Integrand size = 34, antiderivative size = 599 \[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\frac {(d+4 e x) \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}{12 e}-\frac {d^2 (d+4 e x) \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}{2 e \left (\sqrt {5 d^4+256 a e^3}+(d+4 e x)^2\right )}+\frac {d^2 \left (5 d^4+256 a e^3\right )^{3/4} \left (1+\frac {(d+4 e x)^2}{\sqrt {5 d^4+256 a e^3}}\right ) \sqrt {\frac {5 d^4+256 a e^3-6 d^2 (d+4 e x)^2+(d+4 e x)^4}{\left (5 d^4+256 a e^3\right ) \left (1+\frac {(d+4 e x)^2}{\sqrt {5 d^4+256 a e^3}}\right )^2}} E\left (2 \arctan \left (\frac {d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac {1}{2} \left (1+\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}\right )\right )}{64 e^2 \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}}+\frac {\sqrt [4]{5 d^4+256 a e^3} \left (5 d^4+256 a e^3-3 d^2 \sqrt {5 d^4+256 a e^3}\right ) \left (1+\frac {(d+4 e x)^2}{\sqrt {5 d^4+256 a e^3}}\right ) \sqrt {\frac {5 d^4+256 a e^3-6 d^2 (d+4 e x)^2+(d+4 e x)^4}{\left (5 d^4+256 a e^3\right ) \left (1+\frac {(d+4 e x)^2}{\sqrt {5 d^4+256 a e^3}}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {d+4 e x}{\sqrt [4]{5 d^4+256 a e^3}}\right ),\frac {1}{2} \left (1+\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}\right )\right )}{384 e^2 \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4}} \] Output:
1/12*(4*e*x+d)*(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(1/2)/e-1/2*d^2*(4*e* x+d)*(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(1/2)/e/((256*a*e^3+5*d^4)^(1/2 )+(4*e*x+d)^2)+1/64*d^2*(256*a*e^3+5*d^4)^(3/4)*(1+(4*e*x+d)^2/(256*a*e^3+ 5*d^4)^(1/2))*((5*d^4+256*a*e^3-6*d^2*(4*e*x+d)^2+(4*e*x+d)^4)/(256*a*e^3+ 5*d^4)/(1+(4*e*x+d)^2/(256*a*e^3+5*d^4)^(1/2))^2)^(1/2)*EllipticE(sin(2*ar ctan((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))/e^2/(8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(1/2)+1/384*(256*a*e ^3+5*d^4)^(1/4)*(5*d^4+256*a*e^3-3*d^2*(256*a*e^3+5*d^4)^(1/2))*(1+(4*e*x+ d)^2/(256*a*e^3+5*d^4)^(1/2))*((5*d^4+256*a*e^3-6*d^2*(4*e*x+d)^2+(4*e*x+d )^4)/(256*a*e^3+5*d^4)/(1+(4*e*x+d)^2/(256*a*e^3+5*d^4)^(1/2))^2)^(1/2)*In verseJacobiAM(2*arctan((4*e*x+d)/(256*a*e^3+5*d^4)^(1/4)),1/2*(2+6*d^2/(25 6*a*e^3+5*d^4)^(1/2))^(1/2))/e^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. \(2(599)=1198\).
Time = 16.15 (sec) , antiderivative size = 7543, normalized size of antiderivative = 12.59 \[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\text {Result too large to show} \] Input:
Integrate[Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4],x]
Output:
Result too large to show
Time = 1.24 (sec) , antiderivative size = 814, normalized size of antiderivative = 1.36, 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 |
| \(\displaystyle \frac {\sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2} \left (\frac {d}{4 e}+x\right )}{12 \sqrt {2}}+\frac {\frac {3 \sqrt {5 d^4+256 a e^3} \left (\frac {\sqrt [4]{5 d^4+256 a e^3} \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right ) \sqrt {\frac {5 d^4-96 e^2 \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^4 \left (\frac {d}{4 e}+x\right )^4+256 a e^3}{\left (5 d^4+256 a e^3\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right )^2}} E\left (2 \arctan \left (\frac {4 e \left (\frac {d}{4 e}+x\right )}{\sqrt [4]{5 d^4+256 a e^3}}\right )|\frac {1}{2} \left (\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}+1\right )\right )}{4 e \sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}-\frac {e \left (\frac {d}{4 e}+x\right ) \sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}{\left (5 d^4+256 a e^3\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right )}\right ) d^2}{e}+\frac {\sqrt [4]{5 d^4+256 a e^3} \left (5 d^4-3 \sqrt {5 d^4+256 a e^3} d^2+256 a e^3\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right ) \sqrt {\frac {5 d^4-96 e^2 \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^4 \left (\frac {d}{4 e}+x\right )^4+256 a e^3}{\left (5 d^4+256 a e^3\right ) \left (\frac {16 e^2 \left (\frac {d}{4 e}+x\right )^2}{\sqrt {5 d^4+256 a e^3}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {4 e \left (\frac {d}{4 e}+x\right )}{\sqrt [4]{5 d^4+256 a e^3}}\right ),\frac {1}{2} \left (\frac {3 d^2}{\sqrt {5 d^4+256 a e^3}}+1\right )\right )}{8 e^2 \sqrt {\frac {5 d^4}{e}-96 e \left (\frac {d}{4 e}+x\right )^2 d^2+256 e^3 \left (\frac {d}{4 e}+x\right )^4+256 a e^2}}}{6 \sqrt {2}}\) |
Input:
Int[Sqrt[8*a*e^2 - d^3*x + 8*d*e^2*x^3 + 8*e^3*x^4],x]
Output:
((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])
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. \(2(556)=1112\).
Time = 6.77 (sec) , antiderivative size = 7887, normalized size of antiderivative = 13.17
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(7887\) |
| elliptic | \(\text {Expression too large to display}\) | \(7887\) |
| risch | \(\text {Expression too large to display}\) | \(9561\) |
Input:
int((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
result too large to display
\[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\int { \sqrt {8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}} \,d x } \] Input:
integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(1/2),x, algorithm="fricas ")
Output:
integral(sqrt(8*e^3*x^4 + 8*d*e^2*x^3 - d^3*x + 8*a*e^2), x)
\[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\int \sqrt {8 a e^{2} - d^{3} x + 8 d e^{2} x^{3} + 8 e^{3} x^{4}}\, dx \] Input:
integrate((8*e**3*x**4+8*d*e**2*x**3-d**3*x+8*a*e**2)**(1/2),x)
Output:
Integral(sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4), x)
\[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\int { \sqrt {8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}} \,d x } \] Input:
integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(1/2),x, algorithm="maxima ")
Output:
integrate(sqrt(8*e^3*x^4 + 8*d*e^2*x^3 - d^3*x + 8*a*e^2), x)
\[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\int { \sqrt {8 \, e^{3} x^{4} + 8 \, d e^{2} x^{3} - d^{3} x + 8 \, a e^{2}} \,d x } \] Input:
integrate((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(8*e^3*x^4 + 8*d*e^2*x^3 - d^3*x + 8*a*e^2), x)
Timed out. \[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\int \sqrt {-d^3\,x+8\,d\,e^2\,x^3+8\,e^3\,x^4+8\,a\,e^2} \,d x \] Input:
int((8*a*e^2 - d^3*x + 8*e^3*x^4 + 8*d*e^2*x^3)^(1/2),x)
Output:
int((8*a*e^2 - d^3*x + 8*e^3*x^4 + 8*d*e^2*x^3)^(1/2), x)
\[ \int \sqrt {8 a e^2-d^3 x+8 d e^2 x^3+8 e^3 x^4} \, dx=\frac {4 \sqrt {8 e^{3} x^{4}+8 d \,e^{2} x^{3}-d^{3} x +8 a \,e^{2}}\, d e +16 \sqrt {8 e^{3} x^{4}+8 d \,e^{2} x^{3}-d^{3} x +8 a \,e^{2}}\, e^{2} x -3 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (-2 \sqrt {e}\, \sqrt {8 e^{3} x^{4}+8 d \,e^{2} x^{3}-d^{3} x +8 a \,e^{2}}\, \sqrt {2}+d^{2}-4 d e x -8 e^{2} x^{2}\right ) d^{3}-3 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {8 e^{3} x^{4}+8 d \,e^{2} x^{3}-d^{3} x +8 a \,e^{2}}\, \sqrt {2}+d^{2}-4 d e x -8 e^{2} x^{2}\right ) d^{3}+256 \left (\int \frac {\sqrt {8 e^{3} x^{4}+8 d \,e^{2} x^{3}-d^{3} x +8 a \,e^{2}}}{8 e^{3} x^{4}+8 d \,e^{2} x^{3}-d^{3} x +8 a \,e^{2}}d x \right ) a \,e^{4}+2 \left (\int \frac {\sqrt {8 e^{3} x^{4}+8 d \,e^{2} x^{3}-d^{3} x +8 a \,e^{2}}}{8 e^{3} x^{4}+8 d \,e^{2} x^{3}-d^{3} x +8 a \,e^{2}}d x \right ) d^{4} e -48 \left (\int \frac {\sqrt {8 e^{3} x^{4}+8 d \,e^{2} x^{3}-d^{3} x +8 a \,e^{2}}\, x^{2}}{8 e^{3} x^{4}+8 d \,e^{2} x^{3}-d^{3} x +8 a \,e^{2}}d x \right ) d^{2} e^{3}-24 \left (\int \frac {\sqrt {8 e^{3} x^{4}+8 d \,e^{2} x^{3}-d^{3} x +8 a \,e^{2}}\, x}{8 e^{3} x^{4}+8 d \,e^{2} x^{3}-d^{3} x +8 a \,e^{2}}d x \right ) d^{3} e^{2}}{48 e^{2}} \] Input:
int((8*e^3*x^4+8*d*e^2*x^3-d^3*x+8*a*e^2)^(1/2),x)
Output:
(4*sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*d*e + 16*sqrt(8*a *e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*e**2*x - 3*sqrt(e)*sqrt(2)*l og( - 2*sqrt(e)*sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*sqrt (2) + d**2 - 4*d*e*x - 8*e**2*x**2)*d**3 - 3*sqrt(e)*sqrt(2)*log(2*sqrt(e) *sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*sqrt(2) + d**2 - 4* d*e*x - 8*e**2*x**2)*d**3 + 256*int(sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)/(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4),x)*a*e** 4 + 2*int(sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)/(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4),x)*d**4*e - 48*int((sqrt(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x**4)*x**2)/(8*a*e**2 - d**3*x + 8*d*e** 2*x**3 + 8*e**3*x**4),x)*d**2*e**3 - 24*int((sqrt(8*a*e**2 - d**3*x + 8*d* e**2*x**3 + 8*e**3*x**4)*x)/(8*a*e**2 - d**3*x + 8*d*e**2*x**3 + 8*e**3*x* *4),x)*d**3*e**2)/(48*e**2)