Integrand size = 32, antiderivative size = 135 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\frac {2 b \left (a+\frac {c}{d^2}\right ) d^2+\left (c^2+b^2 d^2+2 a c d^2+a^2 d^4\right ) x}{d^4 \sqrt {1-d^2 x^2}}+\frac {2 b c \sqrt {1-d^2 x^2}}{d^4}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4}-\frac {\left (2 b^2+c \left (4 a+\frac {3 c}{d^2}\right )\right ) \arcsin (d x)}{2 d^3} \]
-1/2*(2*b^2+c*(4*a+3*c/d^2))*arcsin(d*x)/d^3+(2*b*(a+c/d^2)*d^2+(a^2*d^4+2 *a*c*d^2+b^2*d^2+c^2)*x)/d^4/(-d^2*x^2+1)^(1/2)+2*b*c*(-d^2*x^2+1)^(1/2)/d ^4+1/2*c^2*x*(-d^2*x^2+1)^(1/2)/d^4
Time = 0.51 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\frac {\sqrt {1-d^2 x^2} \left (-8 b c-4 a b d^2-3 c^2 x-2 b^2 d^2 x-4 a c d^2 x-2 a^2 d^4 x+4 b c d^2 x^2+c^2 d^2 x^3\right )}{2 d^4 \left (-1+d^2 x^2\right )}+\frac {\left (-3 c^2-2 b^2 d^2-4 a c d^2\right ) \arctan \left (\frac {d x}{-1+\sqrt {1-d^2 x^2}}\right )}{d^5} \]
(Sqrt[1 - d^2*x^2]*(-8*b*c - 4*a*b*d^2 - 3*c^2*x - 2*b^2*d^2*x - 4*a*c*d^2 *x - 2*a^2*d^4*x + 4*b*c*d^2*x^2 + c^2*d^2*x^3))/(2*d^4*(-1 + d^2*x^2)) + ((-3*c^2 - 2*b^2*d^2 - 4*a*c*d^2)*ArcTan[(d*x)/(-1 + Sqrt[1 - d^2*x^2])])/ d^5
Time = 0.42 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1188, 2345, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (d x+1)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1188 |
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^2}{\left (1-d^2 x^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {x \left (a^2 d^4+2 a c d^2+b^2 d^2+c^2\right )+2 b d^2 \left (a+\frac {c}{d^2}\right )}{d^4 \sqrt {1-d^2 x^2}}-\int \frac {\frac {c^2 x^2}{d^2}+\frac {2 b c x}{d^2}+\frac {c^2+2 a d^2 c+b^2 d^2}{d^4}}{\sqrt {1-d^2 x^2}}dx\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {\int -\frac {2 b^2+4 c x b+c \left (4 a+\frac {3 c}{d^2}\right )}{\sqrt {1-d^2 x^2}}dx}{2 d^2}+\frac {x \left (a^2 d^4+2 a c d^2+b^2 d^2+c^2\right )+2 b d^2 \left (a+\frac {c}{d^2}\right )}{d^4 \sqrt {1-d^2 x^2}}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {2 b^2+4 c x b+c \left (4 a+\frac {3 c}{d^2}\right )}{\sqrt {1-d^2 x^2}}dx}{2 d^2}+\frac {x \left (a^2 d^4+2 a c d^2+b^2 d^2+c^2\right )+2 b d^2 \left (a+\frac {c}{d^2}\right )}{d^4 \sqrt {1-d^2 x^2}}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle -\frac {\left (c \left (4 a+\frac {3 c}{d^2}\right )+2 b^2\right ) \int \frac {1}{\sqrt {1-d^2 x^2}}dx-\frac {4 b c \sqrt {1-d^2 x^2}}{d^2}}{2 d^2}+\frac {x \left (a^2 d^4+2 a c d^2+b^2 d^2+c^2\right )+2 b d^2 \left (a+\frac {c}{d^2}\right )}{d^4 \sqrt {1-d^2 x^2}}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {x \left (a^2 d^4+2 a c d^2+b^2 d^2+c^2\right )+2 b d^2 \left (a+\frac {c}{d^2}\right )}{d^4 \sqrt {1-d^2 x^2}}-\frac {\frac {\arcsin (d x) \left (c \left (4 a+\frac {3 c}{d^2}\right )+2 b^2\right )}{d}-\frac {4 b c \sqrt {1-d^2 x^2}}{d^2}}{2 d^2}+\frac {c^2 x \sqrt {1-d^2 x^2}}{2 d^4}\) |
(2*b*(a + c/d^2)*d^2 + (c^2 + b^2*d^2 + 2*a*c*d^2 + a^2*d^4)*x)/(d^4*Sqrt[ 1 - d^2*x^2]) + (c^2*x*Sqrt[1 - d^2*x^2])/(2*d^4) - ((-4*b*c*Sqrt[1 - d^2* x^2])/d^2 + ((2*b^2 + c*(4*a + (3*c)/d^2))*ArcSin[d*x])/d)/(2*d^2)
3.8.98.3.1 Defintions of rubi rules used
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. \(345\) vs. \(2(125)=250\).
Time = 0.62 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.56
| method | result | size |
| risch | \(-\frac {c \left (c x +4 b \right ) \left (d x -1\right ) \sqrt {d x +1}\, \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{2 d^{4} \sqrt {-\left (d x -1\right ) \left (d x +1\right )}\, \sqrt {-d x +1}}-\frac {\left (\frac {3 c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{\sqrt {d^{2}}}+\frac {2 b^{2} d^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{\sqrt {d^{2}}}+\frac {4 c \,d^{2} a \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{\sqrt {d^{2}}}+\frac {\left (a^{2} d^{4}+2 b a \,d^{3}+2 c \,d^{2} a +b^{2} d^{2}+2 b c d +c^{2}\right ) \sqrt {-d^{2} \left (x -\frac {1}{d}\right )^{2}-2 d \left (x -\frac {1}{d}\right )}}{d^{2} \left (x -\frac {1}{d}\right )}-\frac {\left (-a^{2} d^{4}+2 b a \,d^{3}-2 c \,d^{2} a -b^{2} d^{2}+2 b c d -c^{2}\right ) \sqrt {-d^{2} \left (x +\frac {1}{d}\right )^{2}+2 d \left (x +\frac {1}{d}\right )}}{d^{2} \left (x +\frac {1}{d}\right )}\right ) \sqrt {\left (-d x +1\right ) \left (d x +1\right )}}{2 d^{4} \sqrt {-d x +1}\, \sqrt {d x +1}}\) | \(346\) |
| default | \(-\frac {\sqrt {-d x +1}\, \left (2 \,\operatorname {csgn}\left (d \right ) d^{5} \sqrt {-d^{2} x^{2}+1}\, a^{2} x -\operatorname {csgn}\left (d \right ) c^{2} d^{3} x^{3} \sqrt {-d^{2} x^{2}+1}-4 \,\operatorname {csgn}\left (d \right ) b c \,d^{3} x^{2} \sqrt {-d^{2} x^{2}+1}+4 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a c \,d^{4} x^{2}+2 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) b^{2} d^{4} x^{2}+4 \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} a c x +2 \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d^{3} b^{2} x +4 \,\operatorname {csgn}\left (d \right ) d^{3} \sqrt {-d^{2} x^{2}+1}\, a b +3 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) c^{2} d^{2} x^{2}+3 \sqrt {-d^{2} x^{2}+1}\, \operatorname {csgn}\left (d \right ) d \,c^{2} x +8 \,\operatorname {csgn}\left (d \right ) d \sqrt {-d^{2} x^{2}+1}\, b c -4 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) a c \,d^{2}-2 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) b^{2} d^{2}-3 \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) c^{2}\right ) \operatorname {csgn}\left (d \right )}{2 \left (d x -1\right ) \sqrt {-d^{2} x^{2}+1}\, d^{5} \sqrt {d x +1}}\) | \(381\) |
-1/2*c*(c*x+4*b)*(d*x-1)*(d*x+1)^(1/2)/d^4/(-(d*x-1)*(d*x+1))^(1/2)*((-d*x +1)*(d*x+1))^(1/2)/(-d*x+1)^(1/2)-1/2/d^4*(3*c^2/(d^2)^(1/2)*arctan((d^2)^ (1/2)*x/(-d^2*x^2+1)^(1/2))+2*b^2*d^2/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d ^2*x^2+1)^(1/2))+4*c*d^2*a/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*x^2+1)^( 1/2))+(a^2*d^4+2*a*b*d^3+2*a*c*d^2+b^2*d^2+2*b*c*d+c^2)/d^2/(x-1/d)*(-d^2* (x-1/d)^2-2*d*(x-1/d))^(1/2)-(-a^2*d^4+2*a*b*d^3-2*a*c*d^2-b^2*d^2+2*b*c*d -c^2)/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 = 0.29 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.51 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=-\frac {4 \, a b d^{3} + 8 \, b c d - 4 \, {\left (a b d^{5} + 2 \, b c d^{3}\right )} x^{2} - {\left (c^{2} d^{3} x^{3} + 4 \, b c d^{3} x^{2} - 4 \, a b d^{3} - 8 \, b c d - {\left (2 \, a^{2} d^{5} + 2 \, {\left (b^{2} + 2 \, a c\right )} d^{3} + 3 \, c^{2} d\right )} x\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 2 \, {\left (2 \, {\left (b^{2} + 2 \, a c\right )} d^{2} - {\left (2 \, {\left (b^{2} + 2 \, a c\right )} d^{4} + 3 \, c^{2} d^{2}\right )} x^{2} + 3 \, c^{2}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{2 \, {\left (d^{7} x^{2} - d^{5}\right )}} \]
-1/2*(4*a*b*d^3 + 8*b*c*d - 4*(a*b*d^5 + 2*b*c*d^3)*x^2 - (c^2*d^3*x^3 + 4 *b*c*d^3*x^2 - 4*a*b*d^3 - 8*b*c*d - (2*a^2*d^5 + 2*(b^2 + 2*a*c)*d^3 + 3* c^2*d)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 2*(2*(b^2 + 2*a*c)*d^2 - (2*(b^2 + 2*a*c)*d^4 + 3*c^2*d^2)*x^2 + 3*c^2)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1 ) - 1)/(d*x)))/(d^7*x^2 - d^5)
\[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{2}}{\left (- d x + 1\right )^{\frac {3}{2}} \left (d x + 1\right )^{\frac {3}{2}}}\, dx \]
Time = 0.30 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\frac {a^{2} x}{\sqrt {-d^{2} x^{2} + 1}} - \frac {c^{2} x^{3}}{2 \, \sqrt {-d^{2} x^{2} + 1} d^{2}} - \frac {2 \, b c x^{2}}{\sqrt {-d^{2} x^{2} + 1} d^{2}} + \frac {2 \, a b}{\sqrt {-d^{2} x^{2} + 1} d^{2}} + \frac {{\left (b^{2} + 2 \, a c\right )} x}{\sqrt {-d^{2} x^{2} + 1} d^{2}} - \frac {{\left (b^{2} + 2 \, a c\right )} \arcsin \left (d x\right )}{d^{3}} + \frac {3 \, c^{2} x}{2 \, \sqrt {-d^{2} x^{2} + 1} d^{4}} - \frac {3 \, c^{2} \arcsin \left (d x\right )}{2 \, d^{5}} + \frac {4 \, b c}{\sqrt {-d^{2} x^{2} + 1} d^{4}} \]
a^2*x/sqrt(-d^2*x^2 + 1) - 1/2*c^2*x^3/(sqrt(-d^2*x^2 + 1)*d^2) - 2*b*c*x^ 2/(sqrt(-d^2*x^2 + 1)*d^2) + 2*a*b/(sqrt(-d^2*x^2 + 1)*d^2) + (b^2 + 2*a*c )*x/(sqrt(-d^2*x^2 + 1)*d^2) - (b^2 + 2*a*c)*arcsin(d*x)/d^3 + 3/2*c^2*x/( sqrt(-d^2*x^2 + 1)*d^4) - 3/2*c^2*arcsin(d*x)/d^5 + 4*b*c/(sqrt(-d^2*x^2 + 1)*d^4)
Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (125) = 250\).
Time = 0.32 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.90 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\frac {\frac {2 \, \sqrt {d x + 1} \sqrt {-d x + 1} {\left ({\left (d x + 1\right )} {\left (\frac {{\left (d x + 1\right )} c^{2}}{d^{4}} + \frac {4 \, b c d^{13} - 3 \, c^{2} d^{12}}{d^{16}}\right )} - \frac {a^{2} d^{16} + 2 \, a b d^{15} + b^{2} d^{14} + 2 \, a c d^{14} + 10 \, b c d^{13} - c^{2} d^{12}}{d^{16}}\right )}}{d x - 1} - \frac {4 \, {\left (2 \, b^{2} d^{2} + 4 \, a c d^{2} + 3 \, c^{2}\right )} \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {d x + 1}\right )}{d^{4}} + \frac {\frac {a^{2} d^{4} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {2 \, a b d^{3} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {b^{2} d^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {2 \, a c d^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} - \frac {2 \, b c d {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}} + \frac {c^{2} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}{\sqrt {d x + 1}}}{d^{4}} - \frac {{\left (a^{2} d^{4} - 2 \, a b d^{3} + b^{2} d^{2} + 2 \, a c d^{2} - 2 \, b c d + c^{2}\right )} \sqrt {d x + 1}}{d^{4} {\left (\sqrt {2} - \sqrt {-d x + 1}\right )}}}{4 \, d} \]
1/4*(2*sqrt(d*x + 1)*sqrt(-d*x + 1)*((d*x + 1)*((d*x + 1)*c^2/d^4 + (4*b*c *d^13 - 3*c^2*d^12)/d^16) - (a^2*d^16 + 2*a*b*d^15 + b^2*d^14 + 2*a*c*d^14 + 10*b*c*d^13 - c^2*d^12)/d^16)/(d*x - 1) - 4*(2*b^2*d^2 + 4*a*c*d^2 + 3* c^2)*arcsin(1/2*sqrt(2)*sqrt(d*x + 1))/d^4 + (a^2*d^4*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - 2*a*b*d^3*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) + b^2*d^2*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) + 2*a*c*d^2*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1) - 2*b*c*d*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d* x + 1) + c^2*(sqrt(2) - sqrt(-d*x + 1))/sqrt(d*x + 1))/d^4 - (a^2*d^4 - 2* a*b*d^3 + b^2*d^2 + 2*a*c*d^2 - 2*b*c*d + c^2)*sqrt(d*x + 1)/(d^4*(sqrt(2) - sqrt(-d*x + 1))))/d
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^2}{(1-d x)^{3/2} (1+d x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^2}{{\left (1-d\,x\right )}^{3/2}\,{\left (d\,x+1\right )}^{3/2}} \,d x \]