Integrand size = 43, antiderivative size = 486 \[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2}} \, dx=-\frac {2 C}{3 c^3 \sqrt {-64+(b+c x)^3}}-\frac {(b+c x) \left (A c^2-b^2 C+c (B c-2 b C) x\right )}{96 c^3 \sqrt {-64+(b+c x)^3}}-\frac {(B c-2 b C) \sqrt {-64+(b+c x)^3}}{96 c^3 \left (4-4 \sqrt {3}-b-c x\right )}+\frac {\sqrt {2+\sqrt {3}} (B c-2 b C) (4-b-c x) \sqrt {\frac {16+4 (b+c x)+(b+c x)^2}{\left (4-4 \sqrt {3}-b-c x\right )^2}} E\left (\arcsin \left (\frac {4+4 \sqrt {3}-b-c x}{4-4 \sqrt {3}-b-c x}\right )|-7+4 \sqrt {3}\right )}{32\ 3^{3/4} c^3 \sqrt {-\frac {4-b-c x}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \sqrt {-64+(b+c x)^3}}-\frac {\sqrt {2-\sqrt {3}} \left (\left (4+4 \sqrt {3}+b\right ) B c-A c^2-b \left (8+8 \sqrt {3}+b\right ) C\right ) (4-b-c x) \sqrt {\frac {16+4 (b+c x)+(b+c x)^2}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {4+4 \sqrt {3}-b-c x}{4-4 \sqrt {3}-b-c x}\right ),-7+4 \sqrt {3}\right )}{192 \sqrt [4]{3} c^3 \sqrt {-\frac {4-b-c x}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \sqrt {-64+(b+c x)^3}} \] Output:
-2/3*C/c^3/(-64+(c*x+b)^3)^(1/2)-1/96*(c*x+b)*(A*c^2-C*b^2+c*(B*c-2*C*b)*x )/c^3/(-64+(c*x+b)^3)^(1/2)-1/96*(B*c-2*C*b)*(-64+(c*x+b)^3)^(1/2)/c^3/(4- 4*3^(1/2)-b-c*x)+1/96*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)-1/576*(1/2*6^(1/2)-1/2*2^(1/2 ))*((4+4*3^(1/2)+b)*B*c-A*c^2-b*(8+8*3^(1/2)+b)*C)*(-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))*3^(3/4)/c^3/(-(-c*x-b+4)/(4-4*3^(1/2)-b- c*x)^2)^(1/2)/(-64+(c*x+b)^3)^(1/2)
Result contains complex when optimal does not.
Time = 11.13 (sec) , antiderivative size = 928, normalized size of antiderivative = 1.91 \[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2}} \, dx=\frac {-2 \sqrt {16+b^2+4 c x+c^2 x^2+2 b (2+c x)} \left (A c^2 (b+c x)+B c^2 x (b+c x)-C \left (-64+b^3+3 b^2 c x+2 b c^2 x^2\right )\right )+2 \sqrt {2} b B c \left (-2 i+2 \sqrt {3}-i b-i c x\right ) \sqrt {-\frac {i (-4+b+c x)}{3 i+\sqrt {3}}} \left (2-2 i \sqrt {3}+b+c x\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 )+\sqrt {2} A c^2 \left (2 i+2 \sqrt {3}+i b+i c x\right ) \sqrt {-\frac {i (-4+b+c x)}{3 i+\sqrt {3}}} \left (2+2 i \sqrt {3}+b+c x\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 )+3 \sqrt {2} b^2 C \left (2 i+2 \sqrt {3}+i b+i c x\right ) \sqrt {-\frac {i (-4+b+c x)}{3 i+\sqrt {3}}} \left (2+2 i \sqrt {3}+b+c x\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 )-\sqrt {2} B c \left (-2-2 i \sqrt {3}-b-c x\right ) \left (2 i+2 \sqrt {3}+i b+i c x\right ) \sqrt {-\frac {i (-4+b+c x)}{3 i+\sqrt {3}}} \left (\left (6-2 i \sqrt {3}\right ) 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 )+(-4+b) \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 )+2 \sqrt {2} b C \left (-2-2 i \sqrt {3}-b-c x\right ) \left (2 i+2 \sqrt {3}+i b+i c x\right ) \sqrt {-\frac {i (-4+b+c x)}{3 i+\sqrt {3}}} \left (\left (6-2 i \sqrt {3}\right ) 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 )+(-4+b) \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 )}{192 c^3 \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3} \sqrt {16+b^2+4 c x+c^2 x^2+2 b (2+c x)}} \] Input:
Integrate[(A + B*x + C*x^2)/(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3 )^(3/2),x]
Output:
(-2*Sqrt[16 + b^2 + 4*c*x + c^2*x^2 + 2*b*(2 + c*x)]*(A*c^2*(b + c*x) + B* c^2*x*(b + c*x) - C*(-64 + b^3 + 3*b^2*c*x + 2*b*c^2*x^2)) + 2*Sqrt[2]*b*B *c*(-2*I + 2*Sqrt[3] - I*b - I*c*x)*Sqrt[((-I)*(-4 + b + c*x))/(3*I + Sqrt [3])]*(2 - (2*I)*Sqrt[3] + b + c*x)*EllipticF[ArcSin[Sqrt[2*I + 2*Sqrt[3] + I*b + I*c*x]/(2*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + Sqrt[2]*A*c^2* (2*I + 2*Sqrt[3] + I*b + I*c*x)*Sqrt[((-I)*(-4 + b + c*x))/(3*I + Sqrt[3]) ]*(2 + (2*I)*Sqrt[3] + b + c*x)*EllipticF[ArcSin[Sqrt[2*I + 2*Sqrt[3] + I* b + I*c*x]/(2*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + 3*Sqrt[2]*b^2*C*(2 *I + 2*Sqrt[3] + I*b + I*c*x)*Sqrt[((-I)*(-4 + b + c*x))/(3*I + Sqrt[3])]* (2 + (2*I)*Sqrt[3] + b + c*x)*EllipticF[ArcSin[Sqrt[2*I + 2*Sqrt[3] + I*b + I*c*x]/(2*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] - Sqrt[2]*B*c*(-2 - (2 *I)*Sqrt[3] - b - c*x)*(2*I + 2*Sqrt[3] + I*b + I*c*x)*Sqrt[((-I)*(-4 + b + c*x))/(3*I + Sqrt[3])]*((6 - (2*I)*Sqrt[3])*EllipticE[ArcSin[Sqrt[2*I + 2*Sqrt[3] + I*b + I*c*x]/(2*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + (-4 + b)*EllipticF[ArcSin[Sqrt[2*I + 2*Sqrt[3] + I*b + I*c*x]/(2*3^(1/4))], (2 *Sqrt[3])/(3*I + Sqrt[3])]) + 2*Sqrt[2]*b*C*(-2 - (2*I)*Sqrt[3] - b - c*x) *(2*I + 2*Sqrt[3] + I*b + I*c*x)*Sqrt[((-I)*(-4 + b + c*x))/(3*I + Sqrt[3] )]*((6 - (2*I)*Sqrt[3])*EllipticE[ArcSin[Sqrt[2*I + 2*Sqrt[3] + I*b + I*c* x]/(2*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + (-4 + b)*EllipticF[ArcSin[ Sqrt[2*I + 2*Sqrt[3] + I*b + I*c*x]/(2*3^(1/4))], (2*Sqrt[3])/(3*I + Sq...
Time = 1.29 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.21, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {2459, 2393, 27, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{\left (b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3-64\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2459 |
| \(\displaystyle \int \frac {A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )+C \left (\frac {b}{c}+x\right )^2}{\left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}d\left (\frac {b}{c}+x\right )\) |
| \(\Big \downarrow \) 2393 |
| \(\displaystyle \frac {1}{96} \int -\frac {A-\frac {b (B c-b C)}{c^2}-\left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )}{2 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{96 c^3 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{192} \int \frac {A-\frac {b (B c-b C)}{c^2}-\left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{96 c^3 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}\) |
| \(\Big \downarrow \) 2419 |
| \(\displaystyle \frac {1}{192} \left (\frac {\left (-A c^2+\left (b+4 \sqrt {3}+4\right ) B c-b \left (b+8 \sqrt {3}+8\right ) C\right ) \int \frac {1}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )}{c^2}-\frac {(B c-2 b C) \int \frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )}{c^2}\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{96 c^3 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {1}{192} \left (-\frac {(B c-2 b C) \int \frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )}{c^2}-\frac {\sqrt {2-\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \left (-A c^2+\left (b+4 \sqrt {3}+4\right ) B c-b \left (b+8 \sqrt {3}+8\right ) C\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} c^3 \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{96 c^3 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {1}{192} \left (-\frac {\sqrt {2-\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \left (-A c^2+\left (b+4 \sqrt {3}+4\right ) B c-b \left (b+8 \sqrt {3}+8\right ) C\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} c^3 \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}-\frac {(B c-2 b C) \left (\frac {2 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}{c \left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )}-\frac {2 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} E\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right )|-7+4 \sqrt {3}\right )}{c \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}\right )}{c^2}\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{96 c^3 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}\) |
Input:
Int[(A + B*x + C*x^2)/(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3)^(3/2 ),x]
Output:
-1/96*(64*C + c^3*(b/c + x)*(A - (b*(B*c - b*C))/c^2 + (B - (2*b*C)/c)*(b/ c + x)))/(c^3*Sqrt[-64 + c^3*(b/c + x)^3]) + (-(((B*c - 2*b*C)*((2*Sqrt[-6 4 + c^3*(b/c + x)^3])/(c*(4*(1 - Sqrt[3]) - c*(b/c + x))) - (2*3^(1/4)*Sqr t[2 + Sqrt[3]]*(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]*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) - (Sqrt[2 - Sqrt[3]]*((4 + 4*Sqrt[3] + b)*B*c - A*c^2 - b* (8 + 8*Sqrt[3] + 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 + Sqr t[3]) - c*(b/c + x))/(4*(1 - Sqrt[3]) - c*(b/c + x))], -7 + 4*Sqrt[3]])/(3 ^(1/4)*c^3*Sqrt[-((4 - c*(b/c + x))/(4*(1 - Sqrt[3]) - c*(b/c + x))^2)]*Sq rt[-64 + c^3*(b/c + x)^3]))/192
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[(a*Coeff[Pq, x, q] - b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q , x])*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) In t[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^( p + 1), x], x] /; q == n - 1] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n , 0] && LtQ[p, -1]
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]
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]
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])
Time = 1.69 (sec) , antiderivative size = 851, normalized size of antiderivative = 1.75
| method | result | size |
| elliptic | \(-\frac {2 c^{3} \left (\frac {\left (B c -2 b C \right ) x^{2}}{192 c^{4}}+\frac {\left (A \,c^{2}+b B c -3 b^{2} C \right ) x}{192 c^{5}}+\frac {c^{2} b A -C \,b^{3}+64 C}{192 c^{6}}\right )}{\sqrt {\left (x^{3}+\frac {3 b \,x^{2}}{c}+\frac {3 b^{2} x}{c^{2}}+\frac {b^{3}-64}{c^{3}}\right ) c^{3}}}+\frac {2 \left (-\frac {A \,c^{2}-b^{2} C}{64 c^{2}}+\frac {A \,c^{2}+b B c -3 b^{2} C}{96 c^{2}}\right ) \left (\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}\right ) \sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}\, \sqrt {\frac {x -\frac {-b -2+2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\, \sqrt {\frac {x -\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}, \sqrt {\frac {-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\right )}{\sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}+\frac {\left (B c -2 b C \right ) \left (\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}\right ) \sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}\, \sqrt {\frac {x -\frac {-b -2+2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\, \sqrt {\frac {x -\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}}\, \left (\left (-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}, \sqrt {\frac {-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\right )+\frac {\left (-b -2+2 i \sqrt {3}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}, \sqrt {\frac {-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\right )}{c}\right )}{96 c \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}\) | \(851\) |
| default | \(\text {Expression too large to display}\) | \(1884\) |
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)^(3/2),x,method=_R ETURNVERBOSE)
Output:
-2*c^3*(1/192*(B*c-2*C*b)/c^4*x^2+1/192*(A*c^2+B*b*c-3*C*b^2)/c^5*x+1/192* (A*b*c^2-C*b^3+64*C)/c^6)/((x^3+3*b/c*x^2+3*b^2/c^2*x+(b^3-64)/c^3)*c^3)^( 1/2)+2*(-1/64*(A*c^2-C*b^2)/c^2+1/96/c^2*(A*c^2+B*b*c-3*C*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)*EllipticF(((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))+1/96/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-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)*((-(b-4)/ c-(-b-2+2*I*3^(1/2))/c)*EllipticE(((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))+(-b-2+2*I*3^(1/2))/c*EllipticF(((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)))
Time = 0.09 (sec) , antiderivative size = 398, normalized size of antiderivative = 0.82 \[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2}} \, dx=-\frac {{\left (C b^{5} + {\left (C b^{2} c^{3} - B b c^{4} + A c^{5}\right )} x^{3} - 64 \, C b^{2} + {\left (A b^{3} - 64 \, A\right )} c^{2} + 3 \, {\left (C b^{3} c^{2} - B b^{2} c^{3} + A b c^{4}\right )} x^{2} - {\left (B b^{4} - 64 \, B b\right )} c + 3 \, {\left (C b^{4} c - B b^{3} c^{2} + A b^{2} c^{3}\right )} x\right )} \sqrt {c^{3}} {\rm weierstrassPInverse}\left (0, \frac {256}{c^{3}}, \frac {c x + b}{c}\right ) - {\left ({\left (2 \, C b c^{4} - B c^{5}\right )} x^{3} - {\left (B b^{3} - 64 \, B\right )} c^{2} + 3 \, {\left (2 \, C b^{2} c^{3} - B b c^{4}\right )} x^{2} + 2 \, {\left (C b^{4} - 64 \, C b\right )} c + 3 \, {\left (2 \, C b^{3} c^{2} - B b^{2} c^{3}\right )} x\right )} \sqrt {c^{3}} {\rm weierstrassZeta}\left (0, \frac {256}{c^{3}}, {\rm weierstrassPInverse}\left (0, \frac {256}{c^{3}}, \frac {c x + b}{c}\right )\right ) + {\left (A b c^{4} - {\left (C b^{3} - 64 \, C\right )} c^{2} - {\left (2 \, C b c^{4} - B c^{5}\right )} x^{2} - {\left (3 \, C b^{2} c^{3} - B b c^{4} - A c^{5}\right )} x\right )} \sqrt {c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64}}{96 \, {\left (c^{8} x^{3} + 3 \, b c^{7} x^{2} + 3 \, b^{2} c^{6} x + {\left (b^{3} - 64\right )} c^{5}\right )}} \] 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)^(3/2),x, al gorithm="fricas")
Output:
-1/96*((C*b^5 + (C*b^2*c^3 - B*b*c^4 + A*c^5)*x^3 - 64*C*b^2 + (A*b^3 - 64 *A)*c^2 + 3*(C*b^3*c^2 - B*b^2*c^3 + A*b*c^4)*x^2 - (B*b^4 - 64*B*b)*c + 3 *(C*b^4*c - B*b^3*c^2 + A*b^2*c^3)*x)*sqrt(c^3)*weierstrassPInverse(0, 256 /c^3, (c*x + b)/c) - ((2*C*b*c^4 - B*c^5)*x^3 - (B*b^3 - 64*B)*c^2 + 3*(2* C*b^2*c^3 - B*b*c^4)*x^2 + 2*(C*b^4 - 64*C*b)*c + 3*(2*C*b^3*c^2 - B*b^2*c ^3)*x)*sqrt(c^3)*weierstrassZeta(0, 256/c^3, weierstrassPInverse(0, 256/c^ 3, (c*x + b)/c)) + (A*b*c^4 - (C*b^3 - 64*C)*c^2 - (2*C*b*c^4 - B*c^5)*x^2 - (3*C*b^2*c^3 - B*b*c^4 - A*c^5)*x)*sqrt(c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c *x + b^3 - 64))/(c^8*x^3 + 3*b*c^7*x^2 + 3*b^2*c^6*x + (b^3 - 64)*c^5)
\[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2}} \, dx=\int \frac {A + B x + C x^{2}}{\left (\left (b + c x - 4\right ) \left (b^{2} + 2 b c x + 4 b + c^{2} x^{2} + 4 c x + 16\right )\right )^{\frac {3}{2}}}\, dx \] 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)**(3/ 2),x)
Output:
Integral((A + B*x + C*x**2)/((b + c*x - 4)*(b**2 + 2*b*c*x + 4*b + c**2*x* *2 + 4*c*x + 16))**(3/2), x)
\[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{\frac {3}{2}}} \,d x } \] 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)^(3/2),x, al gorithm="maxima")
Output:
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) ^(3/2), x)
\[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{\frac {3}{2}}} \,d x } \] 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)^(3/2),x, al gorithm="giac")
Output:
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) ^(3/2), x)
Timed out. \[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2}} \, dx=\int \frac {C\,x^2+B\,x+A}{{\left (b^3+3\,b^2\,c\,x+3\,b\,c^2\,x^2+c^3\,x^3-64\right )}^{3/2}} \,d x \] 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)^(3/2) ,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)^(3/2) , x)
\[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2}} \, dx=\text {too large to display} \] 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)^(3/2),x)
Output:
(12*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*a*c**2*x - 4* sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**3 - 6*sqrt(b** 3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**2*c*x + 256*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64) + 6*int((sqrt(b**3 + 3*b**2 *c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*x**3)/(b**9 + 6*b**8*c*x + 15*b**7* c**2*x**2 + 20*b**6*c**3*x**3 - 192*b**6 + 15*b**5*c**4*x**4 - 768*b**5*c* x + 6*b**4*c**5*x**5 - 1344*b**4*c**2*x**2 + b**3*c**6*x**6 - 1408*b**3*c* *3*x**3 + 12288*b**3 - 960*b**2*c**4*x**4 + 24576*b**2*c*x - 384*b*c**5*x* *5 + 24576*b*c**2*x**2 - 64*c**6*x**6 + 8192*c**3*x**3 - 262144),x)*a*b**6 *c**5 + 18*int((sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*x **3)/(b**9 + 6*b**8*c*x + 15*b**7*c**2*x**2 + 20*b**6*c**3*x**3 - 192*b**6 + 15*b**5*c**4*x**4 - 768*b**5*c*x + 6*b**4*c**5*x**5 - 1344*b**4*c**2*x* *2 + b**3*c**6*x**6 - 1408*b**3*c**3*x**3 + 12288*b**3 - 960*b**2*c**4*x** 4 + 24576*b**2*c*x - 384*b*c**5*x**5 + 24576*b*c**2*x**2 - 64*c**6*x**6 + 8192*c**3*x**3 - 262144),x)*a*b**5*c**6*x + 18*int((sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*x**3)/(b**9 + 6*b**8*c*x + 15*b**7*c**2 *x**2 + 20*b**6*c**3*x**3 - 192*b**6 + 15*b**5*c**4*x**4 - 768*b**5*c*x + 6*b**4*c**5*x**5 - 1344*b**4*c**2*x**2 + b**3*c**6*x**6 - 1408*b**3*c**3*x **3 + 12288*b**3 - 960*b**2*c**4*x**4 + 24576*b**2*c*x - 384*b*c**5*x**5 + 24576*b*c**2*x**2 - 64*c**6*x**6 + 8192*c**3*x**3 - 262144),x)*a*b**4*...