
&& Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] &&  !LtQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {125}{9} x^3 \left (2+3 x^2+x^4\right )^{3/2}+\frac {1}{9} \int \sqrt {2+3 x^2+x^4} \left (3087+5865 x^2+2475 x^4\right ) \, dx \\ & = \frac {275}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac {125}{9} x^3 \left (2+3 x^2+x^4\right )^{3/2}+\frac {1}{63} \int \left (16659+11355 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx \\ & = \frac {1}{21} x \left (2608+757 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {275}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac {125}{9} x^3 \left (2+3 x^2+x^4\right )^{3/2}+\frac {1}{945} \int \frac {265050+181755 x^2}{\sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {1}{21} x \left (2608+757 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {275}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac {125}{9} x^3 \left (2+3 x^2+x^4\right )^{3/2}+\frac {577}{3} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {5890}{21} \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx \\ & = \frac {577 x \left (2+x^2\right )}{3 \sqrt {2+3 x^2+x^4}}+\frac {1}{21} x \left (2608+757 x^2\right ) \sqrt {2+3 x^2+x^4}+\frac {275}{7} x \left (2+3 x^2+x^4\right )^{3/2}+\frac {125}{9} x^3 \left (2+3 x^2+x^4\right )^{3/2}-\frac {577 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {2+3 x^2+x^4}}+\frac {2945 \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 = 6.63 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.62 \[ \int \left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4} \, dx=\frac {25548 x+61214 x^3+57312 x^5+28496 x^7+7725 x^9+875 x^{11}-12117 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-5553 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{63 \sqrt {2+3 x^2+x^4}} \]

[In]

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

[Out]

(25548*x + 61214*x^3 + 57312*x^5 + 28496*x^7 + 7725*x^9 + 875*x^11 - (12117*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*Ell
ipticE[I*ArcSinh[x/Sqrt[2]], 2] - (5553*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(63
*Sqrt[2 + 3*x^2 + x^4])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 5.77 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.72

method	result	size
risch	\(\frac {x \left (875 x^{6}+5100 x^{4}+11446 x^{2}+12774\right ) \sqrt {x^{4}+3 x^{2}+2}}{63}-\frac {2945 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{21 \sqrt {x^{4}+3 x^{2}+2}}+\frac {577 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{6 \sqrt {x^{4}+3 x^{2}+2}}\)	\(138\)
default	\(\frac {4258 x \sqrt {x^{4}+3 x^{2}+2}}{21}-\frac {2945 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{21 \sqrt {x^{4}+3 x^{2}+2}}+\frac {577 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{6 \sqrt {x^{4}+3 x^{2}+2}}+\frac {125 x^{7} \sqrt {x^{4}+3 x^{2}+2}}{9}+\frac {1700 x^{5} \sqrt {x^{4}+3 x^{2}+2}}{21}+\frac {11446 x^{3} \sqrt {x^{4}+3 x^{2}+2}}{63}\)	\(172\)
elliptic	\(\frac {4258 x \sqrt {x^{4}+3 x^{2}+2}}{21}-\frac {2945 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{21 \sqrt {x^{4}+3 x^{2}+2}}+\frac {577 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{6 \sqrt {x^{4}+3 x^{2}+2}}+\frac {125 x^{7} \sqrt {x^{4}+3 x^{2}+2}}{9}+\frac {1700 x^{5} \sqrt {x^{4}+3 x^{2}+2}}{21}+\frac {11446 x^{3} \sqrt {x^{4}+3 x^{2}+2}}{63}\)	\(172\)

[In]

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

[Out]

1/63*x*(875*x^6+5100*x^4+11446*x^2+12774)*(x^4+3*x^2+2)^(1/2)-2945/21*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))+577/6*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 = 0.33 \[ \int \left (7+5 x^2\right )^3 \sqrt {2+3 x^2+x^4} \, dx=\frac {-12117 i \, x E(\arcsin \left (\frac {i}{x}\right )\,|\,2) + 29787 i \, x F(\arcsin \left (\frac {i}{x}\right )\,|\,2) + {\left (875 \, x^{8} + 5100 \, x^{6} + 11446 \, x^{4} + 12774 \, x^{2} + 12117\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}}{63 \, x} \]

[In]

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

[Out]

1/63*(-12117*I*x*elliptic_e(arcsin(I/x), 2) + 29787*I*x*elliptic_f(arcsin(I/x), 2) + (875*x^8 + 5100*x^6 + 114
46*x^4 + 12774*x^2 + 12117)*sqrt(x^4 + 3*x^2 + 2))/x

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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