Integrand size = 31, antiderivative size = 585 \[ \int \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=\frac {(c+d x) \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{3 d}-\frac {2 c^2 (c+d x) \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}{3 d \left (\sqrt {c} \sqrt {c^3+4 a d^2}+(c+d x)^2\right )}+\frac {2 c^{9/4} \left (c^3+4 a d^2\right )^{3/4} \sqrt {\frac {d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (c^3+4 a d^2\right ) \left (\sqrt {c}+\frac {(c+d x)^2}{\sqrt {c^3+4 a d^2}}\right )^2}} \left (\sqrt {c}+\frac {(c+d x)^2}{\sqrt {c^3+4 a d^2}}\right ) E\left (2 \arctan \left (\frac {c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac {1}{2} \left (1+\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}\right )\right )}{3 d^3 \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}}+\frac {c^{3/4} \sqrt [4]{c^3+4 a d^2} \left (c^3+4 a d^2-c^{3/2} \sqrt {c^3+4 a d^2}\right ) \sqrt {\frac {d^2 \left (4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4\right )}{\left (c^3+4 a d^2\right ) \left (\sqrt {c}+\frac {(c+d x)^2}{\sqrt {c^3+4 a d^2}}\right )^2}} \left (\sqrt {c}+\frac {(c+d x)^2}{\sqrt {c^3+4 a d^2}}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {c+d x}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right ),\frac {1}{2} \left (1+\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}\right )\right )}{3 d^3 \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4}} \] Output:
1/3*(d*x+c)*(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(1/2)/d-2/3*c^2*(d*x+c)*(d ^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(1/2)/d/(c^(1/2)*(4*a*d^2+c^3)^(1/2)+(d* x+c)^2)+2/3*c^(9/4)*(4*a*d^2+c^3)^(3/4)*(d^2*(d^2*x^4+4*c*d*x^3+4*c^2*x^2+ 4*a*c)/(4*a*d^2+c^3)/(c^(1/2)+(d*x+c)^2/(4*a*d^2+c^3)^(1/2))^2)^(1/2)*(c^( 1/2)+(d*x+c)^2/(4*a*d^2+c^3)^(1/2))*EllipticE(sin(2*arctan((d*x+c)/c^(1/4) /(4*a*d^2+c^3)^(1/4))),1/2*(2+2*c^(3/2)/(4*a*d^2+c^3)^(1/2))^(1/2))/d^3/(d ^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(1/2)+1/3*c^(3/4)*(4*a*d^2+c^3)^(1/4)*(c ^3+4*a*d^2-c^(3/2)*(4*a*d^2+c^3)^(1/2))*(d^2*(d^2*x^4+4*c*d*x^3+4*c^2*x^2+ 4*a*c)/(4*a*d^2+c^3)/(c^(1/2)+(d*x+c)^2/(4*a*d^2+c^3)^(1/2))^2)^(1/2)*(c^( 1/2)+(d*x+c)^2/(4*a*d^2+c^3)^(1/2))*InverseJacobiAM(2*arctan((d*x+c)/c^(1/ 4)/(4*a*d^2+c^3)^(1/4)),1/2*(2+2*c^(3/2)/(4*a*d^2+c^3)^(1/2))^(1/2))/d^3/( d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(1/2)
Result contains complex when optimal does not.
Time = 16.12 (sec) , antiderivative size = 5218, normalized size of antiderivative = 8.92 \[ \int \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=\text {Result too large to show} \] Input:
Integrate[Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4],x]
Output:
Result too large to show
Time = 0.67 (sec) , antiderivative size = 729, normalized size of antiderivative = 1.25, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2458, 1404, 27, 1511, 27, 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.
| \(\displaystyle \int \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4}d\left (\frac {c}{d}+x\right )\) |
| \(\Big \downarrow \) 1404 |
| \(\displaystyle \frac {1}{3} \int \frac {2 c \left (\frac {c^3}{d^2}-\left (\frac {c}{d}+x\right )^2 c+4 a\right )}{\sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}d\left (\frac {c}{d}+x\right )+\frac {1}{3} \left (\frac {c}{d}+x\right ) \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} c \int \frac {\frac {c^3}{d^2}-\left (\frac {c}{d}+x\right )^2 c+4 a}{\sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}d\left (\frac {c}{d}+x\right )+\frac {1}{3} \left (\frac {c}{d}+x\right ) \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {2}{3} c \left (\frac {c^{3/2} \sqrt {4 a d^2+c^3} \int \frac {\sqrt {c}-\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {c^3+4 a d^2}}}{\sqrt {c} \sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}d\left (\frac {c}{d}+x\right )}{d^2}+\frac {\left (-c^{3/2} \sqrt {4 a d^2+c^3}+4 a d^2+c^3\right ) \int \frac {1}{\sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}d\left (\frac {c}{d}+x\right )}{d^2}\right )+\frac {1}{3} \left (\frac {c}{d}+x\right ) \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{3} c \left (\frac {c \sqrt {4 a d^2+c^3} \int \frac {\sqrt {c}-\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {c^3+4 a d^2}}}{\sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}d\left (\frac {c}{d}+x\right )}{d^2}+\frac {\left (-c^{3/2} \sqrt {4 a d^2+c^3}+4 a d^2+c^3\right ) \int \frac {1}{\sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}d\left (\frac {c}{d}+x\right )}{d^2}\right )+\frac {1}{3} \left (\frac {c}{d}+x\right ) \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2}{3} c \left (\frac {c \sqrt {4 a d^2+c^3} \int \frac {\sqrt {c}-\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {c^3+4 a d^2}}}{\sqrt {d^2 \left (\frac {c}{d}+x\right )^4-2 c^2 \left (\frac {c}{d}+x\right )^2+c \left (\frac {c^3}{d^2}+4 a\right )}}d\left (\frac {c}{d}+x\right )}{d^2}+\frac {\sqrt [4]{4 a d^2+c^3} \left (-c^{3/2} \sqrt {4 a d^2+c^3}+4 a d^2+c^3\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right ) \sqrt {\frac {d^2 \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {d \left (\frac {c}{d}+x\right )}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right ),\frac {1}{2} \left (\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}+1\right )\right )}{2 \sqrt [4]{c} d^3 \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4}}\right )+\frac {1}{3} \left (\frac {c}{d}+x\right ) \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {2}{3} c \left (\frac {c \sqrt {4 a d^2+c^3} \left (\frac {\sqrt [4]{c} \sqrt [4]{4 a d^2+c^3} \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right ) \sqrt {\frac {d^2 \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {d \left (\frac {c}{d}+x\right )}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right )|\frac {1}{2} \left (\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}+1\right )\right )}{d \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4}}-\frac {\left (\frac {c}{d}+x\right ) \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4}}{\left (4 a+\frac {c^3}{d^2}\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right )}\right )}{d^2}+\frac {\sqrt [4]{4 a d^2+c^3} \left (-c^{3/2} \sqrt {4 a d^2+c^3}+4 a d^2+c^3\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right ) \sqrt {\frac {d^2 \left (c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4\right )}{\left (4 a d^2+c^3\right ) \left (\frac {d^2 \left (\frac {c}{d}+x\right )^2}{\sqrt {4 a d^2+c^3}}+\sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {d \left (\frac {c}{d}+x\right )}{\sqrt [4]{c} \sqrt [4]{c^3+4 a d^2}}\right ),\frac {1}{2} \left (\frac {c^{3/2}}{\sqrt {c^3+4 a d^2}}+1\right )\right )}{2 \sqrt [4]{c} d^3 \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4}}\right )+\frac {1}{3} \left (\frac {c}{d}+x\right ) \sqrt {c \left (4 a+\frac {c^3}{d^2}\right )-2 c^2 \left (\frac {c}{d}+x\right )^2+d^2 \left (\frac {c}{d}+x\right )^4}\) |
Input:
Int[Sqrt[4*a*c + 4*c^2*x^2 + 4*c*d*x^3 + d^2*x^4],x]
Output:
((c/d + x)*Sqrt[c*(4*a + c^3/d^2) - 2*c^2*(c/d + x)^2 + d^2*(c/d + x)^4])/ 3 + (2*c*((c*Sqrt[c^3 + 4*a*d^2]*(-(((c/d + x)*Sqrt[c*(4*a + c^3/d^2) - 2* c^2*(c/d + x)^2 + d^2*(c/d + x)^4])/((4*a + c^3/d^2)*(Sqrt[c] + (d^2*(c/d + x)^2)/Sqrt[c^3 + 4*a*d^2]))) + (c^(1/4)*(c^3 + 4*a*d^2)^(1/4)*(Sqrt[c] + (d^2*(c/d + x)^2)/Sqrt[c^3 + 4*a*d^2])*Sqrt[(d^2*(c*(4*a + c^3/d^2) - 2*c ^2*(c/d + x)^2 + d^2*(c/d + x)^4))/((c^3 + 4*a*d^2)*(Sqrt[c] + (d^2*(c/d + x)^2)/Sqrt[c^3 + 4*a*d^2])^2)]*EllipticE[2*ArcTan[(d*(c/d + x))/(c^(1/4)* (c^3 + 4*a*d^2)^(1/4))], (1 + c^(3/2)/Sqrt[c^3 + 4*a*d^2])/2])/(d*Sqrt[c*( 4*a + c^3/d^2) - 2*c^2*(c/d + x)^2 + d^2*(c/d + x)^4])))/d^2 + ((c^3 + 4*a *d^2)^(1/4)*(c^3 + 4*a*d^2 - c^(3/2)*Sqrt[c^3 + 4*a*d^2])*(Sqrt[c] + (d^2* (c/d + x)^2)/Sqrt[c^3 + 4*a*d^2])*Sqrt[(d^2*(c*(4*a + c^3/d^2) - 2*c^2*(c/ d + x)^2 + d^2*(c/d + x)^4))/((c^3 + 4*a*d^2)*(Sqrt[c] + (d^2*(c/d + x)^2) /Sqrt[c^3 + 4*a*d^2])^2)]*EllipticF[2*ArcTan[(d*(c/d + x))/(c^(1/4)*(c^3 + 4*a*d^2)^(1/4))], (1 + c^(3/2)/Sqrt[c^3 + 4*a*d^2])/2])/(2*c^(1/4)*d^3*Sq rt[c*(4*a + c^3/d^2) - 2*c^2*(c/d + x)^2 + d^2*(c/d + x)^4])))/3
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. \(4864\) vs. \(2(520)=1040\).
Time = 7.23 (sec) , antiderivative size = 4865, normalized size of antiderivative = 8.32
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(4865\) |
| default | \(\text {Expression too large to display}\) | \(4890\) |
| elliptic | \(\text {Expression too large to display}\) | \(4890\) |
Input:
int((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3*(d*x+c)*(d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(1/2)/d+2/3*c/d*(-c*d*((x- (-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)*(x-(-c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2)) /d)*(x+(c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)+((c+(-2*d*(-a*c)^(1/2)+c^2)^(1 /2))/d+(-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)*((-(c+(-2*d*(-a*c)^(1/2)+c^2)^ (1/2))/d+(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)*(x-(-c+(2*d*(-a*c)^(1/2)+c^2) ^(1/2))/d)/(-(c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(2*d*(-a*c)^(1/2)+c^2 )^(1/2))/d)/(x+(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d))^(1/2)*(x+(c+(2*d*(-a*c )^(1/2)+c^2)^(1/2))/d)^2*((-(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(2*d*(- a*c)^(1/2)+c^2)^(1/2))/d)*(x-(-c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)/((-c+(- 2*d*(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)/(x+(c+ (2*d*(-a*c)^(1/2)+c^2)^(1/2))/d))^(1/2)*((-(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2) )/d-(-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d)*(x+(c+(-2*d*(-a*c)^(1/2)+c^2)^(1/ 2))/d)/(-(c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(2*d*(-a*c)^(1/2)+c^2)^(1 /2))/d)/(x+(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d))^(1/2)*((-(c+(2*d*(-a*c)^(1 /2)+c^2)^(1/2))/d^2*(-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))+(c+(-2*d*(-a*c)^(1/2 )+c^2)^(1/2))/d^2*(-c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))+(c+(-2*d*(-a*c)^(1/2)+ c^2)^(1/2))/d^2*(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))+(c+(2*d*(-a*c)^(1/2)+c^2) ^(1/2))^2/d^2)/(-(c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d+(c+(2*d*(-a*c)^(1/2)+ c^2)^(1/2))/d)/(-(c+(2*d*(-a*c)^(1/2)+c^2)^(1/2))/d-(-c+(2*d*(-a*c)^(1/2)+ c^2)^(1/2))/d)*EllipticF(((-(c+(-2*d*(-a*c)^(1/2)+c^2)^(1/2))/d+(c+(2*d...
\[ \int \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=\int { \sqrt {d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c} \,d x } \] Input:
integrate((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c), x)
\[ \int \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=\int \sqrt {4 a c + 4 c^{2} x^{2} + 4 c d x^{3} + d^{2} x^{4}}\, dx \] Input:
integrate((d**2*x**4+4*c*d*x**3+4*c**2*x**2+4*a*c)**(1/2),x)
Output:
Integral(sqrt(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4), x)
\[ \int \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=\int { \sqrt {d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c} \,d x } \] Input:
integrate((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c), x)
\[ \int \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=\int { \sqrt {d^{2} x^{4} + 4 \, c d x^{3} + 4 \, c^{2} x^{2} + 4 \, a c} \,d x } \] Input:
integrate((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(d^2*x^4 + 4*c*d*x^3 + 4*c^2*x^2 + 4*a*c), x)
Timed out. \[ \int \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=\int \sqrt {4\,c^2\,x^2+4\,c\,d\,x^3+4\,a\,c+d^2\,x^4} \,d x \] Input:
int((4*a*c + 4*c^2*x^2 + d^2*x^4 + 4*c*d*x^3)^(1/2),x)
Output:
int((4*a*c + 4*c^2*x^2 + d^2*x^4 + 4*c*d*x^3)^(1/2), x)
\[ \int \sqrt {4 a c+4 c^2 x^2+4 c d x^3+d^2 x^4} \, dx=\frac {2 \sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, c +3 \sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, d x +24 \left (\int \frac {\sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}}{d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}d x \right ) a c d +2 \left (\int \frac {\sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, x^{3}}{d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}d x \right ) c \,d^{2}-8 \left (\int \frac {\sqrt {d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}\, x}{d^{2} x^{4}+4 c d \,x^{3}+4 c^{2} x^{2}+4 a c}d x \right ) c^{3}}{9 d} \] Input:
int((d^2*x^4+4*c*d*x^3+4*c^2*x^2+4*a*c)^(1/2),x)
Output:
(2*sqrt(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4)*c + 3*sqrt(4*a*c + 4 *c**2*x**2 + 4*c*d*x**3 + d**2*x**4)*d*x + 24*int(sqrt(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4)/(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4), x)*a*c*d + 2*int((sqrt(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4)*x**3) /(4*a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4),x)*c*d**2 - 8*int((sqrt(4* a*c + 4*c**2*x**2 + 4*c*d*x**3 + d**2*x**4)*x)/(4*a*c + 4*c**2*x**2 + 4*c* d*x**3 + d**2*x**4),x)*c**3)/(9*d)