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

Optimal. Leaf size=198 \[ \frac{13879 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{385 \sqrt{x^4+3 x^2+2}}+\frac{25}{11} x \left (x^4+3 x^2+2\right )^{5/2}+\frac{1}{693} x \left (2240 x^2+7281\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac{x \left (10643 x^2+36783\right ) \sqrt{x^4+3 x^2+2}}{1155}+\frac{742 x \left (x^2+2\right )}{15 \sqrt{x^4+3 x^2+2}}-\frac{742 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{15 \sqrt{x^4+3 x^2+2}} \]

[Out]

(742*x*(2 + x^2))/(15*Sqrt[2 + 3*x^2 + x^4]) + (x*(36783 + 10643*x^2)*Sqrt[2 + 3*x^2 + x^4])/1155 + (x*(7281 +
2240*x^2)*(2 + 3*x^2 + x^4)^(3/2))/693 + (25*x*(2 + 3*x^2 + x^4)^(5/2))/11 - (742*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]) + (13879*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^
2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(385*Sqrt[2 + 3*x^2 + x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.0859281, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1206, 1176, 1189, 1099, 1135} \[ \frac{25}{11} x \left (x^4+3 x^2+2\right )^{5/2}+\frac{1}{693} x \left (2240 x^2+7281\right ) \left (x^4+3 x^2+2\right )^{3/2}+\frac{x \left (10643 x^2+36783\right ) \sqrt{x^4+3 x^2+2}}{1155}+\frac{742 x \left (x^2+2\right )}{15 \sqrt{x^4+3 x^2+2}}+\frac{13879 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{385 \sqrt{x^4+3 x^2+2}}-\frac{742 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{15 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(742*x*(2 + x^2))/(15*Sqrt[2 + 3*x^2 + x^4]) + (x*(36783 + 10643*x^2)*Sqrt[2 + 3*x^2 + x^4])/1155 + (x*(7281 +
2240*x^2)*(2 + 3*x^2 + x^4)^(3/2))/693 + (25*x*(2 + 3*x^2 + x^4)^(5/2))/11 - (742*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]) + (13879*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^
2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(385*Sqrt[2 + 3*x^2 + x^4])

Rule 1206

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 1176

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 1189

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 1099

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)]*EllipticF[ArcTan[Rt[(b + q)/(2*a), 2]*x], (2*q)/(b + q)]
)/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]), 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 1135

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)]*EllipticE[ArcTan[Rt[(b + q)/(2*a), 2]*x], (2*q)/(b + q)])/(2*c*Sqrt[a + b*x^2
+ c*x^4]), 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]

Rubi steps

\begin{align*} \int \left (7+5 x^2\right )^2 \left (2+3 x^2+x^4\right )^{3/2} \, dx &=\frac{25}{11} x \left (2+3 x^2+x^4\right )^{5/2}+\frac{1}{11} \int \left (489+320 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2} \, dx\\ &=\frac{1}{693} x \left (7281+2240 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac{25}{11} x \left (2+3 x^2+x^4\right )^{5/2}+\frac{1}{231} \int \left (15684+10643 x^2\right ) \sqrt{2+3 x^2+x^4} \, dx\\ &=\frac{x \left (36783+10643 x^2\right ) \sqrt{2+3 x^2+x^4}}{1155}+\frac{1}{693} x \left (7281+2240 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac{25}{11} x \left (2+3 x^2+x^4\right )^{5/2}+\frac{\int \frac{249822+171402 x^2}{\sqrt{2+3 x^2+x^4}} \, dx}{3465}\\ &=\frac{x \left (36783+10643 x^2\right ) \sqrt{2+3 x^2+x^4}}{1155}+\frac{1}{693} x \left (7281+2240 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac{25}{11} x \left (2+3 x^2+x^4\right )^{5/2}+\frac{742}{15} \int \frac{x^2}{\sqrt{2+3 x^2+x^4}} \, dx+\frac{27758}{385} \int \frac{1}{\sqrt{2+3 x^2+x^4}} \, dx\\ &=\frac{742 x \left (2+x^2\right )}{15 \sqrt{2+3 x^2+x^4}}+\frac{x \left (36783+10643 x^2\right ) \sqrt{2+3 x^2+x^4}}{1155}+\frac{1}{693} x \left (7281+2240 x^2\right ) \left (2+3 x^2+x^4\right )^{3/2}+\frac{25}{11} x \left (2+3 x^2+x^4\right )^{5/2}-\frac{742 \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{13879 \sqrt{2} \left (1+x^2\right ) \sqrt{\frac{2+x^2}{1+x^2}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{385 \sqrt{2+3 x^2+x^4}}\\ \end{align*}

Mathematica [F]  time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

________________________________________________________________________________________

Maple [C]  time = 0.008, size = 189, normalized size = 1. \begin{align*}{\frac{25\,{x}^{9}}{11}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{1670\,{x}^{7}}{99}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{11492\,{x}^{5}}{231}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{258044\,{x}^{3}}{3465}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{23851\,x}{385}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{{\frac{371\,i}{15}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{{\frac{13879\,i}{385}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/11*x^9*(x^4+3*x^2+2)^(1/2)+1670/99*x^7*(x^4+3*x^2+2)^(1/2)+11492/231*x^5*(x^4+3*x^2+2)^(1/2)+258044/3465*x^
3*(x^4+3*x^2+2)^(1/2)+23851/385*x*(x^4+3*x^2+2)^(1/2)+371/15*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*x*2^(1/2),2^(1/2))-EllipticE(1/2*I*x*2^(1/2),2^(1/2)))-13879/385*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*x*2^(1/2),2^(1/2))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+7)^2*(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)^2, x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (25 \, x^{8} + 145 \, x^{6} + 309 \, x^{4} + 287 \, x^{2} + 98\right )} \sqrt{x^{4} + 3 \, x^{2} + 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((25*x^8 + 145*x^6 + 309*x^4 + 287*x^2 + 98)*sqrt(x^4 + 3*x^2 + 2), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\left (x^{2} + 1\right ) \left (x^{2} + 2\right )\right )^{\frac{3}{2}} \left (5 x^{2} + 7\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x^{4} + 3 \, x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+7)^2*(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)^2, x)