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

Optimal. Leaf size=242 \[ \frac{33159 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ),\frac{1}{8}\right )}{11 \sqrt{2} \sqrt{x^4+3 x^2+4}}+\frac{625}{11} \left (x^4+3 x^2+4\right )^{3/2} x^5+\frac{23500}{99} \left (x^4+3 x^2+4\right )^{3/2} x^3+\frac{3050}{11} \left (x^4+3 x^2+4\right )^{3/2} x+\frac{1}{33} \left (4516 x^2+18727\right ) \sqrt{x^4+3 x^2+4} x+\frac{51665 \sqrt{x^4+3 x^2+4} x}{33 \left (x^2+2\right )}-\frac{51665 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{33 \sqrt{x^4+3 x^2+4}} \]

[Out]

(51665*x*Sqrt[4 + 3*x^2 + x^4])/(33*(2 + x^2)) + (x*(18727 + 4516*x^2)*Sqrt[4 + 3*x^2 + x^4])/33 + (3050*x*(4
+ 3*x^2 + x^4)^(3/2))/11 + (23500*x^3*(4 + 3*x^2 + x^4)^(3/2))/99 + (625*x^5*(4 + 3*x^2 + x^4)^(3/2))/11 - (51
665*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(33*Sqrt[4 + 3*
x^2 + x^4]) + (33159*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(11*Sq
rt[2]*Sqrt[4 + 3*x^2 + x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.150363, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1206, 1679, 1176, 1197, 1103, 1195} \[ \frac{625}{11} \left (x^4+3 x^2+4\right )^{3/2} x^5+\frac{23500}{99} \left (x^4+3 x^2+4\right )^{3/2} x^3+\frac{3050}{11} \left (x^4+3 x^2+4\right )^{3/2} x+\frac{1}{33} \left (4516 x^2+18727\right ) \sqrt{x^4+3 x^2+4} x+\frac{51665 \sqrt{x^4+3 x^2+4} x}{33 \left (x^2+2\right )}+\frac{33159 \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{11 \sqrt{2} \sqrt{x^4+3 x^2+4}}-\frac{51665 \sqrt{2} \left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{33 \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(51665*x*Sqrt[4 + 3*x^2 + x^4])/(33*(2 + x^2)) + (x*(18727 + 4516*x^2)*Sqrt[4 + 3*x^2 + x^4])/33 + (3050*x*(4
+ 3*x^2 + x^4)^(3/2))/11 + (23500*x^3*(4 + 3*x^2 + x^4)^(3/2))/99 + (625*x^5*(4 + 3*x^2 + x^4)^(3/2))/11 - (51
665*Sqrt[2]*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticE[2*ArcTan[x/Sqrt[2]], 1/8])/(33*Sqrt[4 + 3*
x^2 + x^4]) + (33159*(2 + x^2)*Sqrt[(4 + 3*x^2 + x^4)/(2 + x^2)^2]*EllipticF[2*ArcTan[x/Sqrt[2]], 1/8])/(11*Sq
rt[2]*Sqrt[4 + 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 1679

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]

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 1197

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

Rule 1103

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

Rule 1195

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

Rubi steps

\begin{align*} \int \left (7+5 x^2\right )^4 \sqrt{4+3 x^2+x^4} \, dx &=\frac{625}{11} x^5 \left (4+3 x^2+x^4\right )^{3/2}+\frac{1}{11} \int \sqrt{4+3 x^2+x^4} \left (26411+75460 x^2+68350 x^4+23500 x^6\right ) \, dx\\ &=\frac{23500}{99} x^3 \left (4+3 x^2+x^4\right )^{3/2}+\frac{625}{11} x^5 \left (4+3 x^2+x^4\right )^{3/2}+\frac{1}{99} \int \sqrt{4+3 x^2+x^4} \left (237699+397140 x^2+192150 x^4\right ) \, dx\\ &=\frac{3050}{11} x \left (4+3 x^2+x^4\right )^{3/2}+\frac{23500}{99} x^3 \left (4+3 x^2+x^4\right )^{3/2}+\frac{625}{11} x^5 \left (4+3 x^2+x^4\right )^{3/2}+\frac{1}{693} \int \left (895293+474180 x^2\right ) \sqrt{4+3 x^2+x^4} \, dx\\ &=\frac{1}{33} x \left (18727+4516 x^2\right ) \sqrt{4+3 x^2+x^4}+\frac{3050}{11} x \left (4+3 x^2+x^4\right )^{3/2}+\frac{23500}{99} x^3 \left (4+3 x^2+x^4\right )^{3/2}+\frac{625}{11} x^5 \left (4+3 x^2+x^4\right )^{3/2}+\frac{\int \frac{30121560+16274475 x^2}{\sqrt{4+3 x^2+x^4}} \, dx}{10395}\\ &=\frac{1}{33} x \left (18727+4516 x^2\right ) \sqrt{4+3 x^2+x^4}+\frac{3050}{11} x \left (4+3 x^2+x^4\right )^{3/2}+\frac{23500}{99} x^3 \left (4+3 x^2+x^4\right )^{3/2}+\frac{625}{11} x^5 \left (4+3 x^2+x^4\right )^{3/2}-\frac{103330}{33} \int \frac{1-\frac{x^2}{2}}{\sqrt{4+3 x^2+x^4}} \, dx+\frac{66318}{11} \int \frac{1}{\sqrt{4+3 x^2+x^4}} \, dx\\ &=\frac{51665 x \sqrt{4+3 x^2+x^4}}{33 \left (2+x^2\right )}+\frac{1}{33} x \left (18727+4516 x^2\right ) \sqrt{4+3 x^2+x^4}+\frac{3050}{11} x \left (4+3 x^2+x^4\right )^{3/2}+\frac{23500}{99} x^3 \left (4+3 x^2+x^4\right )^{3/2}+\frac{625}{11} x^5 \left (4+3 x^2+x^4\right )^{3/2}-\frac{51665 \sqrt{2} \left (2+x^2\right ) \sqrt{\frac{4+3 x^2+x^4}{\left (2+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{33 \sqrt{4+3 x^2+x^4}}+\frac{33159 \left (2+x^2\right ) \sqrt{\frac{4+3 x^2+x^4}{\left (2+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{11 \sqrt{2} \sqrt{4+3 x^2+x^4}}\\ \end{align*}

Mathematica [C]  time = 0.587995, size = 354, normalized size = 1.46 \[ \frac{3 \sqrt{2} \left (51665 \sqrt{7}-36253 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{\sqrt{7}-3 i}} x\right ),\frac{-\sqrt{7}+3 i}{\sqrt{7}+3 i}\right )+4 \sqrt{-\frac{i}{\sqrt{7}-3 i}} x \left (5625 x^{12}+57250 x^{10}+264075 x^8+712748 x^6+1217475 x^4+1257535 x^2+663924\right )-154995 \sqrt{2} \left (\sqrt{7}+3 i\right ) \sqrt{\frac{-2 i x^2+\sqrt{7}-3 i}{\sqrt{7}-3 i}} \sqrt{\frac{2 i x^2+\sqrt{7}+3 i}{\sqrt{7}+3 i}} E\left (i \sinh ^{-1}\left (\sqrt{-\frac{2 i}{-3 i+\sqrt{7}}} x\right )|\frac{3 i-\sqrt{7}}{3 i+\sqrt{7}}\right )}{396 \sqrt{-\frac{i}{\sqrt{7}-3 i}} \sqrt{x^4+3 x^2+4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[(-I)/(-3*I + Sqrt[7])]*x*(663924 + 1257535*x^2 + 1217475*x^4 + 712748*x^6 + 264075*x^8 + 57250*x^10 +
5625*x^12) - 154995*Sqrt[2]*(3*I + Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(3*I + Sq
rt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticE[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[7])/(3*I
+ Sqrt[7])] + 3*Sqrt[2]*(-36253*I + 51665*Sqrt[7])*Sqrt[(-3*I + Sqrt[7] - (2*I)*x^2)/(-3*I + Sqrt[7])]*Sqrt[(
3*I + Sqrt[7] + (2*I)*x^2)/(3*I + Sqrt[7])]*EllipticF[I*ArcSinh[Sqrt[(-2*I)/(-3*I + Sqrt[7])]*x], (3*I - Sqrt[
7])/(3*I + Sqrt[7])])/(396*Sqrt[(-I)/(-3*I + Sqrt[7])]*Sqrt[4 + 3*x^2 + x^4])

________________________________________________________________________________________

Maple [C]  time = 0.148, size = 292, normalized size = 1.2 \begin{align*}{\frac{625\,{x}^{9}}{11}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{40375\,{x}^{7}}{99}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{189898\,{x}^{3}}{99}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{55327\,x}{33}\sqrt{{x}^{4}+3\,{x}^{2}+4}}+{\frac{382496}{33\,\sqrt{-6+2\,i\sqrt{7}}}\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}-{\frac{1653280}{33\,\sqrt{-6+2\,i\sqrt{7}} \left ( i\sqrt{7}+3 \right ) }\sqrt{1- \left ( -{\frac{3}{8}}+{\frac{i}{8}}\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{3}{8}}-{\frac{i}{8}}\sqrt{7} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-6+2\,i\sqrt{7}}}{4}},{\frac{\sqrt{2+6\,i\sqrt{7}}}{4}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+4}}}}+{\frac{3650\,{x}^{5}}{3}\sqrt{{x}^{4}+3\,{x}^{2}+4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

625/11*x^9*(x^4+3*x^2+4)^(1/2)+40375/99*x^7*(x^4+3*x^2+4)^(1/2)+189898/99*x^3*(x^4+3*x^2+4)^(1/2)+55327/33*x*(
x^4+3*x^2+4)^(1/2)+382496/33/(-6+2*I*7^(1/2))^(1/2)*(1-(-3/8+1/8*I*7^(1/2))*x^2)^(1/2)*(1-(-3/8-1/8*I*7^(1/2))
*x^2)^(1/2)/(x^4+3*x^2+4)^(1/2)*EllipticF(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2))-1653280/33/(
-6+2*I*7^(1/2))^(1/2)*(1-(-3/8+1/8*I*7^(1/2))*x^2)^(1/2)*(1-(-3/8-1/8*I*7^(1/2))*x^2)^(1/2)/(x^4+3*x^2+4)^(1/2
)/(I*7^(1/2)+3)*(EllipticF(1/4*x*(-6+2*I*7^(1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2))-EllipticE(1/4*x*(-6+2*I*7^(
1/2))^(1/2),1/4*(2+6*I*7^(1/2))^(1/2)))+3650/3*x^5*(x^4+3*x^2+4)^(1/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (625 \, x^{8} + 3500 \, x^{6} + 7350 \, x^{4} + 6860 \, x^{2} + 2401\right )} \sqrt{x^{4} + 3 \, x^{2} + 4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((625*x^8 + 3500*x^6 + 7350*x^4 + 6860*x^2 + 2401)*sqrt(x^4 + 3*x^2 + 4), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\left (x^{2} - x + 2\right ) \left (x^{2} + x + 2\right )} \left (5 x^{2} + 7\right )^{4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt((x**2 - x + 2)*(x**2 + x + 2))*(5*x**2 + 7)**4, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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