Integrand size = 28, antiderivative size = 430 \[ \int x^2 \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\frac {1}{2} \sqrt {3+a-2 (1-x)^2-(1-x)^4} \left (1+(-1+x)^2\right )-\frac {1}{15} \left (7+3 (1-x)^2\right ) \sqrt {3+a-2 (1-x)^2-(1-x)^4} (1-x)+\frac {1}{2} (4+a) \arctan \left (\frac {1+(-1+x)^2}{\sqrt {3+a-2 (1-x)^2-(1-x)^4}}\right )-\frac {2 (8+3 a) \sqrt {-1+\sqrt {4+a}} \left (1+\sqrt {4+a}\right ) \sqrt {1+\frac {(1-x)^2}{1-\sqrt {4+a}}} \sqrt {1+\frac {(1-x)^2}{1+\sqrt {4+a}}} E\left (\arcsin \left (\frac {1-x}{\sqrt {-1+\sqrt {4+a}}}\right )|\frac {1-\sqrt {4+a}}{1+\sqrt {4+a}}\right )}{15 \sqrt {3+a-2 (1-x)^2-(1-x)^4}}-\frac {2 \sqrt {-1+\sqrt {4+a}} \left (4+a-\sqrt {4+a} (8+3 a)\right ) \sqrt {1+\frac {(1-x)^2}{1-\sqrt {4+a}}} \sqrt {1+\frac {(1-x)^2}{1+\sqrt {4+a}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-x}{\sqrt {-1+\sqrt {4+a}}}\right ),\frac {1-\sqrt {4+a}}{1+\sqrt {4+a}}\right )}{15 \sqrt {3+a-2 (1-x)^2-(1-x)^4}} \] Output:
1/2*(3+a-2*(1-x)^2-(1-x)^4)^(1/2)*(1+(-1+x)^2)-1/15*(7+3*(1-x)^2)*(3+a-2*( 1-x)^2-(1-x)^4)^(1/2)*(1-x)+1/2*(4+a)*arctan((1+(-1+x)^2)/(3+a-2*(1-x)^2-( 1-x)^4)^(1/2))-2/15*(8+3*a)*(-1+(4+a)^(1/2))^(1/2)*(1+(4+a)^(1/2))*(1+(1-x )^2/(1-(4+a)^(1/2)))^(1/2)*(1+(1-x)^2/(1+(4+a)^(1/2)))^(1/2)*EllipticE((1- x)/(-1+(4+a)^(1/2))^(1/2),((1-(4+a)^(1/2))/(1+(4+a)^(1/2)))^(1/2))/(3+a-2* (1-x)^2-(1-x)^4)^(1/2)-2/15*(-1+(4+a)^(1/2))^(1/2)*(4+a-(4+a)^(1/2)*(8+3*a ))*(1+(1-x)^2/(1-(4+a)^(1/2)))^(1/2)*(1+(1-x)^2/(1+(4+a)^(1/2)))^(1/2)*Ell ipticF((1-x)/(-1+(4+a)^(1/2))^(1/2),((1-(4+a)^(1/2))/(1+(4+a)^(1/2)))^(1/2 ))/(3+a-2*(1-x)^2-(1-x)^4)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(5647\) vs. \(2(430)=860\).
Time = 16.59 (sec) , antiderivative size = 5647, normalized size of antiderivative = 13.13 \[ \int x^2 \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\text {Result too large to show} \] Input:
Integrate[x^2*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4],x]
Output:
Result too large to show
Time = 1.33 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.29, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {2459, 2006, 2202, 27, 1432, 1087, 1092, 217, 1490, 27, 1514, 406, 320, 388, 313}
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.
\(\displaystyle \int x^2 \sqrt {a-x^4+4 x^3-8 x^2+8 x} \, dx\) |
\(\Big \downarrow \) 2459 |
\(\displaystyle \int \left ((x-1)^2+2 (x-1)+1\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}d(x-1)\) |
\(\Big \downarrow \) 2006 |
\(\displaystyle \int x^2 \sqrt {a-(x-1)^4-2 (x-1)^2+3}d(x-1)\) |
\(\Big \downarrow \) 2202 |
\(\displaystyle \int \left ((x-1)^2+1\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\int 2 \sqrt {-(x-1)^4-2 (x-1)^2+a+3} (x-1)d(x-1)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \left ((x-1)^2+1\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+2 \int \sqrt {-(x-1)^4-2 (x-1)^2+a+3} (x-1)d(x-1)\) |
\(\Big \downarrow \) 1432 |
\(\displaystyle \int \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)^2+\int \left ((x-1)^2+1\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {1}{2} (a+4) \int \frac {1}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)^2+\int \left ((x-1)^2+1\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle (a+4) \int \frac {1}{-(x-1)^4-4}d\left (-\frac {2 x}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}\right )+\int \left ((x-1)^2+1\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \int \left ((x-1)^2+1\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\) |
\(\Big \downarrow \) 1490 |
\(\displaystyle -\frac {1}{15} \int -\frac {2 \left ((3 a+8) (x-1)^2+4 (a+3)\right )}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)+\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2}{15} \int \frac {(3 a+8) (x-1)^2+4 (a+3)}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)+\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\) |
\(\Big \downarrow \) 1514 |
\(\displaystyle \frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \int \frac {(3 a+8) (x-1)^2+4 (a+3)}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\) |
\(\Big \downarrow \) 406 |
\(\displaystyle \frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (4 (a+3) \int \frac {1}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)+(3 a+8) \int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\) |
\(\Big \downarrow \) 320 |
\(\displaystyle \frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left ((3 a+8) \int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)+\frac {4 (a+3) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\) |
\(\Big \downarrow \) 388 |
\(\displaystyle \frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left ((3 a+8) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\left (1-\sqrt {a+4}\right ) \int \frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\left (\frac {(x-1)^2}{\sqrt {a+4}+1}+1\right )^{3/2}}d(x-1)\right )+\frac {4 (a+3) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\) |
\(\Big \downarrow \) 313 |
\(\displaystyle \frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {4 (a+3) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}+(3 a+8) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\frac {\left (1-\sqrt {a+4}\right ) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} E\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} \left (3 (x-1)^2+7\right ) (x-1) \sqrt {a-(x-1)^4-2 (x-1)^2+3}+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\) |
Input:
Int[x^2*Sqrt[a + 8*x - 8*x^2 + 4*x^3 - x^4],x]
Output:
((7 + 3*(-1 + x)^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/15 + (Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]*x)/2 + ((4 + a)*ArcTan[x/Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]])/2 + (2*Sqrt[1 + (-1 + x)^2/(1 - Sqrt[4 + a])]*Sqrt[1 + (-1 + x)^2/(1 + Sqrt[4 + a])]*((8 + 3*a)*(((1 - Sqrt[4 + a ])*Sqrt[1 + (-1 + x)^2/(1 - Sqrt[4 + a])]*(-1 + x))/Sqrt[1 + (-1 + x)^2/(1 + Sqrt[4 + a])] - ((1 - Sqrt[4 + a])*Sqrt[1 + Sqrt[4 + a]]*Sqrt[1 + (-1 + x)^2/(1 - Sqrt[4 + a])]*EllipticE[ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])])/(Sqrt[(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 + Sqrt[4 + a]))]*Sqrt[1 + (-1 + x)^2/(1 + Sqrt[4 + a])])) + (4*(3 + a)*Sqrt[1 + Sqrt[4 + a]]*Sqrt[1 + (-1 + x)^2/(1 - Sqrt[4 + a])]*EllipticF[ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a]) /(1 - Sqrt[4 + a])])/(Sqrt[(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))/(1 + (-1 + x )^2/(1 + Sqrt[4 + a]))]*Sqrt[1 + (-1 + x)^2/(1 + Sqrt[4 + a])])))/(15*Sqrt [3 + a - 2*(-1 + x)^2 - (-1 + x)^4])
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)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> 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] + Simp[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]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*(Sqrt [1 + 2*c*(x^2/(b + q))]/Sqrt[a + b*x^2 + c*x^4]) Int[(d + e*x^2)/(Sqrt[1 + 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(2581\) vs. \(2(368)=736\).
Time = 3.85 (sec) , antiderivative size = 2582, normalized size of antiderivative = 6.00
method | result | size |
default | \(\text {Expression too large to display}\) | \(2582\) |
elliptic | \(\text {Expression too large to display}\) | \(2582\) |
risch | \(\text {Expression too large to display}\) | \(3044\) |
Input:
int(x^2*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/5*x^3*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)-1/10*x^2*(-x^4+4*x^3-8*x^2+a+8*x)^( 1/2)+1/15*x*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+1/3*(-x^4+4*x^3-8*x^2+a+8*x)^(1 /2)-(-1/15*a-4/3)*((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*((-(-1-( a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1+(a+4)^(1/2))^(1/2))/(-( -1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1/2)) )^(1/2)*(x-1+(-1+(a+4)^(1/2))^(1/2))^2*(-2*(-1+(a+4)^(1/2))^(1/2)*(x-1-(-1 -(a+4)^(1/2))^(1/2))/((-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+ (-1+(a+4)^(1/2))^(1/2)))^(1/2)*(-2*(-1+(a+4)^(1/2))^(1/2)*(x-1+(-1-(a+4)^( 1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+ 4)^(1/2))^(1/2)))^(1/2)/(-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/( -1+(a+4)^(1/2))^(1/2)/(-(x-1-(-1+(a+4)^(1/2))^(1/2))*(x-1+(-1+(a+4)^(1/2)) ^(1/2))*(x-1-(-1-(a+4)^(1/2))^(1/2))*(x-1+(-1-(a+4)^(1/2))^(1/2)))^(1/2)*E llipticF(((-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1+(a+4)^ (1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a +4)^(1/2))^(1/2)))^(1/2),((-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2)) *((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)+ (-1+(a+4)^(1/2))^(1/2))/((-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2)))^( 1/2))-(1/5*a+28/15)*((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*((-(-1 -(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1+(a+4)^(1/2))^(1/2))/( -(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(...
\[ \int x^2 \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\int { \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x} x^{2} \,d x } \] Input:
integrate(x^2*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)*x^2, x)
\[ \int x^2 \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\int x^{2} \sqrt {a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x}\, dx \] Input:
integrate(x**2*(-x**4+4*x**3-8*x**2+a+8*x)**(1/2),x)
Output:
Integral(x**2*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x), x)
\[ \int x^2 \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\int { \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x} x^{2} \,d x } \] Input:
integrate(x^2*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)*x^2, x)
\[ \int x^2 \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\int { \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x} x^{2} \,d x } \] Input:
integrate(x^2*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)*x^2, x)
Timed out. \[ \int x^2 \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\int x^2\,\sqrt {-x^4+4\,x^3-8\,x^2+8\,x+a} \,d x \] Input:
int(x^2*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(1/2),x)
Output:
int(x^2*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(1/2), x)
\[ \int x^2 \sqrt {a+8 x-8 x^2+4 x^3-x^4} \, dx=\frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, a}{15}+\frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x^{3}}{5}-\frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x^{2}}{10}+\frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x}{15}+\frac {23 \sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}}{45}-\frac {\left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right ) a}{3}-\frac {92 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right )}{45}+\frac {2 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x^{3}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right ) a}{15}+\frac {16 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x^{3}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right )}{45}+\frac {11 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right ) a}{15}+\frac {148 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right )}{45} \] Input:
int(x^2*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2),x)
Output:
(6*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*a + 18*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x**3 - 9*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x**2 + 6*sqr t(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x + 46*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x) - 30*int(sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x)*a - 184*int(sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x) + 12*int((sqrt(a - x**4 + 4*x**3 - 8*x* *2 + 8*x)*x**3)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x)*a + 32*int((sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x**3)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x) + 66*int((sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x)*a + 296*int((sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x))/90