Integrand size = 32, antiderivative size = 276 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\frac {b \left (3 a^2+\frac {3 c^2}{d^4}+\frac {b^2}{d^2}+\frac {6 a c}{d^2}\right ) d^4+\left (c+a d^2\right ) \left (c^2+3 b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^6 \sqrt {1-d^2 x^2}}+\frac {b \left (5 c^2+b^2 d^2+6 a c d^2\right ) \sqrt {1-d^2 x^2}}{d^6}+\frac {c \left (7 c^2+12 b^2 d^2+12 a c d^2\right ) x \sqrt {1-d^2 x^2}}{8 d^6}+\frac {b c^2 x^2 \sqrt {1-d^2 x^2}}{d^4}+\frac {c^3 x^3 \sqrt {1-d^2 x^2}}{4 d^4}-\frac {3 \left (5 c^3+12 b^2 c d^2+12 a c^2 d^2+8 a b^2 d^4+8 a^2 c d^4\right ) \arcsin (d x)}{8 d^7} \]
-3/8*(8*a^2*c*d^4+8*a*b^2*d^4+12*a*c^2*d^2+12*b^2*c*d^2+5*c^3)*arcsin(d*x) /d^7+(b*(3*a^2+3*c^2/d^4+b^2/d^2+6*a*c/d^2)*d^4+(a*d^2+c)*(a^2*d^4+2*a*c*d ^2+3*b^2*d^2+c^2)*x)/d^6/(-d^2*x^2+1)^(1/2)+b*(6*a*c*d^2+b^2*d^2+5*c^2)*(- d^2*x^2+1)^(1/2)/d^6+1/8*c*(12*a*c*d^2+12*b^2*d^2+7*c^2)*x*(-d^2*x^2+1)^(1 /2)/d^6+b*c^2*x^2*(-d^2*x^2+1)^(1/2)/d^4+1/4*c^3*x^3*(-d^2*x^2+1)^(1/2)/d^ 4
Time = 0.94 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=-\frac {\frac {d \left (8 b^3 d^2 \left (-2+d^2 x^2\right )+12 b^2 d^2 x \left (-2 a d^2+c \left (-3+d^2 x^2\right )\right )+8 b \left (-3 a^2 d^4+6 a c d^2 \left (-2+d^2 x^2\right )+c^2 \left (-8+4 d^2 x^2+d^4 x^4\right )\right )+x \left (-24 a^2 c d^4-8 a^3 d^6+12 a c^2 d^2 \left (-3+d^2 x^2\right )+c^3 \left (-15+5 d^2 x^2+2 d^4 x^4\right )\right )\right )}{\sqrt {1-d^2 x^2}}+6 \left (5 c^3+12 b^2 c d^2+12 a c^2 d^2+8 a b^2 d^4+8 a^2 c d^4\right ) \arctan \left (\frac {d x}{-1+\sqrt {1-d^2 x^2}}\right )}{8 d^7} \]
-1/8*((d*(8*b^3*d^2*(-2 + d^2*x^2) + 12*b^2*d^2*x*(-2*a*d^2 + c*(-3 + d^2* x^2)) + 8*b*(-3*a^2*d^4 + 6*a*c*d^2*(-2 + d^2*x^2) + c^2*(-8 + 4*d^2*x^2 + d^4*x^4)) + x*(-24*a^2*c*d^4 - 8*a^3*d^6 + 12*a*c^2*d^2*(-3 + d^2*x^2) + c^3*(-15 + 5*d^2*x^2 + 2*d^4*x^4))))/Sqrt[1 - d^2*x^2] + 6*(5*c^3 + 12*b^2 *c*d^2 + 12*a*c^2*d^2 + 8*a*b^2*d^4 + 8*a^2*c*d^4)*ArcTan[(d*x)/(-1 + Sqrt [1 - d^2*x^2])])/d^7
Time = 1.02 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {1188, 2345, 2346, 25, 2346, 27, 2346, 25, 455, 223}
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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{(1-d x)^{3/2} (d x+1)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1188 |
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{\left (1-d^2 x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b d^4 \left (3 a^2+\frac {6 a c}{d^2}+\frac {b^2}{d^2}+\frac {3 c^2}{d^4}\right )}{d^6 \sqrt {1-d^2 x^2}}-\int \frac {\frac {c^3 x^4}{d^2}+\frac {3 b c^2 x^3}{d^2}+\frac {c \left (3 b^2+c \left (3 a+\frac {c}{d^2}\right )\right ) x^2}{d^2}+\frac {b \left (b^2+3 c \left (2 a+\frac {c}{d^2}\right )\right ) x}{d^2}+\frac {3 a b^2 d^4+3 a c^2 d^2+3 c \left (b^2+a^2 d^2\right ) d^2+c^3}{d^6}}{\sqrt {1-d^2 x^2}}dx\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {\int -\frac {12 b c^2 x^3+c \left (12 b^2+c \left (12 a+\frac {7 c}{d^2}\right )\right ) x^2+4 b \left (b^2+3 c \left (2 a+\frac {c}{d^2}\right )\right ) x+\frac {4 \left (3 a b^2 d^4+3 a c^2 d^2+3 c \left (b^2+a^2 d^2\right ) d^2+c^3\right )}{d^4}}{\sqrt {1-d^2 x^2}}dx}{4 d^2}+\frac {x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b d^4 \left (3 a^2+\frac {6 a c}{d^2}+\frac {b^2}{d^2}+\frac {3 c^2}{d^4}\right )}{d^6 \sqrt {1-d^2 x^2}}+\frac {c^3 x^3 \sqrt {1-d^2 x^2}}{4 d^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {12 b c^2 x^3+c \left (12 b^2+c \left (12 a+\frac {7 c}{d^2}\right )\right ) x^2+4 b \left (b^2+3 c \left (2 a+\frac {c}{d^2}\right )\right ) x+\frac {4 \left (3 a b^2 d^4+3 a c^2 d^2+3 c \left (b^2+a^2 d^2\right ) d^2+c^3\right )}{d^4}}{\sqrt {1-d^2 x^2}}dx}{4 d^2}+\frac {x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b d^4 \left (3 a^2+\frac {6 a c}{d^2}+\frac {b^2}{d^2}+\frac {3 c^2}{d^4}\right )}{d^6 \sqrt {1-d^2 x^2}}+\frac {c^3 x^3 \sqrt {1-d^2 x^2}}{4 d^4}\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle -\frac {-\frac {\int -\frac {3 \left (c \left (7 c^2+12 a d^2 c+12 b^2 d^2\right ) x^2+4 b \left (5 c^2+6 a d^2 c+b^2 d^2\right ) x+\frac {4 \left (3 a b^2 d^4+3 a c^2 d^2+3 c \left (b^2+a^2 d^2\right ) d^2+c^3\right )}{d^2}\right )}{\sqrt {1-d^2 x^2}}dx}{3 d^2}-\frac {4 b c^2 x^2 \sqrt {1-d^2 x^2}}{d^2}}{4 d^2}+\frac {x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b d^4 \left (3 a^2+\frac {6 a c}{d^2}+\frac {b^2}{d^2}+\frac {3 c^2}{d^4}\right )}{d^6 \sqrt {1-d^2 x^2}}+\frac {c^3 x^3 \sqrt {1-d^2 x^2}}{4 d^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {c \left (7 c^2+12 a d^2 c+12 b^2 d^2\right ) x^2+4 b \left (5 c^2+6 a d^2 c+b^2 d^2\right ) x+4 \left (3 \left (a d^2+c\right ) b^2+c \left (\frac {c^2}{d^2}+3 a c+3 a^2 d^2\right )\right )}{\sqrt {1-d^2 x^2}}dx}{d^2}-\frac {4 b c^2 x^2 \sqrt {1-d^2 x^2}}{d^2}}{4 d^2}+\frac {x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b d^4 \left (3 a^2+\frac {6 a c}{d^2}+\frac {b^2}{d^2}+\frac {3 c^2}{d^4}\right )}{d^6 \sqrt {1-d^2 x^2}}+\frac {c^3 x^3 \sqrt {1-d^2 x^2}}{4 d^4}\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle -\frac {\frac {-\frac {\int -\frac {8 b \left (5 c^2+6 a d^2 c+b^2 d^2\right ) x d^2+3 \left (8 a b^2 d^4+8 a^2 c d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right )}{\sqrt {1-d^2 x^2}}dx}{2 d^2}-\frac {c x \sqrt {1-d^2 x^2} \left (12 a c d^2+12 b^2 d^2+7 c^2\right )}{2 d^2}}{d^2}-\frac {4 b c^2 x^2 \sqrt {1-d^2 x^2}}{d^2}}{4 d^2}+\frac {x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b d^4 \left (3 a^2+\frac {6 a c}{d^2}+\frac {b^2}{d^2}+\frac {3 c^2}{d^4}\right )}{d^6 \sqrt {1-d^2 x^2}}+\frac {c^3 x^3 \sqrt {1-d^2 x^2}}{4 d^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {8 b \left (5 c^2+6 a d^2 c+b^2 d^2\right ) x d^2+3 \left (8 a b^2 d^4+8 a^2 c d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right )}{\sqrt {1-d^2 x^2}}dx}{2 d^2}-\frac {c x \sqrt {1-d^2 x^2} \left (12 a c d^2+12 b^2 d^2+7 c^2\right )}{2 d^2}}{d^2}-\frac {4 b c^2 x^2 \sqrt {1-d^2 x^2}}{d^2}}{4 d^2}+\frac {x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b d^4 \left (3 a^2+\frac {6 a c}{d^2}+\frac {b^2}{d^2}+\frac {3 c^2}{d^4}\right )}{d^6 \sqrt {1-d^2 x^2}}+\frac {c^3 x^3 \sqrt {1-d^2 x^2}}{4 d^4}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -\frac {\frac {\frac {3 \left (8 a^2 c d^4+8 a b^2 d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right ) \int \frac {1}{\sqrt {1-d^2 x^2}}dx-8 b \sqrt {1-d^2 x^2} \left (6 a c d^2+b^2 d^2+5 c^2\right )}{2 d^2}-\frac {c x \sqrt {1-d^2 x^2} \left (12 a c d^2+12 b^2 d^2+7 c^2\right )}{2 d^2}}{d^2}-\frac {4 b c^2 x^2 \sqrt {1-d^2 x^2}}{d^2}}{4 d^2}+\frac {x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b d^4 \left (3 a^2+\frac {6 a c}{d^2}+\frac {b^2}{d^2}+\frac {3 c^2}{d^4}\right )}{d^6 \sqrt {1-d^2 x^2}}+\frac {c^3 x^3 \sqrt {1-d^2 x^2}}{4 d^4}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {\frac {\frac {\frac {3 \arcsin (d x) \left (8 a^2 c d^4+8 a b^2 d^4+12 a c^2 d^2+12 b^2 c d^2+5 c^3\right )}{d}-8 b \sqrt {1-d^2 x^2} \left (6 a c d^2+b^2 d^2+5 c^2\right )}{2 d^2}-\frac {c x \sqrt {1-d^2 x^2} \left (12 a c d^2+12 b^2 d^2+7 c^2\right )}{2 d^2}}{d^2}-\frac {4 b c^2 x^2 \sqrt {1-d^2 x^2}}{d^2}}{4 d^2}+\frac {x \left (a d^2+c\right ) \left (a^2 d^4+2 a c d^2+3 b^2 d^2+c^2\right )+b d^4 \left (3 a^2+\frac {6 a c}{d^2}+\frac {b^2}{d^2}+\frac {3 c^2}{d^4}\right )}{d^6 \sqrt {1-d^2 x^2}}+\frac {c^3 x^3 \sqrt {1-d^2 x^2}}{4 d^4}\) |
(b*(3*a^2 + (3*c^2)/d^4 + b^2/d^2 + (6*a*c)/d^2)*d^4 + (c + a*d^2)*(c^2 + 3*b^2*d^2 + 2*a*c*d^2 + a^2*d^4)*x)/(d^6*Sqrt[1 - d^2*x^2]) + (c^3*x^3*Sqr t[1 - d^2*x^2])/(4*d^4) - ((-4*b*c^2*x^2*Sqrt[1 - d^2*x^2])/d^2 + (-1/2*(c *(7*c^2 + 12*b^2*d^2 + 12*a*c*d^2)*x*Sqrt[1 - d^2*x^2])/d^2 + (-8*b*(5*c^2 + b^2*d^2 + 6*a*c*d^2)*Sqrt[1 - d^2*x^2] + (3*(5*c^3 + 12*b^2*c*d^2 + 12* a*c^2*d^2 + 8*a*b^2*d^4 + 8*a^2*c*d^4)*ArcSin[d*x])/d)/(2*d^2))/d^2)/(4*d^ 2)
3.8.97.3.1 Defintions of rubi rules used
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_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[((d_) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d*f + e*g*x^2)^m*(a + b*x + c*x^2 )^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[m, n] && EqQ[e*f + d*g, 0] && (IntegerQ[m] || (GtQ[d, 0] && GtQ[f, 0]))
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(566\) vs. \(2(260)=520\).
Time = 0.62 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.05
| method | result | size |
| risch | \(-\frac {\left (2 c^{3} d^{2} x^{3}+8 b \,c^{2} d^{2} x^{2}+12 a \,c^{2} d^{2} x +12 b^{2} c \,d^{2} x +48 a b c \,d^{2}+8 b^{3} d^{2}+7 c^{3} x +40 b \,c^{2}\right ) \left (d x -1\right ) \sqrt {d x +1}\, \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{8 d^{6} \sqrt {-\left (d x -1\right ) \left (d x +1\right )}\, \sqrt {-d x +1}}-\frac {\left (\frac {15 c^{3} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{\sqrt {d^{2}}}+\frac {24 a^{2} c \,d^{4} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{\sqrt {d^{2}}}+\frac {24 b^{2} d^{4} a \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{\sqrt {d^{2}}}+\frac {36 a \,c^{2} d^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{\sqrt {d^{2}}}+\frac {36 b^{2} c \,d^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{\sqrt {d^{2}}}+\frac {\left (4 a^{3} d^{6}+12 b \,a^{2} d^{5}+12 a^{2} c \,d^{4}+12 b^{2} d^{4} a +24 a b c \,d^{3}+4 b^{3} d^{3}+12 a \,c^{2} d^{2}+12 b^{2} c \,d^{2}+12 b d \,c^{2}+4 c^{3}\right ) \sqrt {-d^{2} \left (x -\frac {1}{d}\right )^{2}-2 d \left (x -\frac {1}{d}\right )}}{d^{2} \left (x -\frac {1}{d}\right )}-\frac {\left (-4 a^{3} d^{6}+12 b \,a^{2} d^{5}-12 a^{2} c \,d^{4}-12 b^{2} d^{4} a +24 a b c \,d^{3}+4 b^{3} d^{3}-12 a \,c^{2} d^{2}-12 b^{2} c \,d^{2}+12 b d \,c^{2}-4 c^{3}\right ) \sqrt {-d^{2} \left (x +\frac {1}{d}\right )^{2}+2 d \left (x +\frac {1}{d}\right )}}{d^{2} \left (x +\frac {1}{d}\right )}\right ) \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{8 d^{6} \sqrt {-d x +1}\, \sqrt {d x +1}}\) | \(567\) |
| default | \(-\frac {\sqrt {-d x +1}\, \left (96 \,\operatorname {csgn}\left (d \right ) d^{3} \sqrt {-d^{2} x^{2}+1}\, a b c +8 \,\operatorname {csgn}\left (d \right ) d^{7} \sqrt {-d^{2} x^{2}+1}\, a^{3} x -12 \,\operatorname {csgn}\left (d \right ) b^{2} c \,d^{5} x^{3} \sqrt {-d^{2} x^{2}+1}+24 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a^{2} c \,d^{6} x^{2}+24 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a \,b^{2} d^{6} x^{2}+15 \,\operatorname {csgn}\left (d \right ) d \sqrt {-d^{2} x^{2}+1}\, c^{3} x -2 \,\operatorname {csgn}\left (d \right ) c^{3} d^{5} x^{5} \sqrt {-d^{2} x^{2}+1}-8 \,\operatorname {csgn}\left (d \right ) b^{3} d^{5} x^{2} \sqrt {-d^{2} x^{2}+1}-5 \,\operatorname {csgn}\left (d \right ) d^{3} \sqrt {-d^{2} x^{2}+1}\, c^{3} x^{3}+24 \,\operatorname {csgn}\left (d \right ) d^{5} \sqrt {-d^{2} x^{2}+1}\, a^{2} c x +24 \,\operatorname {csgn}\left (d \right ) d^{5} \sqrt {-d^{2} x^{2}+1}\, a \,b^{2} x +36 \,\operatorname {csgn}\left (d \right ) d^{3} \sqrt {-d^{2} x^{2}+1}\, a \,c^{2} x +36 \,\operatorname {csgn}\left (d \right ) d^{3} \sqrt {-d^{2} x^{2}+1}\, b^{2} c x +16 \,\operatorname {csgn}\left (d \right ) d^{3} \sqrt {-d^{2} x^{2}+1}\, b^{3}-24 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a^{2} c \,d^{4}-24 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a \,b^{2} d^{4}-36 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a \,c^{2} d^{2}-36 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) b^{2} c \,d^{2}+15 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) c^{3} d^{2} x^{2}-32 \,\operatorname {csgn}\left (d \right ) d^{3} \sqrt {-d^{2} x^{2}+1}\, b \,c^{2} x^{2}-48 \,\operatorname {csgn}\left (d \right ) a b c \,d^{5} x^{2} \sqrt {-d^{2} x^{2}+1}+36 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a \,c^{2} d^{4} x^{2}+36 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) b^{2} c \,d^{4} x^{2}+24 \,\operatorname {csgn}\left (d \right ) d^{5} \sqrt {-d^{2} x^{2}+1}\, a^{2} b +64 \,\operatorname {csgn}\left (d \right ) d \sqrt {-d^{2} x^{2}+1}\, b \,c^{2}-15 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) c^{3}-8 \,\operatorname {csgn}\left (d \right ) b \,c^{2} d^{5} x^{4} \sqrt {-d^{2} x^{2}+1}-12 \,\operatorname {csgn}\left (d \right ) a \,c^{2} d^{5} x^{3} \sqrt {-d^{2} x^{2}+1}\right ) \operatorname {csgn}\left (d \right )}{8 \left (d x -1\right ) \sqrt {-d^{2} x^{2}+1}\, d^{7} \sqrt {d x +1}}\) | \(755\) |
-1/8*(2*c^3*d^2*x^3+8*b*c^2*d^2*x^2+12*a*c^2*d^2*x+12*b^2*c*d^2*x+48*a*b*c *d^2+8*b^3*d^2+7*c^3*x+40*b*c^2)*(d*x-1)*(d*x+1)^(1/2)/d^6/(-(d*x-1)*(d*x+ 1))^(1/2)*((-d*x+1)*(d*x+1))^(1/2)/(-d*x+1)^(1/2)-1/8/d^6*(15*c^3/(d^2)^(1 /2)*arctan((d^2)^(1/2)*x/(-d^2*x^2+1)^(1/2))+24*a^2*c*d^4/(d^2)^(1/2)*arct an((d^2)^(1/2)*x/(-d^2*x^2+1)^(1/2))+24*b^2*d^4*a/(d^2)^(1/2)*arctan((d^2) ^(1/2)*x/(-d^2*x^2+1)^(1/2))+36*a*c^2*d^2/(d^2)^(1/2)*arctan((d^2)^(1/2)*x /(-d^2*x^2+1)^(1/2))+36*b^2*c*d^2/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*x ^2+1)^(1/2))+(4*a^3*d^6+12*a^2*b*d^5+12*a^2*c*d^4+12*a*b^2*d^4+24*a*b*c*d^ 3+4*b^3*d^3+12*a*c^2*d^2+12*b^2*c*d^2+12*b*c^2*d+4*c^3)/d^2/(x-1/d)*(-d^2* (x-1/d)^2-2*d*(x-1/d))^(1/2)-(-4*a^3*d^6+12*a^2*b*d^5-12*a^2*c*d^4-12*a*b^ 2*d^4+24*a*b*c*d^3+4*b^3*d^3-12*a*c^2*d^2-12*b^2*c*d^2+12*b*c^2*d-4*c^3)/d ^2/(x+1/d)*(-d^2*(x+1/d)^2+2*d*(x+1/d))^(1/2))*((-d*x+1)*(d*x+1))^(1/2)/(- d*x+1)^(1/2)/(d*x+1)^(1/2)
Time = 0.31 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=-\frac {24 \, a^{2} b d^{5} + 64 \, b c^{2} d + 16 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} - 8 \, {\left (3 \, a^{2} b d^{7} + 8 \, b c^{2} d^{3} + 2 \, {\left (b^{3} + 6 \, a b c\right )} d^{5}\right )} x^{2} - {\left (2 \, c^{3} d^{5} x^{5} + 8 \, b c^{2} d^{5} x^{4} - 24 \, a^{2} b d^{5} - 64 \, b c^{2} d - 16 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} + {\left (5 \, c^{3} d^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{5}\right )} x^{3} + 8 \, {\left (4 \, b c^{2} d^{3} + {\left (b^{3} + 6 \, a b c\right )} d^{5}\right )} x^{2} - {\left (8 \, a^{3} d^{7} + 24 \, {\left (a b^{2} + a^{2} c\right )} d^{5} + 15 \, c^{3} d + 36 \, {\left (b^{2} c + a c^{2}\right )} d^{3}\right )} x\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 6 \, {\left (8 \, {\left (a b^{2} + a^{2} c\right )} d^{4} + 5 \, c^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} - {\left (8 \, {\left (a b^{2} + a^{2} c\right )} d^{6} + 5 \, c^{3} d^{2} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{4}\right )} x^{2}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{8 \, {\left (d^{9} x^{2} - d^{7}\right )}} \]
-1/8*(24*a^2*b*d^5 + 64*b*c^2*d + 16*(b^3 + 6*a*b*c)*d^3 - 8*(3*a^2*b*d^7 + 8*b*c^2*d^3 + 2*(b^3 + 6*a*b*c)*d^5)*x^2 - (2*c^3*d^5*x^5 + 8*b*c^2*d^5* x^4 - 24*a^2*b*d^5 - 64*b*c^2*d - 16*(b^3 + 6*a*b*c)*d^3 + (5*c^3*d^3 + 12 *(b^2*c + a*c^2)*d^5)*x^3 + 8*(4*b*c^2*d^3 + (b^3 + 6*a*b*c)*d^5)*x^2 - (8 *a^3*d^7 + 24*(a*b^2 + a^2*c)*d^5 + 15*c^3*d + 36*(b^2*c + a*c^2)*d^3)*x)* sqrt(d*x + 1)*sqrt(-d*x + 1) + 6*(8*(a*b^2 + a^2*c)*d^4 + 5*c^3 + 12*(b^2* c + a*c^2)*d^2 - (8*(a*b^2 + a^2*c)*d^6 + 5*c^3*d^2 + 12*(b^2*c + a*c^2)*d ^4)*x^2)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/(d^9*x^2 - d^7)
\[ \int \frac {\left (a+b x+c x^2\right )^3}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{3}}{\left (- d x + 1\right )^{\frac {3}{2}} \left (d x + 1\right )^{\frac {3}{2}}}\, dx \]
Time = 0.32 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=-\frac {c^{3} x^{5}}{4 \, \sqrt {-d^{2} x^{2} + 1} d^{2}} - \frac {b c^{2} x^{4}}{\sqrt {-d^{2} x^{2} + 1} d^{2}} + \frac {a^{3} x}{\sqrt {-d^{2} x^{2} + 1}} - \frac {5 \, c^{3} x^{3}}{8 \, \sqrt {-d^{2} x^{2} + 1} d^{4}} - \frac {3 \, {\left (b^{2} c + a c^{2}\right )} x^{3}}{2 \, \sqrt {-d^{2} x^{2} + 1} d^{2}} + \frac {3 \, a^{2} b}{\sqrt {-d^{2} x^{2} + 1} d^{2}} - \frac {4 \, b c^{2} x^{2}}{\sqrt {-d^{2} x^{2} + 1} d^{4}} - \frac {{\left (b^{3} + 6 \, a b c\right )} x^{2}}{\sqrt {-d^{2} x^{2} + 1} d^{2}} + \frac {3 \, {\left (a b^{2} + a^{2} c\right )} x}{\sqrt {-d^{2} x^{2} + 1} d^{2}} - \frac {3 \, {\left (a b^{2} + a^{2} c\right )} \arcsin \left (d x\right )}{d^{3}} + \frac {15 \, c^{3} x}{8 \, \sqrt {-d^{2} x^{2} + 1} d^{6}} + \frac {9 \, {\left (b^{2} c + a c^{2}\right )} x}{2 \, \sqrt {-d^{2} x^{2} + 1} d^{4}} - \frac {15 \, c^{3} \arcsin \left (d x\right )}{8 \, d^{7}} - \frac {9 \, {\left (b^{2} c + a c^{2}\right )} \arcsin \left (d x\right )}{2 \, d^{5}} + \frac {8 \, b c^{2}}{\sqrt {-d^{2} x^{2} + 1} d^{6}} + \frac {2 \, {\left (b^{3} + 6 \, a b c\right )}}{\sqrt {-d^{2} x^{2} + 1} d^{4}} \]
-1/4*c^3*x^5/(sqrt(-d^2*x^2 + 1)*d^2) - b*c^2*x^4/(sqrt(-d^2*x^2 + 1)*d^2) + a^3*x/sqrt(-d^2*x^2 + 1) - 5/8*c^3*x^3/(sqrt(-d^2*x^2 + 1)*d^4) - 3/2*( b^2*c + a*c^2)*x^3/(sqrt(-d^2*x^2 + 1)*d^2) + 3*a^2*b/(sqrt(-d^2*x^2 + 1)* d^2) - 4*b*c^2*x^2/(sqrt(-d^2*x^2 + 1)*d^4) - (b^3 + 6*a*b*c)*x^2/(sqrt(-d ^2*x^2 + 1)*d^2) + 3*(a*b^2 + a^2*c)*x/(sqrt(-d^2*x^2 + 1)*d^2) - 3*(a*b^2 + a^2*c)*arcsin(d*x)/d^3 + 15/8*c^3*x/(sqrt(-d^2*x^2 + 1)*d^6) + 9/2*(b^2 *c + a*c^2)*x/(sqrt(-d^2*x^2 + 1)*d^4) - 15/8*c^3*arcsin(d*x)/d^7 - 9/2*(b ^2*c + a*c^2)*arcsin(d*x)/d^5 + 8*b*c^2/(sqrt(-d^2*x^2 + 1)*d^6) + 2*(b^3 + 6*a*b*c)/(sqrt(-d^2*x^2 + 1)*d^4)
Leaf count of result is larger than twice the leaf count of optimal. 736 vs. \(2 (260) = 520\).
Time = 0.36 (sec) , antiderivative size = 736, normalized size of antiderivative = 2.67 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\frac {\frac {{\left ({\left ({\left (d x + 1\right )} {\left (2 \, {\left (d x + 1\right )} {\left (\frac {{\left (d x + 1\right )} c^{3}}{d^{6}} + \frac {4 \, b c^{2} d^{31} - 5 \, c^{3} d^{30}}{d^{36}}\right )} + \frac {12 \, b^{2} c d^{32} + 12 \, a c^{2} d^{32} - 32 \, b c^{2} d^{31} + 25 \, c^{3} d^{30}}{d^{36}}\right )} + \frac {8 \, b^{3} d^{33} + 48 \, a b c d^{33} - 36 \, b^{2} c d^{32} - 36 \, a c^{2} d^{32} + 80 \, b c^{2} d^{31} - 35 \, c^{3} d^{30}}{d^{36}}\right )} {\left (d x + 1\right )} - \frac {2 \, {\left (2 \, a^{3} d^{36} + 6 \, a^{2} b d^{35} + 6 \, a b^{2} d^{34} + 6 \, a^{2} c d^{34} + 10 \, b^{3} d^{33} + 60 \, a b c d^{33} - 6 \, b^{2} c d^{32} - 6 \, a c^{2} d^{32} + 54 \, b c^{2} d^{31} - 7 \, c^{3} d^{30}\right )}}{d^{36}}\right )} \sqrt {d x + 1} \sqrt {-d x + 1}}{d x - 1} - \frac {6 \, {\left (8 \, a b^{2} d^{4} + 8 \, a^{2} c d^{4} + 12 \, b^{2} c d^{2} + 12 \, a c^{2} d^{2} + 5 \, c^{3}\right )} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{d^{6}} + \frac {2 \, {\left (\frac {a^{3} d^{6} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {3 \, a^{2} b d^{5} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {3 \, a b^{2} d^{4} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {3 \, a^{2} c d^{4} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {b^{3} d^{3} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {6 \, a b c d^{3} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {3 \, b^{2} c d^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {3 \, a c^{2} d^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {3 \, b c^{2} d {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {c^{3} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}}\right )}}{d^{6}} - \frac {2 \, {\left (a^{3} d^{6} - 3 \, a^{2} b d^{5} + 3 \, a b^{2} d^{4} + 3 \, a^{2} c d^{4} - b^{3} d^{3} - 6 \, a b c d^{3} + 3 \, b^{2} c d^{2} + 3 \, a c^{2} d^{2} - 3 \, b c^{2} d + c^{3}\right )} \sqrt {d x + 1}}{d^{6} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}}{8 \, d} \]
1/8*((((d*x + 1)*(2*(d*x + 1)*((d*x + 1)*c^3/d^6 + (4*b*c^2*d^31 - 5*c^3*d ^30)/d^36) + (12*b^2*c*d^32 + 12*a*c^2*d^32 - 32*b*c^2*d^31 + 25*c^3*d^30) /d^36) + (8*b^3*d^33 + 48*a*b*c*d^33 - 36*b^2*c*d^32 - 36*a*c^2*d^32 + 80* b*c^2*d^31 - 35*c^3*d^30)/d^36)*(d*x + 1) - 2*(2*a^3*d^36 + 6*a^2*b*d^35 + 6*a*b^2*d^34 + 6*a^2*c*d^34 + 10*b^3*d^33 + 60*a*b*c*d^33 - 6*b^2*c*d^32 - 6*a*c^2*d^32 + 54*b*c^2*d^31 - 7*c^3*d^30)/d^36)*sqrt(d*x + 1)*sqrt(-d*x + 1)/(d*x - 1) - 6*(8*a*b^2*d^4 + 8*a^2*c*d^4 + 12*b^2*c*d^2 + 12*a*c^2*d ^2 + 5*c^3)*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^6 + 2*(a^3*d^6*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - 3*a^2*b*d^5*(sqrt(2) - sqrt(-d*x + 1))/sqr t(d*x + 1) + 3*a*b^2*d^4*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) + 3*a^2* c*d^4*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - b^3*d^3*(sqrt(2) - sqrt(- d*x + 1))/sqrt(d*x + 1) - 6*a*b*c*d^3*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) + 3*b^2*c*d^2*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) + 3*a*c^2*d^2* (sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - 3*b*c^2*d*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) + c^3*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1))/d^6 - 2*(a^3*d^6 - 3*a^2*b*d^5 + 3*a*b^2*d^4 + 3*a^2*c*d^4 - b^3*d^3 - 6*a*b*c*d ^3 + 3*b^2*c*d^2 + 3*a*c^2*d^2 - 3*b*c^2*d + c^3)*sqrt(d*x + 1)/(d^6*(sqrt (2) - sqrt(-d*x + 1))))/d
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^3}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^3}{{\left (1-d\,x\right )}^{3/2}\,{\left (d\,x+1\right )}^{3/2}} \,d x \]