3.8.64 \(\int (8 x-8 x^2+4 x^3-x^4)^{3/2} \, dx\) [764]

3.8.64.1 Optimal result

Integrand size = 23, antiderivative size = 102 \[ \int \left (8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\frac {2}{35} \left (13-3 (-1+x)^2\right ) \sqrt {3-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {1}{7} \left (3-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac {16}{5} \sqrt {3} E\left (\arcsin (1-x)\left |-\frac {1}{3}\right .\right )-\frac {176}{35} \sqrt {3} \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right ) \]

output

1/7*(3-2*(-1+x)^2-(-1+x)^4)^(3/2)*(-1+x)-16/5*EllipticE(-1+x,1/3*I*3^(1/2) 
)*3^(1/2)+176/35*EllipticF(-1+x,1/3*I*3^(1/2))*3^(1/2)+2/35*(13-3*(-1+x)^2 
)*(-1+x)*(3-2*(-1+x)^2-(-1+x)^4)^(1/2)
 

3.8.64.2 Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 22.48 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.73 \[ \int \left (8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\frac {896-1056 x+352 x^2+848 x^3-1420 x^4+1152 x^5-602 x^6+206 x^7-45 x^8+5 x^9+\frac {112 i \sqrt {2} (-2+x) x \sqrt {\frac {4-2 x+x^2}{x^2}} E\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )}{\sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}}}-304 i \sqrt {2} \sqrt {-\frac {i (-2+x)}{\left (-i+\sqrt {3}\right ) x}} x^2 \sqrt {\frac {4-2 x+x^2}{x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {i+\sqrt {3}-\frac {4 i}{x}}}{\sqrt {2} \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{-i+\sqrt {3}}\right )}{35 \sqrt {-x \left (-8+8 x-4 x^2+x^3\right )}} \]

input

Integrate[(8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]
 

output

(896 - 1056*x + 352*x^2 + 848*x^3 - 1420*x^4 + 1152*x^5 - 602*x^6 + 206*x^ 
7 - 45*x^8 + 5*x^9 + ((112*I)*Sqrt[2]*(-2 + x)*x*Sqrt[(4 - 2*x + x^2)/x^2] 
*EllipticE[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqrt[ 
3])/(-I + Sqrt[3])])/Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)] - (304*I)*Sq 
rt[2]*Sqrt[((-I)*(-2 + x))/((-I + Sqrt[3])*x)]*x^2*Sqrt[(4 - 2*x + x^2)/x^ 
2]*EllipticF[ArcSin[Sqrt[I + Sqrt[3] - (4*I)/x]/(Sqrt[2]*3^(1/4))], (2*Sqr 
t[3])/(-I + Sqrt[3])])/(35*Sqrt[-(x*(-8 + 8*x - 4*x^2 + x^3))])
 

3.8.64.3 Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2458, 1404, 27, 1490, 27, 1494, 27, 399, 321, 327}

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.

\(\displaystyle \int \left (-x^4+4 x^3-8 x^2+8 x\right )^{3/2} \, dx\)

\(\Big \downarrow \) 2458

\(\displaystyle \int \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}d(x-1)\)

\(\Big \downarrow \) 1404

\(\displaystyle \frac {3}{7} \int 2 \left (3-(x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3}d(x-1)+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{7} \int \left (3-(x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3}d(x-1)+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {6}{7} \left (\frac {1}{15} \left (13-3 (x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)-\frac {1}{15} \int -\frac {8 \left (12-7 (x-1)^2\right )}{\sqrt {-(x-1)^4-2 (x-1)^2+3}}d(x-1)\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \int \frac {12-7 (x-1)^2}{\sqrt {-(x-1)^4-2 (x-1)^2+3}}d(x-1)+\frac {1}{15} \left (13-3 (x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 1494

\(\displaystyle \frac {6}{7} \left (\frac {16}{15} \int \frac {12-7 (x-1)^2}{2 \sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)+\frac {1}{15} \left (13-3 (x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \int \frac {12-7 (x-1)^2}{\sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)+\frac {1}{15} \left (13-3 (x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \left (33 \int \frac {1}{\sqrt {1-(x-1)^2} \sqrt {(x-1)^2+3}}d(x-1)-7 \int \frac {\sqrt {(x-1)^2+3}}{\sqrt {1-(x-1)^2}}d(x-1)\right )+\frac {1}{15} \left (13-3 (x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \left (-7 \int \frac {\sqrt {(x-1)^2+3}}{\sqrt {1-(x-1)^2}}d(x-1)-11 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )\right )+\frac {1}{15} \left (13-3 (x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {6}{7} \left (\frac {8}{15} \left (7 \sqrt {3} E\left (\arcsin (1-x)\left |-\frac {1}{3}\right .\right )-11 \sqrt {3} \operatorname {EllipticF}\left (\arcsin (1-x),-\frac {1}{3}\right )\right )+\frac {1}{15} \left (13-3 (x-1)^2\right ) \sqrt {-(x-1)^4-2 (x-1)^2+3} (x-1)\right )+\frac {1}{7} (x-1) \left (-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

input

Int[(8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]
 

output

((3 - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(-1 + x))/7 + (6*(((13 - 3*(-1 + x) 
^2)*Sqrt[3 - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/15 + (8*(7*Sqrt[3]*Ellip 
ticE[ArcSin[1 - x], -1/3] - 11*Sqrt[3]*EllipticF[ArcSin[1 - x], -1/3]))/15 
))/7
 

3.8.64.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 1404

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]
 

rule 1490

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]
 

rule 1494

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[2*Sqrt[-c]   Int[(d + e*x^2)/(Sqr 
t[b + q + 2*c*x^2]*Sqrt[-b + q - 2*c*x^2]), x], x]] /; FreeQ[{a, b, c, d, e 
}, x] && GtQ[b^2 - 4*a*c, 0] && LtQ[c, 0]
 

rule 2458

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]
 

3.8.64.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 957 vs. \(2 (86 ) = 172\).

Time = 2.64 (sec) , antiderivative size = 958, normalized size of antiderivative = 9.39

method	result	size
risch	\(\text {Expression too large to display}\)	\(958\)
default	\(\text {Expression too large to display}\)	\(1050\)
elliptic	\(\text {Expression too large to display}\)	\(1050\)

input

int((-x^4+4*x^3-8*x^2+8*x)^(3/2),x,method=_RETURNVERBOSE)
 

output

1/35*(5*x^5-25*x^4+66*x^3-98*x^2+32*x+20)*x*(x^3-4*x^2+8*x-8)/(-x*(x^3-4*x 
^2+8*x-8))^(1/2)+32/7*(-1-I*3^(1/2))*((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2) 
)^(1/2)*(x-2)^2*((x-1+I*3^(1/2))/(1-I*3^(1/2))/(x-2))^(1/2)*((x-1-I*3^(1/2 
))/(1+I*3^(1/2))/(x-2))^(1/2)/(-1+I*3^(1/2))/(-x*(x-2)*(x-1+I*3^(1/2))*(x- 
1-I*3^(1/2)))^(1/2)*EllipticF(((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(1/2) 
,((1+I*3^(1/2))*(-1-I*3^(1/2))/(-1+I*3^(1/2))/(1-I*3^(1/2)))^(1/2))+64/5*( 
-1-I*3^(1/2))*((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(1/2)*(x-2)^2*((x-1+I 
*3^(1/2))/(1-I*3^(1/2))/(x-2))^(1/2)*((x-1-I*3^(1/2))/(1+I*3^(1/2))/(x-2)) 
^(1/2)/(-1+I*3^(1/2))/(-x*(x-2)*(x-1+I*3^(1/2))*(x-1-I*3^(1/2)))^(1/2)*(2* 
EllipticF(((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(1/2),((1+I*3^(1/2))*(-1- 
I*3^(1/2))/(-1+I*3^(1/2))/(1-I*3^(1/2)))^(1/2))-2*EllipticPi(((-1+I*3^(1/2 
))*x/(1+I*3^(1/2))/(x-2))^(1/2),(1+I*3^(1/2))/(-1+I*3^(1/2)),((1+I*3^(1/2) 
)*(-1-I*3^(1/2))/(-1+I*3^(1/2))/(1-I*3^(1/2)))^(1/2)))-16/5*(x*(x-1+I*3^(1 
/2))*(x-1-I*3^(1/2))+2*(-1-I*3^(1/2))*((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2 
))^(1/2)*(x-2)^2*((x-1+I*3^(1/2))/(1-I*3^(1/2))/(x-2))^(1/2)*((x-1-I*3^(1/ 
2))/(1+I*3^(1/2))/(x-2))^(1/2)*(1/2*(6+2*I*3^(1/2))/(-1+I*3^(1/2))*Ellipti 
cF(((-1+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(1/2),((1+I*3^(1/2))*(-1-I*3^(1/ 
2))/(-1+I*3^(1/2))/(1-I*3^(1/2)))^(1/2))+1/2*(-1+I*3^(1/2))*EllipticE(((-1 
+I*3^(1/2))*x/(1+I*3^(1/2))/(x-2))^(1/2),((1+I*3^(1/2))*(-1-I*3^(1/2))/(-1 
+I*3^(1/2))/(1-I*3^(1/2)))^(1/2))-4/(-1+I*3^(1/2))*EllipticPi(((-1+I*3^...
 

3.8.64.5 Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.88 \[ \int \left (8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=-\frac {112 \, {\left (-i \, x + i\right )} E(\arcsin \left (\frac {1}{x - 1}\right )\,|\,-3) + 80 \, {\left (-i \, x + i\right )} F(\arcsin \left (\frac {1}{x - 1}\right )\,|\,-3) + {\left (5 \, x^{6} - 30 \, x^{5} + 91 \, x^{4} - 164 \, x^{3} + 130 \, x^{2} - 12 \, x - 132\right )} \sqrt {-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x}}{35 \, {\left (x - 1\right )}} \]

input

integrate((-x^4+4*x^3-8*x^2+8*x)^(3/2),x, algorithm="fricas")
 

output

-1/35*(112*(-I*x + I)*elliptic_e(arcsin(1/(x - 1)), -3) + 80*(-I*x + I)*el 
liptic_f(arcsin(1/(x - 1)), -3) + (5*x^6 - 30*x^5 + 91*x^4 - 164*x^3 + 130 
*x^2 - 12*x - 132)*sqrt(-x^4 + 4*x^3 - 8*x^2 + 8*x))/(x - 1)
 

3.8.64.6 Sympy [F]

\[ \int \left (8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int \left (- x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac {3}{2}}\, dx \]

input

integrate((-x**4+4*x**3-8*x**2+8*x)**(3/2),x)
 

output

Integral((-x**4 + 4*x**3 - 8*x**2 + 8*x)**(3/2), x)
 

3.8.64.7 Maxima [F]

\[ \int \left (8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x\right )}^{\frac {3}{2}} \,d x } \]

input

integrate((-x^4+4*x^3-8*x^2+8*x)^(3/2),x, algorithm="maxima")
 

output

integrate((-x^4 + 4*x^3 - 8*x^2 + 8*x)^(3/2), x)
 

3.8.64.8 Giac [F]

\[ \int \left (8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + 8 \, x\right )}^{\frac {3}{2}} \,d x } \]

input

integrate((-x^4+4*x^3-8*x^2+8*x)^(3/2),x, algorithm="giac")
 

output

integrate((-x^4 + 4*x^3 - 8*x^2 + 8*x)^(3/2), x)
 

3.8.64.9 Mupad [F(-1)]

Timed out. \[ \int \left (8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int {\left (-x^4+4\,x^3-8\,x^2+8\,x\right )}^{3/2} \,d x \]

input

int((8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x)
 

output

int((8*x - 8*x^2 + 4*x^3 - x^4)^(3/2), x)