3.1.27 \(\int x^4 (2+3 x^2) (5+x^4)^{3/2} \, dx\) [27]

Optimal. Leaf size=235 \[ \frac {200}{77} x \sqrt {5+x^4}+\frac {20}{13} x^3 \sqrt {5+x^4}-\frac {300 x \sqrt {5+x^4}}{13 \left (\sqrt {5}+x^2\right )}+\frac {10 x^5 \left (78+77 x^2\right ) \sqrt {5+x^4}}{1001}+\frac {1}{143} x^5 \left (26+33 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {300 \sqrt [4]{5} \left (\sqrt {5}+x^2\right ) \sqrt {\frac {5+x^4}{\left (\sqrt {5}+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{13 \sqrt {5+x^4}}-\frac {50 \sqrt [4]{5} \left (231+26 \sqrt {5}\right ) \left (\sqrt {5}+x^2\right ) \sqrt {\frac {5+x^4}{\left (\sqrt {5}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{1001 \sqrt {5+x^4}} \]

[Out]

1/143*x^5*(33*x^2+26)*(x^4+5)^(3/2)+200/77*x*(x^4+5)^(1/2)+20/13*x^3*(x^4+5)^(1/2)+10/1001*x^5*(77*x^2+78)*(x^
4+5)^(1/2)-300/13*x*(x^4+5)^(1/2)/(x^2+5^(1/2))+300/13*5^(1/4)*(cos(2*arctan(1/5*x*5^(3/4)))^2)^(1/2)/cos(2*ar
ctan(1/5*x*5^(3/4)))*EllipticE(sin(2*arctan(1/5*x*5^(3/4))),1/2*2^(1/2))*(x^2+5^(1/2))*((x^4+5)/(x^2+5^(1/2))^
2)^(1/2)/(x^4+5)^(1/2)-50/1001*5^(1/4)*(cos(2*arctan(1/5*x*5^(3/4)))^2)^(1/2)/cos(2*arctan(1/5*x*5^(3/4)))*Ell
ipticF(sin(2*arctan(1/5*x*5^(3/4))),1/2*2^(1/2))*(x^2+5^(1/2))*(231+26*5^(1/2))*((x^4+5)/(x^2+5^(1/2))^2)^(1/2
)/(x^4+5)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1288, 1294, 1212, 226, 1210} \begin {gather*} -\frac {50 \sqrt [4]{5} \left (231+26 \sqrt {5}\right ) \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{1001 \sqrt {x^4+5}}+\frac {300 \sqrt [4]{5} \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} E\left (2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{13 \sqrt {x^4+5}}+\frac {200}{77} \sqrt {x^4+5} x+\frac {20}{13} \sqrt {x^4+5} x^3-\frac {300 \sqrt {x^4+5} x}{13 \left (x^2+\sqrt {5}\right )}+\frac {1}{143} \left (33 x^2+26\right ) \left (x^4+5\right )^{3/2} x^5+\frac {10 \left (77 x^2+78\right ) \sqrt {x^4+5} x^5}{1001} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(200*x*Sqrt[5 + x^4])/77 + (20*x^3*Sqrt[5 + x^4])/13 - (300*x*Sqrt[5 + x^4])/(13*(Sqrt[5] + x^2)) + (10*x^5*(7
8 + 77*x^2)*Sqrt[5 + x^4])/1001 + (x^5*(26 + 33*x^2)*(5 + x^4)^(3/2))/143 + (300*5^(1/4)*(Sqrt[5] + x^2)*Sqrt[
(5 + x^4)/(Sqrt[5] + x^2)^2]*EllipticE[2*ArcTan[x/5^(1/4)], 1/2])/(13*Sqrt[5 + x^4]) - (50*5^(1/4)*(231 + 26*S
qrt[5])*(Sqrt[5] + x^2)*Sqrt[(5 + x^4)/(Sqrt[5] + x^2)^2]*EllipticF[2*ArcTan[x/5^(1/4)], 1/2])/(1001*Sqrt[5 +
x^4])

Rule 226

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

Rule 1210

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

Rule 1212

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

Rule 1288

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(a +
c*x^4)^p*((c*d*(m + 4*p + 3) + c*e*(4*p + m + 1)*x^2)/(c*f*(4*p + m + 1)*(m + 4*p + 3))), x] + Dist[4*a*(p/((
4*p + m + 1)*(m + 4*p + 3))), Int[(f*x)^m*(a + c*x^4)^(p - 1)*Simp[d*(m + 4*p + 3) + e*(4*p + m + 1)*x^2, x],
x], x] /; FreeQ[{a, c, d, e, f, m}, x] && GtQ[p, 0] && NeQ[4*p + m + 1, 0] && NeQ[m + 4*p + 3, 0] && IntegerQ[
2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1294

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*(
(a + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m - 2)*(a + c*x^4)^p*(a*e*
(m - 1) - c*d*(m + 4*p + 3)*x^2), x], x] /; FreeQ[{a, c, d, e, f, p}, x] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] &
& IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rubi steps

\begin {align*} \int x^4 \left (2+3 x^2\right ) \left (5+x^4\right )^{3/2} \, dx &=\frac {1}{143} x^5 \left (26+33 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {30}{143} \int x^4 \left (26+33 x^2\right ) \sqrt {5+x^4} \, dx\\ &=\frac {10 x^5 \left (78+77 x^2\right ) \sqrt {5+x^4}}{1001}+\frac {1}{143} x^5 \left (26+33 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {100 \int \frac {x^4 \left (234+231 x^2\right )}{\sqrt {5+x^4}} \, dx}{3003}\\ &=\frac {20}{13} x^3 \sqrt {5+x^4}+\frac {10 x^5 \left (78+77 x^2\right ) \sqrt {5+x^4}}{1001}+\frac {1}{143} x^5 \left (26+33 x^2\right ) \left (5+x^4\right )^{3/2}-\frac {20 \int \frac {x^2 \left (3465-1170 x^2\right )}{\sqrt {5+x^4}} \, dx}{3003}\\ &=\frac {200}{77} x \sqrt {5+x^4}+\frac {20}{13} x^3 \sqrt {5+x^4}+\frac {10 x^5 \left (78+77 x^2\right ) \sqrt {5+x^4}}{1001}+\frac {1}{143} x^5 \left (26+33 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {20 \int \frac {-5850-10395 x^2}{\sqrt {5+x^4}} \, dx}{9009}\\ &=\frac {200}{77} x \sqrt {5+x^4}+\frac {20}{13} x^3 \sqrt {5+x^4}+\frac {10 x^5 \left (78+77 x^2\right ) \sqrt {5+x^4}}{1001}+\frac {1}{143} x^5 \left (26+33 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {1}{13} \left (300 \sqrt {5}\right ) \int \frac {1-\frac {x^2}{\sqrt {5}}}{\sqrt {5+x^4}} \, dx-\frac {\left (100 \left (130+231 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt {5+x^4}} \, dx}{1001}\\ &=\frac {200}{77} x \sqrt {5+x^4}+\frac {20}{13} x^3 \sqrt {5+x^4}-\frac {300 x \sqrt {5+x^4}}{13 \left (\sqrt {5}+x^2\right )}+\frac {10 x^5 \left (78+77 x^2\right ) \sqrt {5+x^4}}{1001}+\frac {1}{143} x^5 \left (26+33 x^2\right ) \left (5+x^4\right )^{3/2}+\frac {300 \sqrt [4]{5} \left (\sqrt {5}+x^2\right ) \sqrt {\frac {5+x^4}{\left (\sqrt {5}+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{13 \sqrt {5+x^4}}-\frac {50 \sqrt [4]{5} \left (231+26 \sqrt {5}\right ) \left (\sqrt {5}+x^2\right ) \sqrt {\frac {5+x^4}{\left (\sqrt {5}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{1001 \sqrt {5+x^4}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 6.94, size = 74, normalized size = 0.31 \begin {gather*} \frac {1}{143} x \left (\left (26+33 x^2\right ) \left (5+x^4\right )^{5/2}-650 \sqrt {5} \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-\frac {x^4}{5}\right )-825 \sqrt {5} x^2 \, _2F_1\left (-\frac {3}{2},\frac {3}{4};\frac {7}{4};-\frac {x^4}{5}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*((26 + 33*x^2)*(5 + x^4)^(5/2) - 650*Sqrt[5]*Hypergeometric2F1[-3/2, 1/4, 5/4, -1/5*x^4] - 825*Sqrt[5]*x^2*
Hypergeometric2F1[-3/2, 3/4, 7/4, -1/5*x^4]))/143

________________________________________________________________________________________

Maple [C] Result contains complex when optimal does not.
time = 0.14, size = 216, normalized size = 0.92

method	result	size
meijerg	\(\frac {15 \sqrt {5}\, x^{7} \hypergeom \left (\left [-\frac {3}{2}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], -\frac {x^{4}}{5}\right )}{7}+2 \sqrt {5}\, x^{5} \hypergeom \left (\left [-\frac {3}{2}, \frac {5}{4}\right ], \left [\frac {9}{4}\right ], -\frac {x^{4}}{5}\right )\)	\(40\)
risch	\(\frac {x \left (231 x^{10}+182 x^{8}+1925 x^{6}+1690 x^{4}+1540 x^{2}+2600\right ) \sqrt {x^{4}+5}}{1001}-\frac {60 i \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )-\EllipticE \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )\right )}{13 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}-\frac {40 \sqrt {5}\, \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )}{77 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}\)	\(183\)
default	\(\frac {3 x^{11} \sqrt {x^{4}+5}}{13}+\frac {25 x^{7} \sqrt {x^{4}+5}}{13}+\frac {20 x^{3} \sqrt {x^{4}+5}}{13}-\frac {60 i \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )-\EllipticE \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )\right )}{13 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}+\frac {2 x^{9} \sqrt {x^{4}+5}}{11}+\frac {130 x^{5} \sqrt {x^{4}+5}}{77}+\frac {200 x \sqrt {x^{4}+5}}{77}-\frac {40 \sqrt {5}\, \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )}{77 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}\)	\(216\)
elliptic	\(\frac {3 x^{11} \sqrt {x^{4}+5}}{13}+\frac {25 x^{7} \sqrt {x^{4}+5}}{13}+\frac {20 x^{3} \sqrt {x^{4}+5}}{13}-\frac {60 i \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )-\EllipticE \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )\right )}{13 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}+\frac {2 x^{9} \sqrt {x^{4}+5}}{11}+\frac {130 x^{5} \sqrt {x^{4}+5}}{77}+\frac {200 x \sqrt {x^{4}+5}}{77}-\frac {40 \sqrt {5}\, \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )}{77 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}\)	\(216\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/13*x^11*(x^4+5)^(1/2)+25/13*x^7*(x^4+5)^(1/2)+20/13*x^3*(x^4+5)^(1/2)-60/13*I/(I*5^(1/2))^(1/2)*(25-5*I*5^(1
/2)*x^2)^(1/2)*(25+5*I*5^(1/2)*x^2)^(1/2)/(x^4+5)^(1/2)*(EllipticF(1/5*x*5^(1/2)*(I*5^(1/2))^(1/2),I)-Elliptic
E(1/5*x*5^(1/2)*(I*5^(1/2))^(1/2),I))+2/11*x^9*(x^4+5)^(1/2)+130/77*x^5*(x^4+5)^(1/2)+200/77*x*(x^4+5)^(1/2)-4
0/77*5^(1/2)/(I*5^(1/2))^(1/2)*(25-5*I*5^(1/2)*x^2)^(1/2)*(25+5*I*5^(1/2)*x^2)^(1/2)/(x^4+5)^(1/2)*EllipticF(1
/5*x*5^(1/2)*(I*5^(1/2))^(1/2),I)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 1.93, size = 160, normalized size = 0.68 \begin {gather*} \frac {3 \sqrt {5} x^{11} \Gamma \left (\frac {11}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {11}{4} \\ \frac {15}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{4 \Gamma \left (\frac {15}{4}\right )} + \frac {\sqrt {5} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{2 \Gamma \left (\frac {13}{4}\right )} + \frac {15 \sqrt {5} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} + \frac {5 \sqrt {5} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{2 \Gamma \left (\frac {9}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*sqrt(5)*x**11*gamma(11/4)*hyper((-1/2, 11/4), (15/4,), x**4*exp_polar(I*pi)/5)/(4*gamma(15/4)) + sqrt(5)*x**
9*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), x**4*exp_polar(I*pi)/5)/(2*gamma(13/4)) + 15*sqrt(5)*x**7*gamma(7/4)*
hyper((-1/2, 7/4), (11/4,), x**4*exp_polar(I*pi)/5)/(4*gamma(11/4)) + 5*sqrt(5)*x**5*gamma(5/4)*hyper((-1/2, 5
/4), (9/4,), x**4*exp_polar(I*pi)/5)/(2*gamma(9/4))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\left (x^4+5\right )}^{3/2}\,\left (3\,x^2+2\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________