\(\int (7+5 x^2)^3 (2+3 x^2+x^4)^{3/2} \, dx\) [293]

Optimal result

Integrand size = 24, antiderivative size = 219 \[ \int \left (7+5 x^2\right )^3 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {20884 x \left (2+x^2\right )}{65 \sqrt {2+3 x^2+x^4}}+\frac {x \left (1032541+297911 x^2\right ) \sqrt {2+3 x^2+x^4}}{5005}+\frac {x \left (208212+65345 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}}{3003}+\frac {3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}-\frac {20884 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{65 \sqrt {2+3 x^2+x^4}}+\frac {1171349 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{5005 \sqrt {2+3 x^2+x^4}} \]

[Out]

1/3003*x*(65345*x^2+208212)*(x^4+3*x^2+2)^(3/2)+3825/143*x*(x^4+3*x^2+2)^(5/2)+125/13*x^3*(x^4+3*x^2+2)^(5/2)+
20884/65*x*(x^2+2)/(x^4+3*x^2+2)^(1/2)-20884/65*(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)+1171349/5005*(x^2+1)^(3/2)*(1/(x^2+1))^(1/2)*Elli
pticF(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)+1/5005*x*(297911*x^2+10
32541)*(x^4+3*x^2+2)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1220, 1693, 1190, 1203, 1113, 1149} \[ \int \left (7+5 x^2\right )^3 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {1171349 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),\frac {1}{2}\right )}{5005 \sqrt {x^4+3 x^2+2}}-\frac {20884 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {x^2+2}{x^2+1}} E\left (\arctan (x)\left |\frac {1}{2}\right .\right )}{65 \sqrt {x^4+3 x^2+2}}+\frac {3825}{143} \left (x^4+3 x^2+2\right )^{5/2} x+\frac {\left (65345 x^2+208212\right ) \left (x^4+3 x^2+2\right )^{3/2} x}{3003}+\frac {\left (297911 x^2+1032541\right ) \sqrt {x^4+3 x^2+2} x}{5005}+\frac {20884 \left (x^2+2\right ) x}{65 \sqrt {x^4+3 x^2+2}}+\frac {125}{13} \left (x^4+3 x^2+2\right )^{5/2} x^3 \]

[In]

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

[Out]

(20884*x*(2 + x^2))/(65*Sqrt[2 + 3*x^2 + x^4]) + (x*(1032541 + 297911*x^2)*Sqrt[2 + 3*x^2 + x^4])/5005 + (x*(2
08212 + 65345*x^2)*(2 + 3*x^2 + x^4)^(3/2))/3003 + (3825*x*(2 + 3*x^2 + x^4)^(5/2))/143 + (125*x^3*(2 + 3*x^2
+ x^4)^(5/2))/13 - (20884*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(65*Sqrt[2 +
3*x^2 + x^4]) + (1171349*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(5005*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}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac {1}{13} \int \left (2+3 x^2+x^4\right )^{3/2} \left (4459+8805 x^2+3825 x^4\right ) \, dx \\ & = \frac {3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac {1}{143} \int \left (41399+28005 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx \\ & = \frac {x \left (208212+65345 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}}{3003}+\frac {3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac {\int \left (1322334+893733 x^2\right ) \sqrt {2+3 x^2+x^4} \, dx}{3003} \\ & = \frac {x \left (1032541+297911 x^2\right ) \sqrt {2+3 x^2+x^4}}{5005}+\frac {x \left (208212+65345 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}}{3003}+\frac {3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac {\int \frac {21084282+14472612 x^2}{\sqrt {2+3 x^2+x^4}} \, dx}{45045} \\ & = \frac {x \left (1032541+297911 x^2\right ) \sqrt {2+3 x^2+x^4}}{5005}+\frac {x \left (208212+65345 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}}{3003}+\frac {3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}+\frac {20884}{65} \int \frac {x^2}{\sqrt {2+3 x^2+x^4}} \, dx+\frac {2342698 \int \frac {1}{\sqrt {2+3 x^2+x^4}} \, dx}{5005} \\ & = \frac {20884 x \left (2+x^2\right )}{65 \sqrt {2+3 x^2+x^4}}+\frac {x \left (1032541+297911 x^2\right ) \sqrt {2+3 x^2+x^4}}{5005}+\frac {x \left (208212+65345 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}}{3003}+\frac {3825}{143} x \left (2+3 x^2+x^4\right )^{5/2}+\frac {125}{13} x^3 \left (2+3 x^2+x^4\right )^{5/2}-\frac {20884 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} E\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{65 \sqrt {2+3 x^2+x^4}}+\frac {1171349 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac {1}{2}\right )}{5005 \sqrt {2+3 x^2+x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.59 \[ \int \left (7+5 x^2\right )^3 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {13572486 x+40493455 x^3+54938052 x^5+46218643 x^7+25350660 x^9+8705725 x^{11}+1701000 x^{13}+144375 x^{15}-4824204 i \sqrt {1+x^2} \sqrt {2+x^2} E\left (\left .i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right )\right |2\right )-2203890 i \sqrt {1+x^2} \sqrt {2+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {x}{\sqrt {2}}\right ),2\right )}{15015 \sqrt {2+3 x^2+x^4}} \]

[In]

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

[Out]

(13572486*x + 40493455*x^3 + 54938052*x^5 + 46218643*x^7 + 25350660*x^9 + 8705725*x^11 + 1701000*x^13 + 144375
*x^15 - (4824204*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (2203890*I)*Sqrt[1 + x^2]
*Sqrt[2 + x^2]*EllipticF[I*ArcSinh[x/Sqrt[2]], 2])/(15015*Sqrt[2 + 3*x^2 + x^4])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.60 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.68

method	result	size
risch	\(\frac {x \left (144375 x^{10}+1267875 x^{8}+4613350 x^{6}+8974860 x^{4}[phone] x^{2}+6786243\right ) \sqrt {x^{4}+3 x^{2}+2}}{15015}-\frac {1171349 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{5005 \sqrt {x^{4}+3 x^{2}+2}}+\frac {10442 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{65 \sqrt {x^{4}+3 x^{2}+2}}\)	\(148\)
default	\(\frac {598324 x^{5} \sqrt {x^{4}+3 x^{2}+2}}{1001}+\frac {10067363 x^{3} \sqrt {x^{4}+3 x^{2}+2}}{15015}+\frac {2262081 x \sqrt {x^{4}+3 x^{2}+2}}{5005}-\frac {1171349 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{5005 \sqrt {x^{4}+3 x^{2}+2}}+\frac {10442 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{65 \sqrt {x^{4}+3 x^{2}+2}}+\frac {125 x^{11} \sqrt {x^{4}+3 x^{2}+2}}{13}+\frac {12075 x^{9} \sqrt {x^{4}+3 x^{2}+2}}{143}+\frac {131810 x^{7} \sqrt {x^{4}+3 x^{2}+2}}{429}\)	\(206\)
elliptic	\(\frac {598324 x^{5} \sqrt {x^{4}+3 x^{2}+2}}{1001}+\frac {10067363 x^{3} \sqrt {x^{4}+3 x^{2}+2}}{15015}+\frac {2262081 x \sqrt {x^{4}+3 x^{2}+2}}{5005}-\frac {1171349 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )}{5005 \sqrt {x^{4}+3 x^{2}+2}}+\frac {10442 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (F\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-E\left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )\right )}{65 \sqrt {x^{4}+3 x^{2}+2}}+\frac {125 x^{11} \sqrt {x^{4}+3 x^{2}+2}}{13}+\frac {12075 x^{9} \sqrt {x^{4}+3 x^{2}+2}}{143}+\frac {131810 x^{7} \sqrt {x^{4}+3 x^{2}+2}}{429}\)	\(206\)

[In]

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

[Out]

1/15015*x*(144375*x^10+1267875*x^8+4613350*x^6+8974860*x^4+10067363*x^2+6786243)*(x^4+3*x^2+2)^(1/2)-1171349/5
005*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))+10442/65*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 = 73, normalized size of antiderivative = 0.33 \[ \int \left (7+5 x^2\right )^3 \left (2+3 x^2+x^4\right )^{3/2} \, dx=\frac {-4824204 i \, x E(\arcsin \left (\frac {i}{x}\right )\,|\,2) + 11852298 i \, x F(\arcsin \left (\frac {i}{x}\right )\,|\,2) + {\left (144375 \, x^{12} + 1267875 \, x^{10} + 4613350 \, x^{8} + 8974860 \, x^{6} + 10067363 \, x^{4} + 6786243 \, x^{2} + 4824204\right )} \sqrt {x^{4} + 3 \, x^{2} + 2}}{15015 \, x} \]

[In]

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

[Out]

1/15015*(-4824204*I*x*elliptic_e(arcsin(I/x), 2) + 11852298*I*x*elliptic_f(arcsin(I/x), 2) + (144375*x^12 + 12
67875*x^10 + 4613350*x^8 + 8974860*x^6 + 10067363*x^4 + 6786243*x^2 + 4824204)*sqrt(x^4 + 3*x^2 + 2))/x

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((5*x^2+7)^3*(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)^3, x)

Giac [F]

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

[In]

integrate((5*x^2+7)^3*(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)^3, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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