Integrand size = 34, antiderivative size = 663 \[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\frac {1}{3} \left (\frac {d}{4 e}+x\right ) \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}-\frac {2 d^2 \left (\frac {d}{4 e}+x\right ) \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}{\sqrt {5 d^4+256 a e^3} \left (1+\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}\right )}+\frac {d^2 \left (5 d^4+256 a e^3\right )^{3/4} \sqrt {\frac {e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (5 d^4+256 a e^3\right ) \left (1+\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}\right )^2}} \left (1+\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}\right ) E\left (2 \arctan \left (\frac {d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac {1}{2} \left (1+\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}\right )\right )}{8 \sqrt {2} e^2 \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}+\frac {\sqrt [4]{5 d^4+256 a e^3} \left (5 d^4+256 a e^3-3 d^2 \sqrt {5 d^4+256 a e^3}\right ) \sqrt {\frac {e \left (8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4\right )}{\left (5 d^4+256 a e^3\right ) \left (1+\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}\right )^2}} \left (1+\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right ),\frac {1}{2} \left (1+\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}\right )\right )}{48 \sqrt {2} e^2 \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \]
1/3*(1/4*d/e+x)*(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(1/2)-2*d^2*(1/4*d/e +x)*(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(1/2)/(1+16*e^2*(1/4*d/e+x)^2/(2 56*a*e^3+5*d^4)^(1/2))/(256*a*e^3+5*d^4)^(1/2)+1/16*d^2*(256*a*e^3+5*d^4)^ (3/4)*(cos(2*arctan((4*e*x+d)/(256*a*e^3+5*d^4)^(1/4)))^2)^(1/2)/cos(2*arc tan((4*e*x+d)/(256*a*e^3+5*d^4)^(1/4)))*EllipticE(sin(2*arctan((4*e*x+d)/( 256*a*e^3+5*d^4)^(1/4))),1/2*(2+6*d^2/(256*a*e^3+5*d^4)^(1/2))^(1/2))*(1+1 6*e^2*(1/4*d/e+x)^2/(256*a*e^3+5*d^4)^(1/2))*(e*(8*e^3*x^4+8*d*e^2*x^3-d^3 *x+8*a*e^2)/(256*a*e^3+5*d^4)/(1+16*e^2*(1/4*d/e+x)^2/(256*a*e^3+5*d^4)^(1 /2))^2)^(1/2)/e^2*2^(1/2)/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(1/2)+1/96 *(256*a*e^3+5*d^4)^(1/4)*(cos(2*arctan((4*e*x+d)/(256*a*e^3+5*d^4)^(1/4))) ^2)^(1/2)/cos(2*arctan((4*e*x+d)/(256*a*e^3+5*d^4)^(1/4)))*EllipticF(sin(2 *arctan((4*e*x+d)/(256*a*e^3+5*d^4)^(1/4))),1/2*(2+6*d^2/(256*a*e^3+5*d^4) ^(1/2))^(1/2))*(1+16*e^2*(1/4*d/e+x)^2/(256*a*e^3+5*d^4)^(1/2))*(5*d^4+256 *a*e^3-3*d^2*(256*a*e^3+5*d^4)^(1/2))*(e*(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a* e^2)/(256*a*e^3+5*d^4)/(1+16*e^2*(1/4*d/e+x)^2/(256*a*e^3+5*d^4)^(1/2))^2) ^(1/2)/e^2*2^(1/2)/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(7543\) vs. \(2(663)=1326\).
Time = 13.92 (sec) , antiderivative size = 7543, normalized size of antiderivative = 11.38 \[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\text {Result too large to show} \]
Time = 0.77 (sec) , antiderivative size = 814, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2458, 1404, 27, 1511, 1416, 1509}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx\) |
\(\Big \downarrow \) 2458 |
\(\displaystyle \int \sqrt {\frac {1}{32} \left (256 a e^2+\frac {5 d^4}{e}\right )-3 d^2 e \left (\frac {d}{4 e}+x\right )^2+8 e^3 \left (\frac {d}{4 e}+x\right )^4}d\left (\frac {d}{4 e}+x\right )\) |
\(\Big \downarrow \) 1404 |
\(\displaystyle \frac {1}{3} \int \frac {\frac {5 d^4}{e}-48 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 a e^2}{2 \sqrt {2} \sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}d\left (\frac {d}{4 e}+x\right )+\frac {\left (\frac {d}{4 e}+x\right ) \sqrt {256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4}}{12 \sqrt {2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\frac {5 d^4}{e}-48 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 a e^2}{\sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}d\left (\frac {d}{4 e}+x\right )}{6 \sqrt {2}}+\frac {\left (\frac {d}{4 e}+x\right ) \sqrt {256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4}}{12 \sqrt {2}}\) |
\(\Big \downarrow \) 1511 |
\(\displaystyle \frac {\frac {3 d^2 \sqrt {256 a e^3+5 d^4} \int \frac {1-\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}}{\sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}d\left (\frac {d}{4 e}+x\right )}{e}+\frac {\left (-3 d^2 \sqrt {256 a e^3+5 d^4}+256 a e^3+5 d^4\right ) \int \frac {1}{\sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}d\left (\frac {d}{4 e}+x\right )}{e}}{6 \sqrt {2}}+\frac {\left (\frac {d}{4 e}+x\right ) \sqrt {256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4}}{12 \sqrt {2}}\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle \frac {\frac {3 d^2 \sqrt {256 a e^3+5 d^4} \int \frac {1-\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}}{\sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}d\left (\frac {d}{4 e}+x\right )}{e}+\frac {\sqrt [4]{256 a e^3+5 d^4} \left (-3 d^2 \sqrt {256 a e^3+5 d^4}+256 a e^3+5 d^4\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {256 a e^3+5 d^4}}+1\right ) \sqrt {\frac {256 a e^3+5 d^4-96 d^2 e^2 \left (\frac {d}{4 e}+x\right )^2+256 e^4 \left (\frac {d}{4 e}+x\right )^4}{\left (256 a e^3+5 d^4\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {256 a e^3+5 d^4}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {4 e \left (\frac {d}{4 e}+x\right )}{\sqrt [4]{5 d^4+256 a e^3}}\right ),\frac {1}{2} \left (\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}+1\right )\right )}{8 e^2 \sqrt {256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4}}}{6 \sqrt {2}}+\frac {\left (\frac {d}{4 e}+x\right ) \sqrt {256 a e^2+\frac {5 d^4}{e}-96 d^2 e \left (\frac {d}{4 e}+x\right )^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4}}{12 \sqrt {2}}\) |
\(\Big \downarrow \) 1509 |
\(\displaystyle \frac {\sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2} \left (\frac {d}{4 e}+x\right )}{12 \sqrt {2}}+\frac {\frac {3 \sqrt {5 d^4+256 a e^3} \left (\frac {\sqrt [4]{5 d^4+256 a e^3} \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right ) \sqrt {\frac {5 d^4-96 e^2 \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^4 \left (\frac {d}{4 e}+x\right )^4+256 a e^3}{\left (5 d^4+256 a e^3\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right )^2}} E\left (2 \arctan \left (\frac {4 e \left (\frac {d}{4 e}+x\right )}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac {1}{2} \left (\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}+1\right )\right )}{4 e \sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}-\frac {e \left (\frac {d}{4 e}+x\right ) \sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}{\left (5 d^4+256 a e^3\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right )}\right ) d^2}{e}+\frac {\sqrt [4]{5 d^4+256 a e^3} \left (5 d^4-3 \sqrt {5 d^4+256 a e^3} d^2+256 a e^3\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right ) \sqrt {\frac {5 d^4-96 e^2 \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^4 \left (\frac {d}{4 e}+x\right )^4+256 a e^3}{\left (5 d^4+256 a e^3\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {4 e \left (\frac {d}{4 e}+x\right )}{\sqrt [4]{5 d^4+256 a e^3}}\right ),\frac {1}{2} \left (\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}+1\right )\right )}{8 e^2 \sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}}{6 \sqrt {2}}\) |
((d/(4*e) + x)*Sqrt[(5*d^4)/e + 256*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 256 *e^3*(d/(4*e) + x)^4])/(12*Sqrt[2]) + ((3*d^2*Sqrt[5*d^4 + 256*a*e^3]*(-(( e*(d/(4*e) + x)*Sqrt[(5*d^4)/e + 256*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 25 6*e^3*(d/(4*e) + x)^4])/((5*d^4 + 256*a*e^3)*(1 + (16*e^2*(d/(4*e) + x)^2) /Sqrt[5*d^4 + 256*a*e^3]))) + ((5*d^4 + 256*a*e^3)^(1/4)*(1 + (16*e^2*(d/( 4*e) + x)^2)/Sqrt[5*d^4 + 256*a*e^3])*Sqrt[(5*d^4 + 256*a*e^3 - 96*d^2*e^2 *(d/(4*e) + x)^2 + 256*e^4*(d/(4*e) + x)^4)/((5*d^4 + 256*a*e^3)*(1 + (16* e^2*(d/(4*e) + x)^2)/Sqrt[5*d^4 + 256*a*e^3])^2)]*EllipticE[2*ArcTan[(4*e* (d/(4*e) + x))/(5*d^4 + 256*a*e^3)^(1/4)], (1 + (3*d^2)/Sqrt[5*d^4 + 256*a *e^3])/2])/(4*e*Sqrt[(5*d^4)/e + 256*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 25 6*e^3*(d/(4*e) + x)^4])))/e + ((5*d^4 + 256*a*e^3)^(1/4)*(5*d^4 + 256*a*e^ 3 - 3*d^2*Sqrt[5*d^4 + 256*a*e^3])*(1 + (16*e^2*(d/(4*e) + x)^2)/Sqrt[5*d^ 4 + 256*a*e^3])*Sqrt[(5*d^4 + 256*a*e^3 - 96*d^2*e^2*(d/(4*e) + x)^2 + 256 *e^4*(d/(4*e) + x)^4)/((5*d^4 + 256*a*e^3)*(1 + (16*e^2*(d/(4*e) + x)^2)/S qrt[5*d^4 + 256*a*e^3])^2)]*EllipticF[2*ArcTan[(4*e*(d/(4*e) + x))/(5*d^4 + 256*a*e^3)^(1/4)], (1 + (3*d^2)/Sqrt[5*d^4 + 256*a*e^3])/2])/(8*e^2*Sqrt [(5*d^4)/e + 256*a*e^2 - 96*d^2*e*(d/(4*e) + x)^2 + 256*e^3*(d/(4*e) + x)^ 4]))/(6*Sqrt[2])
3.8.78.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*((a + b *x^2 + c*x^4)^p/(4*p + 1)), x] + Simp[2*(p/(4*p + 1)) Int[(2*a + b*x^2)*( a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a* c, 0] && GtQ[p, 0] && IntegerQ[2*p]
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)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> 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)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[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] && Pos Q[c/a]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Leaf count of result is larger than twice the leaf count of optimal. \(7886\) vs. \(2(715)=1430\).
Time = 7.22 (sec) , antiderivative size = 7887, normalized size of antiderivative = 11.90
method | result | size |
default | \(\text {Expression too large to display}\) | \(7887\) |
elliptic | \(\text {Expression too large to display}\) | \(7887\) |
risch | \(\text {Expression too large to display}\) | \(9561\) |
\[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\int { \sqrt {8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}} \,d x } \]
\[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\int \sqrt {8 a e^{2} - d^{3} x + 8 d e^{2} x^{3} + 8 e^{3} x^{4}}\, dx \]
\[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\int { \sqrt {8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}} \,d x } \]
\[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\int { \sqrt {8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}} \,d x } \]
Timed out. \[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\int \sqrt {-d^3\,x+8\,d\,e^2\,x^3+8\,e^3\,x^4+8\,a\,e^2} \,d x \]