\(\int (A+B x+C x^2) \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3} \, dx\) [75]

Optimal result

Integrand size = 43, antiderivative size = 509 \[ \int \left (A+B x+C x^2\right ) \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3} \, dx=\frac {384 (B c-2 b C) \sqrt {-64+(b+c x)^3}}{7 c^3 \left (4-4 \sqrt {3}-b-c x\right )}-\frac {2 \left (\frac {7 \left (b B c-A c^2-b^2 C\right ) (b+c x)}{c}-\frac {5 (B c-2 b C) (b+c x)^2}{c}\right ) \sqrt {-64+(b+c x)^3}}{35 c^2}+\frac {2 C \left (-64+(b+c x)^3\right )^{3/2}}{9 c^3}-\frac {384 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (B c-2 b C) (4-b-c x) \sqrt {\frac {16+4 (b+c x)+(b+c x)^2}{\left (4-4 \sqrt {3}-b-c x\right )^2}} E\left (\arcsin \left (\frac {4+4 \sqrt {3}-b-c x}{4-4 \sqrt {3}-b-c x}\right )|-7+4 \sqrt {3}\right )}{7 c^3 \sqrt {-\frac {4-b-c x}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \sqrt {-64+(b+c x)^3}}+\frac {64\ 3^{3/4} \sqrt {2-\sqrt {3}} \left (20 \left (1+\sqrt {3}\right ) (B c-2 b C)-7 \left (b B c-A c^2-b^2 C\right )\right ) (4-b-c x) \sqrt {\frac {16+4 (b+c x)+(b+c x)^2}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {4+4 \sqrt {3}-b-c x}{4-4 \sqrt {3}-b-c x}\right ),-7+4 \sqrt {3}\right )}{35 c^3 \sqrt {-\frac {4-b-c x}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \sqrt {-64+(b+c x)^3}} \] Output:

384/7*(B*c-2*C*b)*(-64+(c*x+b)^3)^(1/2)/c^3/(4-4*3^(1/2)-b-c*x)-2/35*(7*(- 
A*c^2+B*b*c-C*b^2)*(c*x+b)/c-5*(B*c-2*C*b)*(c*x+b)^2/c)*(-64+(c*x+b)^3)^(1 
/2)/c^2+2/9*C*(-64+(c*x+b)^3)^(3/2)/c^3-384/7*3^(1/4)*(1/2*6^(1/2)+1/2*2^( 
1/2))*(B*c-2*C*b)*(-c*x-b+4)*((16+4*c*x+4*b+(c*x+b)^2)/(4-4*3^(1/2)-b-c*x) 
^2)^(1/2)*EllipticE((4+4*3^(1/2)-b-c*x)/(4-4*3^(1/2)-b-c*x),2*I-I*3^(1/2)) 
/c^3/(-(-c*x-b+4)/(4-4*3^(1/2)-b-c*x)^2)^(1/2)/(-64+(c*x+b)^3)^(1/2)+64/35 
*3^(3/4)*(1/2*6^(1/2)-1/2*2^(1/2))*(20*(1+3^(1/2))*(B*c-2*C*b)+7*A*c^2-7*B 
*b*c+7*C*b^2)*(-c*x-b+4)*((16+4*c*x+4*b+(c*x+b)^2)/(4-4*3^(1/2)-b-c*x)^2)^ 
(1/2)*EllipticF((4+4*3^(1/2)-b-c*x)/(4-4*3^(1/2)-b-c*x),2*I-I*3^(1/2))/c^3 
/(-(-c*x-b+4)/(4-4*3^(1/2)-b-c*x)^2)^(1/2)/(-64+(c*x+b)^3)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.73 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.69 \[ \int \left (A+B x+C x^2\right ) \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3} \, dx=\frac {2 \left (\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right ) \left (8 b^3 C+9 c^3 x (7 A+5 B x)-6 b^2 c (3 B+2 C x)+35 C \left (-64+c^3 x^3\right )+3 b c^2 (21 A+x (9 B+5 C x))\right )+864 \sqrt {2} \sqrt {-\frac {i (-4+b+c x)}{3 i+\sqrt {3}}} \sqrt {16+b^2+4 c x+c^2 x^2+2 b (2+c x)} \left (-10 \left (3 i+\sqrt {3}\right ) (B c-2 b C) E\left (\arcsin \left (\frac {\sqrt {2 i+2 \sqrt {3}+i b+i c x}}{2 \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+i \left ((20-7 b) B c+7 A c^2+b (-40+7 b) C\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2 i+2 \sqrt {3}+i b+i c x}}{2 \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )\right )}{315 c^3 \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3}} \] Input:

Integrate[(A + B*x + C*x^2)*Sqrt[-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3 
*x^3],x]
 

Output:

(2*((-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3)*(8*b^3*C + 9*c^3*x*(7* 
A + 5*B*x) - 6*b^2*c*(3*B + 2*C*x) + 35*C*(-64 + c^3*x^3) + 3*b*c^2*(21*A 
+ x*(9*B + 5*C*x))) + 864*Sqrt[2]*Sqrt[((-I)*(-4 + b + c*x))/(3*I + Sqrt[3 
])]*Sqrt[16 + b^2 + 4*c*x + c^2*x^2 + 2*b*(2 + c*x)]*(-10*(3*I + Sqrt[3])* 
(B*c - 2*b*C)*EllipticE[ArcSin[Sqrt[2*I + 2*Sqrt[3] + I*b + I*c*x]/(2*3^(1 
/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + I*((20 - 7*b)*B*c + 7*A*c^2 + b*(-40 
 + 7*b)*C)*EllipticF[ArcSin[Sqrt[2*I + 2*Sqrt[3] + I*b + I*c*x]/(2*3^(1/4) 
)], (2*Sqrt[3])/(3*I + Sqrt[3])])))/(315*c^3*Sqrt[-64 + b^3 + 3*b^2*c*x + 
3*b*c^2*x^2 + c^3*x^3])
 

Rubi [A] (warning: unable to verify)

Time = 1.52 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.23, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {2459, 2392, 27, 2425, 793, 2419, 760, 2418}

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.

\(\displaystyle \int \sqrt {b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3-64} \left (A+B x+C x^2\right ) \, dx\)

\(\Big \downarrow \) 2459

\(\displaystyle \int \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )+C \left (\frac {b}{c}+x\right )^2\right )d\left (\frac {b}{c}+x\right )\)

\(\Big \downarrow \) 2392

\(\displaystyle \frac {2}{315} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (63 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+45 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+35 C \left (\frac {b}{c}+x\right )^3\right )-96 \int \frac {2 \left (35 C \left (\frac {b}{c}+x\right )^2+45 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )+63 \left (A-\frac {b (B c-b C)}{c^2}\right )\right )}{315 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{315} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (63 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+45 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+35 C \left (\frac {b}{c}+x\right )^3\right )-\frac {64}{105} \int \frac {35 C \left (\frac {b}{c}+x\right )^2+45 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )+63 \left (A-\frac {b (B c-b C)}{c^2}\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )\)

\(\Big \downarrow \) 2425

\(\displaystyle \frac {2}{315} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (63 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+45 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+35 C \left (\frac {b}{c}+x\right )^3\right )-\frac {64}{105} \left (\int \frac {63 \left (A-\frac {b (B c-b C)}{c^2}\right )+45 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )+35 C \int \frac {\left (\frac {b}{c}+x\right )^2}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )\right )\)

\(\Big \downarrow \) 793

\(\displaystyle \frac {2}{315} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (63 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+45 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+35 C \left (\frac {b}{c}+x\right )^3\right )-\frac {64}{105} \left (\int \frac {63 \left (A-\frac {b (B c-b C)}{c^2}\right )+45 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )+\frac {70 C \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}{3 c^3}\right )\)

\(\Big \downarrow \) 2419

\(\displaystyle \frac {2}{315} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (63 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+45 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+35 C \left (\frac {b}{c}+x\right )^3\right )-\frac {64}{105} \left (\frac {9 \left (7 A c^2+\left (20 \left (1+\sqrt {3}\right )-7 b\right ) B c-\left (40 \left (1+\sqrt {3}\right )-7 b\right ) b C\right ) \int \frac {1}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )}{c^2}-\frac {45 (B c-2 b C) \int \frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )}{c^2}+\frac {70 C \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}{3 c^3}\right )\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {2}{315} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (63 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+45 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+35 C \left (\frac {b}{c}+x\right )^3\right )-\frac {64}{105} \left (-\frac {45 (B c-2 b C) \int \frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )}{c^2}-\frac {3\ 3^{3/4} \sqrt {2-\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \left (7 A c^2+\left (20 \left (1+\sqrt {3}\right )-7 b\right ) B c-\left (40 \left (1+\sqrt {3}\right )-7 b\right ) b C\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right ),-7+4 \sqrt {3}\right )}{c^3 \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}+\frac {70 C \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}{3 c^3}\right )\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {2}{315} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (63 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}\right )+45 \left (\frac {b}{c}+x\right )^2 \left (B-\frac {2 b C}{c}\right )+35 C \left (\frac {b}{c}+x\right )^3\right )-\frac {64}{105} \left (-\frac {3\ 3^{3/4} \sqrt {2-\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \left (7 A c^2+\left (20 \left (1+\sqrt {3}\right )-7 b\right ) B c-\left (40 \left (1+\sqrt {3}\right )-7 b\right ) b C\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right ),-7+4 \sqrt {3}\right )}{c^3 \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}-\frac {45 (B c-2 b C) \left (\frac {2 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}{c \left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )}-\frac {2 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} E\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right )|-7+4 \sqrt {3}\right )}{c \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}\right )}{c^2}+\frac {70 C \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}{3 c^3}\right )\)

Input:

Int[(A + B*x + C*x^2)*Sqrt[-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3], 
x]
 

Output:

(2*Sqrt[-64 + c^3*(b/c + x)^3]*(63*(A - (b*(B*c - b*C))/c^2)*(b/c + x) + 4 
5*(B - (2*b*C)/c)*(b/c + x)^2 + 35*C*(b/c + x)^3))/315 - (64*((70*C*Sqrt[- 
64 + c^3*(b/c + x)^3])/(3*c^3) - (45*(B*c - 2*b*C)*((2*Sqrt[-64 + c^3*(b/c 
 + x)^3])/(c*(4*(1 - Sqrt[3]) - c*(b/c + x))) - (2*3^(1/4)*Sqrt[2 + Sqrt[3 
]]*(4 - c*(b/c + x))*Sqrt[(16 + 4*c*(b/c + x) + c^2*(b/c + x)^2)/(4*(1 - S 
qrt[3]) - c*(b/c + x))^2]*EllipticE[ArcSin[(4*(1 + Sqrt[3]) - c*(b/c + x)) 
/(4*(1 - Sqrt[3]) - c*(b/c + x))], -7 + 4*Sqrt[3]])/(c*Sqrt[-((4 - c*(b/c 
+ x))/(4*(1 - Sqrt[3]) - c*(b/c + x))^2)]*Sqrt[-64 + c^3*(b/c + x)^3])))/c 
^2 - (3*3^(3/4)*Sqrt[2 - Sqrt[3]]*((20*(1 + Sqrt[3]) - 7*b)*B*c + 7*A*c^2 
- (40*(1 + Sqrt[3]) - 7*b)*b*C)*(4 - c*(b/c + x))*Sqrt[(16 + 4*c*(b/c + x) 
 + c^2*(b/c + x)^2)/(4*(1 - Sqrt[3]) - c*(b/c + x))^2]*EllipticF[ArcSin[(4 
*(1 + Sqrt[3]) - c*(b/c + x))/(4*(1 - Sqrt[3]) - c*(b/c + x))], -7 + 4*Sqr 
t[3]])/(c^3*Sqrt[-((4 - c*(b/c + x))/(4*(1 - Sqrt[3]) - c*(b/c + x))^2)]*S 
qrt[-64 + c^3*(b/c + x)^3])))/105
 

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 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 793

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n) 
^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && 
 NeQ[p, -1]
 

rule 2392

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Module[{q = Expon[Pq 
, x], i}, Simp[(a + b*x^n)^p*Sum[Coeff[Pq, x, i]*(x^(i + 1)/(n*p + i + 1)), 
 {i, 0, q}], x] + Simp[a*n*p   Int[(a + b*x^n)^(p - 1)*Sum[Coeff[Pq, x, i]* 
(x^i/(n*p + i + 1)), {i, 0, q}], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x 
] && IGtQ[(n - 1)/2, 0] && GtQ[p, 0]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

rule 2419

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(c*r - (1 + Sqrt[3])*d*s)/r 
  Int[1/Sqrt[a + b*x^3], x], x] + Simp[d/r   Int[((1 + Sqrt[3])*s + r*x)/Sq 
rt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && NeQ[b*c^3 - 
2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

rule 2425

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Coeff[Pq, x, n - 
 1]   Int[x^(n - 1)*(a + b*x^n)^p, x], x] + Int[ExpandToSum[Pq - Coeff[Pq, 
x, n - 1]*x^(n - 1), x]*(a + b*x^n)^p, x] /; FreeQ[{a, b, p}, x] && PolyQ[P 
q, x] && IGtQ[n, 0] && Expon[Pq, x] == n - 1
 

rule 2459

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])
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1352 vs. \(2 (451 ) = 902\).

Time = 2.69 (sec) , antiderivative size = 1353, normalized size of antiderivative = 2.66

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1353\)
risch	\(\text {Expression too large to display}\)	\(1397\)
default	\(\text {Expression too large to display}\)	\(2104\)

Input:

int((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2),x,method=_R 
ETURNVERBOSE)
 

Output:

2/9*C*x^3*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2)+2/7*(B*c^3+1/3*C*b* 
c^2)/c^3*x^2*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2)+2/5*(A*c^3+3*B*b 
*c^2+2/3*C*b^2*c-18/7*b/c*(B*c^3+1/3*C*b*c^2))/c^3*x*(c^3*x^3+3*b*c^2*x^2+ 
3*b^2*c*x+b^3-64)^(1/2)+2/3*(3*c^2*b*A+3*B*b^2*c+C*b^3-64*C-2/9*C*(3*b^3-1 
92)-15/7*b^2/c^2*(B*c^3+1/3*C*b*c^2)-12/5*b/c*(A*c^3+3*B*b*c^2+2/3*C*b^2*c 
-18/7*b/c*(B*c^3+1/3*C*b*c^2)))/c^3*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64) 
^(1/2)+2*(A*b^3-64*A-2/5*(b^3-64)/c^3*(A*c^3+3*B*b*c^2+2/3*C*b^2*c-18/7*b/ 
c*(B*c^3+1/3*C*b*c^2))-(3*c^2*b*A+3*B*b^2*c+C*b^3-64*C-2/9*C*(3*b^3-192)-1 
5/7*b^2/c^2*(B*c^3+1/3*C*b*c^2)-12/5*b/c*(A*c^3+3*B*b*c^2+2/3*C*b^2*c-18/7 
*b/c*(B*c^3+1/3*C*b*c^2)))/c^2*b^2)*((-b-2-2*I*3^(1/2))/c+(b-4)/c)*((x+(b- 
4)/c)/((-b-2-2*I*3^(1/2))/c+(b-4)/c))^(1/2)*((x-(-b-2+2*I*3^(1/2))/c)/(-(b 
-4)/c-(-b-2+2*I*3^(1/2))/c))^(1/2)*((x-(-b-2-2*I*3^(1/2))/c)/(-(b-4)/c-(-b 
-2-2*I*3^(1/2))/c))^(1/2)/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2)*Ell 
ipticF(((x+(b-4)/c)/((-b-2-2*I*3^(1/2))/c+(b-4)/c))^(1/2),((-(b-4)/c-(-b-2 
-2*I*3^(1/2))/c)/(-(b-4)/c-(-b-2+2*I*3^(1/2))/c))^(1/2))+2*(3*A*b^2*c+B*b^ 
3-64*B-2/7*(B*c^3+1/3*C*b*c^2)/c^3*(2*b^3-128)-9/5*b^2/c^2*(A*c^3+3*B*b*c^ 
2+2/3*C*b^2*c-18/7*b/c*(B*c^3+1/3*C*b*c^2))-2*(3*c^2*b*A+3*B*b^2*c+C*b^3-6 
4*C-2/9*C*(3*b^3-192)-15/7*b^2/c^2*(B*c^3+1/3*C*b*c^2)-12/5*b/c*(A*c^3+3*B 
*b*c^2+2/3*C*b^2*c-18/7*b/c*(B*c^3+1/3*C*b*c^2)))/c*b)*((-b-2-2*I*3^(1/2)) 
/c+(b-4)/c)*((x+(b-4)/c)/((-b-2-2*I*3^(1/2))/c+(b-4)/c))^(1/2)*((x-(-b-...
                                                                                    
                                                                                    
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.40 \[ \int \left (A+B x+C x^2\right ) \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3} \, dx=-\frac {2 \, {\left (12096 \, {\left (C b^{2} - B b c + A c^{2}\right )} \sqrt {c^{3}} {\rm weierstrassPInverse}\left (0, \frac {256}{c^{3}}, \frac {c x + b}{c}\right ) + 8640 \, {\left (2 \, C b c - B c^{2}\right )} \sqrt {c^{3}} {\rm weierstrassZeta}\left (0, \frac {256}{c^{3}}, {\rm weierstrassPInverse}\left (0, \frac {256}{c^{3}}, \frac {c x + b}{c}\right )\right ) - {\left (35 \, C c^{5} x^{3} - 18 \, B b^{2} c^{3} + 63 \, A b c^{4} + 8 \, {\left (C b^{3} - 280 \, C\right )} c^{2} + 15 \, {\left (C b c^{4} + 3 \, B c^{5}\right )} x^{2} - 3 \, {\left (4 \, C b^{2} c^{3} - 9 \, B b c^{4} - 21 \, A c^{5}\right )} x\right )} \sqrt {c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64}\right )}}{315 \, c^{5}} \] Input:

integrate((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2),x, al 
gorithm="fricas")
 

Output:

-2/315*(12096*(C*b^2 - B*b*c + A*c^2)*sqrt(c^3)*weierstrassPInverse(0, 256 
/c^3, (c*x + b)/c) + 8640*(2*C*b*c - B*c^2)*sqrt(c^3)*weierstrassZeta(0, 2 
56/c^3, weierstrassPInverse(0, 256/c^3, (c*x + b)/c)) - (35*C*c^5*x^3 - 18 
*B*b^2*c^3 + 63*A*b*c^4 + 8*(C*b^3 - 280*C)*c^2 + 15*(C*b*c^4 + 3*B*c^5)*x 
^2 - 3*(4*C*b^2*c^3 - 9*B*b*c^4 - 21*A*c^5)*x)*sqrt(c^3*x^3 + 3*b*c^2*x^2 
+ 3*b^2*c*x + b^3 - 64))/c^5
 

Sympy [F]

\[ \int \left (A+B x+C x^2\right ) \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3} \, dx=\int \sqrt {\left (b + c x - 4\right ) \left (b^{2} + 2 b c x + 4 b + c^{2} x^{2} + 4 c x + 16\right )} \left (A + B x + C x^{2}\right )\, dx \] Input:

integrate((C*x**2+B*x+A)*(c**3*x**3+3*b*c**2*x**2+3*b**2*c*x+b**3-64)**(1/ 
2),x)
 

Output:

Integral(sqrt((b + c*x - 4)*(b**2 + 2*b*c*x + 4*b + c**2*x**2 + 4*c*x + 16 
))*(A + B*x + C*x**2), x)
 

Maxima [F]

\[ \int \left (A+B x+C x^2\right ) \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3} \, dx=\int { \sqrt {c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64} {\left (C x^{2} + B x + A\right )} \,d x } \] Input:

integrate((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2),x, al 
gorithm="maxima")
 

Output:

integrate(sqrt(c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64)*(C*x^2 + B*x 
+ A), x)
 

Giac [F]

\[ \int \left (A+B x+C x^2\right ) \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3} \, dx=\int { \sqrt {c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64} {\left (C x^{2} + B x + A\right )} \,d x } \] Input:

integrate((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2),x, al 
gorithm="giac")
 

Output:

integrate(sqrt(c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64)*(C*x^2 + B*x 
+ A), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (A+B x+C x^2\right ) \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3} \, dx=\int \left (C\,x^2+B\,x+A\right )\,\sqrt {b^3+3\,b^2\,c\,x+3\,b\,c^2\,x^2+c^3\,x^3-64} \,d x \] Input:

int((A + B*x + C*x^2)*(b^3 + c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x - 64)^(1/2) 
,x)
 

Output:

int((A + B*x + C*x^2)*(b^3 + c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x - 64)^(1/2) 
, x)
 

Reduce [F]

\[ \int \left (A+B x+C x^2\right ) \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3} \, dx=\frac {\frac {2 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, a b c}{5}+\frac {2 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, a \,c^{2} x}{5}-\frac {4 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{3}}{63}+\frac {2 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{2} c x}{21}+\frac {8 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b \,c^{2} x^{2}}{21}+\frac {2 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, c^{3} x^{3}}{9}-\frac {320 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}{63}-\frac {192 \left (\int \frac {\sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}{c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}d x \right ) a \,c^{2}}{5}+\frac {96 \left (\int \frac {\sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}{c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}d x \right ) b^{2} c}{7}-\frac {96 \left (\int \frac {\sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, x^{2}}{c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}d x \right ) c^{3}}{7}}{c^{2}} \] Input:

int((C*x^2+B*x+A)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2),x)
 

Output:

(2*(63*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*a*b*c + 63 
*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*a*c**2*x - 10*sq 
rt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**3 + 15*sqrt(b**3 
 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**2*c*x + 60*sqrt(b**3 + 
3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b*c**2*x**2 + 35*sqrt(b**3 + 
3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*c**3*x**3 - 800*sqrt(b**3 + 3 
*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64) - 6048*int(sqrt(b**3 + 3*b**2* 
c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)/(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + 
 c**3*x**3 - 64),x)*a*c**2 + 2160*int(sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x* 
*2 + c**3*x**3 - 64)/(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64), 
x)*b**2*c - 2160*int((sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 
 64)*x**2)/(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64),x)*c**3))/ 
(315*c**2)