Integrand size = 32, antiderivative size = 939 \[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=-\frac {d^2 \left (b \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-\left (2 c^4+b^2 d^4 \left (2 b^2+a^2 d^2\right )-c^2 d^2 \left (b^2+6 a^2 d^2\right )-c \left (6 a b^2 d^4+4 a^3 d^6\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (c-b d+a d^2\right )^2 \left (c+b d+a d^2\right )^2 \sqrt {1-d^2 x^2}}-\frac {b \left (b^2 d^2-c \left (c+3 a d^2\right )\right )-c \left (2 c^2-b^2 d^2+2 a c d^2\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (c+a d^2\right )^2\right ) \left (a+b x+c x^2\right ) \sqrt {1-d^2 x^2}}+\frac {c \left (4 c^5+24 a c^4 d^2+3 a b^3 \left (b+\sqrt {b^2-4 a c}\right ) d^6-c^3 d^2 \left (9 b^2-b \sqrt {b^2-4 a c}-36 a^2 d^2\right )-2 a c^2 d^4 \left (7 b^2+5 b \sqrt {b^2-4 a c}-8 a^2 d^2\right )+b c d^4 \left (2 b^3+2 b^2 \sqrt {b^2-4 a c}-17 a^2 b d^2-11 a^2 \sqrt {b^2-4 a c} d^2\right )\right ) \text {arctanh}\left (\frac {2 c+\left (b-\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} \sqrt {2 c^2+2 a c d^2-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \left (c^2-b^2 d^2+2 a c d^2+a^2 d^4\right )^2}+\frac {c \left (b \left (b+\sqrt {b^2-4 a c}\right ) d^4 \left (c^3+2 b^2 c d^2-10 a c^2 d^2+3 a b^2 d^4-11 a^2 c d^4\right )-2 \left (2 c^5 d^2+12 a c^4 d^4+3 a b^4 d^8+2 b^2 c d^6 \left (b^2-7 a^2 d^2\right )-c^3 \left (4 b^2 d^4-18 a^2 d^6\right )-4 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )\right ) \text {arctanh}\left (\frac {2 c+\left (b+\sqrt {b^2-4 a c}\right ) d^2 x}{\sqrt {2} \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \left (b^2-4 a c\right )^{3/2} d^2 \sqrt {2 c^2+2 a c d^2-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \left (c^2-b^2 d^2+2 a c d^2+a^2 d^4\right )^2} \]
-d^2*(b*(-11*a^2*c*d^4+3*a*b^2*d^4-10*a*c^2*d^2+2*b^2*c*d^2+c^3)-(2*c^4+b^ 2*d^4*(a^2*d^2+2*b^2)-c^2*d^2*(6*a^2*d^2+b^2)-c*(4*a^3*d^6+6*a*b^2*d^4))*x )/(-4*a*c+b^2)/(a*d^2-b*d+c)^2/(a*d^2+b*d+c)^2/(-d^2*x^2+1)^(1/2)+(-b*(b^2 *d^2-c*(3*a*d^2+c))+c*(2*a*c*d^2-b^2*d^2+2*c^2)*x)/(-4*a*c+b^2)/(b^2*d^2-( a*d^2+c)^2)/(c*x^2+b*x+a)/(-d^2*x^2+1)^(1/2)+1/2*c*arctanh(1/2*(2*c+d^2*x* (b-(-4*a*c+b^2)^(1/2)))*2^(1/2)/(-d^2*x^2+1)^(1/2)/(2*c^2+2*a*c*d^2-b*d^2* (b-(-4*a*c+b^2)^(1/2)))^(1/2))*(4*c^5+24*a*c^4*d^2+3*a*b^3*d^6*(b+(-4*a*c+ b^2)^(1/2))-c^3*d^2*(9*b^2-36*a^2*d^2-b*(-4*a*c+b^2)^(1/2))-2*a*c^2*d^4*(7 *b^2-8*a^2*d^2+5*b*(-4*a*c+b^2)^(1/2))+b*c*d^4*(2*b^3-17*a^2*b*d^2+2*b^2*( -4*a*c+b^2)^(1/2)-11*a^2*d^2*(-4*a*c+b^2)^(1/2)))/(-4*a*c+b^2)^(3/2)/(a^2* d^4+2*a*c*d^2-b^2*d^2+c^2)^2*2^(1/2)/(2*c^2+2*a*c*d^2-b*d^2*(b-(-4*a*c+b^2 )^(1/2)))^(1/2)+1/2*c*arctanh(1/2*(2*c+d^2*x*(b+(-4*a*c+b^2)^(1/2)))*2^(1/ 2)/(-d^2*x^2+1)^(1/2)/(2*c^2+2*a*c*d^2-b*d^2*(b+(-4*a*c+b^2)^(1/2)))^(1/2) )*(-4*c^5*d^2-24*a*c^4*d^4-6*a*b^4*d^8-4*b^2*c*d^6*(-7*a^2*d^2+b^2)+2*c^3* (-18*a^2*d^6+4*b^2*d^4)+8*c^2*(-2*a^3*d^8+3*a*b^2*d^6)+b*d^4*(-11*a^2*c*d^ 4+3*a*b^2*d^4-10*a*c^2*d^2+2*b^2*c*d^2+c^3)*(b+(-4*a*c+b^2)^(1/2)))/(-4*a* c+b^2)^(3/2)/d^2/(a^2*d^4+2*a*c*d^2-b^2*d^2+c^2)^2*2^(1/2)/(2*c^2+2*a*c*d^ 2-b*d^2*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 7.47 (sec) , antiderivative size = 3830, normalized size of antiderivative = 4.08 \[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Result too large to show} \]
(-((Sqrt[1 - d^2*x^2]*(b^5*d^4*(-1 + 2*d^2*x^2) + 2*c*(c + a*d^2)^2*x*(-2* a^2*d^4 - 2*a*c*d^4*x^2 + c^2*(-1 + d^2*x^2)) + b^2*d^2*x*(a^3*d^6 + a*c^2 *d^2*(13 - 6*d^2*x^2) + c^3*(2 - d^2*x^2) + a^2*c*d^4*(6 + d^2*x^2)) + b^4 *d^4*x*(-(a*d^2) + c*(-3 + 2*d^2*x^2)) + b*c*(c + a*d^2)*(-4*a^2*d^4*(-2 + d^2*x^2) + c^2*(-1 + d^2*x^2) + a*c*d^2*(-5 + 9*d^2*x^2)) + b^3*(a^2*d^6* (-2 + d^2*x^2) + c^2*(2*d^2 - 3*d^4*x^2) + a*c*(3*d^4 - 9*d^6*x^2))))/((b^ 2 - 4*a*c)*(-1 + d*x)*(1 + d*x)*(a + x*(b + c*x)))) + RootSum[a*d^4 - 2*b* d^2*#1 + 4*c*#1^2 + 2*a*d^2*#1^2 - 2*b*#1^3 + a*#1^4 & , (-4*b^4*c^4*Log[x ] + 20*a*b^2*c^5*Log[x] - 16*a^2*c^6*Log[x] + 8*b^6*c^2*d^2*Log[x] - 56*a* b^4*c^3*d^2*Log[x] + 107*a^2*b^2*c^4*d^2*Log[x] - 46*a^3*c^5*d^2*Log[x] - 4*b^8*d^4*Log[x] + 36*a*b^6*c*d^4*Log[x] - 110*a^2*b^4*c^2*d^4*Log[x] + 13 2*a^3*b^2*c^3*d^4*Log[x] - 44*a^4*c^4*d^4*Log[x] + 3*a^2*b^6*d^6*Log[x] - 22*a^3*b^4*c*d^6*Log[x] + 43*a^4*b^2*c^2*d^6*Log[x] - 14*a^5*c^3*d^6*Log[x ] + 4*b^4*c^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 20*a*b^2*c^5*Log[-1 + S qrt[1 - d^2*x^2] - x*#1] + 16*a^2*c^6*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 8*b^6*c^2*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 56*a*b^4*c^3*d^2*Log[- 1 + Sqrt[1 - d^2*x^2] - x*#1] - 107*a^2*b^2*c^4*d^2*Log[-1 + Sqrt[1 - d^2* x^2] - x*#1] + 46*a^3*c^5*d^2*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] + 4*b^8*d ^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1] - 36*a*b^6*c*d^4*Log[-1 + Sqrt[1 - d ^2*x^2] - x*#1] + 110*a^2*b^4*c^2*d^4*Log[-1 + Sqrt[1 - d^2*x^2] - x*#1...
Time = 4.94 (sec) , antiderivative size = 907, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1188, 1306, 25, 2136, 27, 1367, 488, 219}
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.
| \(\displaystyle \int \frac {1}{(1-d x)^{3/2} (d x+1)^{3/2} \left (a+b x+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 1188 |
| \(\displaystyle \int \frac {1}{\left (1-d^2 x^2\right )^{3/2} \left (a+b x+c x^2\right )^2}dx\) |
| \(\Big \downarrow \) 1306 |
| \(\displaystyle -\frac {\int -\frac {-a b^2 d^4+6 a c^2 d^2-2 c \left (2 c^2+2 a d^2 c-b^2 d^2\right ) x^2 d^2-2 c \left (b^2-2 a^2 d^2\right ) d^2-b \left (c^2+7 a d^2 c-2 b^2 d^2\right ) x d^2+2 c^3}{\left (c x^2+b x+a\right ) \left (1-d^2 x^2\right )^{3/2}}dx}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )}{\sqrt {1-d^2 x^2} \left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-a b^2 d^4+4 a^2 c d^4+6 a c^2 d^2-2 c \left (2 c^2+2 a d^2 c-b^2 d^2\right ) x^2 d^2-2 b^2 c d^2-b \left (c^2+7 a d^2 c-2 b^2 d^2\right ) x d^2+2 c^3}{\left (c x^2+b x+a\right ) \left (1-d^2 x^2\right )^{3/2}}dx}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )}{\sqrt {1-d^2 x^2} \left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 2136 |
| \(\displaystyle \frac {\frac {\int -\frac {2 \left (3 a b^4 d^8+2 b^2 c \left (b^2-7 a^2 d^2\right ) d^6+12 a c^4 d^4+b c \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right ) x d^4+2 c^5 d^2-2 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-4 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right )}{\left (c x^2+b x+a\right ) \sqrt {1-d^2 x^2}}dx}{2 d^2 \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {d^2 \left (b \left (-11 a^2 c d^4+3 a b^2 d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right )-x \left (-4 a^3 c d^6+a^2 b^2 d^6-6 a^2 c^2 d^4-6 a b^2 c d^4+2 b^4 d^4-b^2 c^2 d^2+2 c^4\right )\right )}{\sqrt {1-d^2 x^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )}{\sqrt {1-d^2 x^2} \left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {3 a b^4 d^8+2 b^2 c \left (b^2-7 a^2 d^2\right ) d^6+12 a c^4 d^4+b c \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right ) x d^4+2 c^5 d^2-c^3 \left (4 b^2 d^4-18 a^2 d^6\right )-4 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )}{\left (c x^2+b x+a\right ) \sqrt {1-d^2 x^2}}dx}{d^2 \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {d^2 \left (b \left (-11 a^2 c d^4+3 a b^2 d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right )-x \left (-4 a^3 c d^6+a^2 b^2 d^6-6 a^2 c^2 d^4-6 a b^2 c d^4+2 b^4 d^4-b^2 c^2 d^2+2 c^4\right )\right )}{\sqrt {1-d^2 x^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )}{\sqrt {1-d^2 x^2} \left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 1367 |
| \(\displaystyle \frac {-\frac {\frac {c \left (-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-b d^4 \left (b-\sqrt {b^2-4 a c}\right ) \left (-11 a^2 c d^4+3 a b^2 d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right )+4 b^2 c d^6 \left (b^2-7 a^2 d^2\right )+6 a b^4 d^8+24 a c^4 d^4+4 c^5 d^2\right ) \int \frac {1}{\left (b+2 c x-\sqrt {b^2-4 a c}\right ) \sqrt {1-d^2 x^2}}dx}{\sqrt {b^2-4 a c}}-\frac {c \left (-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-b d^4 \left (\sqrt {b^2-4 a c}+b\right ) \left (-11 a^2 c d^4+3 a b^2 d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right )+4 b^2 c d^6 \left (b^2-7 a^2 d^2\right )+6 a b^4 d^8+24 a c^4 d^4+4 c^5 d^2\right ) \int \frac {1}{\left (b+2 c x+\sqrt {b^2-4 a c}\right ) \sqrt {1-d^2 x^2}}dx}{\sqrt {b^2-4 a c}}}{d^2 \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {d^2 \left (b \left (-11 a^2 c d^4+3 a b^2 d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right )-x \left (-4 a^3 c d^6+a^2 b^2 d^6-6 a^2 c^2 d^4-6 a b^2 c d^4+2 b^4 d^4-b^2 c^2 d^2+2 c^4\right )\right )}{\sqrt {1-d^2 x^2} \left (b^2 d^2-\left (a d^2+c\right )^2\right )}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c x \left (2 a c d^2-b^2 d^2+2 c^2\right )}{\sqrt {1-d^2 x^2} \left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (a+b x+c x^2\right )}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {-\frac {\left (b \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right )-\left (a^2 b^2 d^6-4 a^3 c d^6+2 b^4 d^4-6 a^2 c^2 d^4-6 a b^2 c d^4-b^2 c^2 d^2+2 c^4\right ) x\right ) d^2}{\left (b^2 d^2-\left (a d^2+c\right )^2\right ) \sqrt {1-d^2 x^2}}-\frac {\frac {c \left (6 a b^4 d^8+4 b^2 c \left (b^2-7 a^2 d^2\right ) d^6+24 a c^4 d^4-b \left (b+\sqrt {b^2-4 a c}\right ) \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right ) d^4+4 c^5 d^2-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right ) \int \frac {1}{4 c^2-\left (b+\sqrt {b^2-4 a c}\right )^2 d^2-\frac {\left (\left (b+\sqrt {b^2-4 a c}\right ) x d^2+2 c\right )^2}{1-d^2 x^2}}d\frac {\left (b+\sqrt {b^2-4 a c}\right ) x d^2+2 c}{\sqrt {1-d^2 x^2}}}{\sqrt {b^2-4 a c}}-\frac {c \left (6 a b^4 d^8+4 b^2 c \left (b^2-7 a^2 d^2\right ) d^6+24 a c^4 d^4-b \left (b-\sqrt {b^2-4 a c}\right ) \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right ) d^4+4 c^5 d^2-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right ) \int \frac {1}{4 c^2-\left (b-\sqrt {b^2-4 a c}\right )^2 d^2-\frac {\left (\left (b-\sqrt {b^2-4 a c}\right ) x d^2+2 c\right )^2}{1-d^2 x^2}}d\frac {\left (b-\sqrt {b^2-4 a c}\right ) x d^2+2 c}{\sqrt {1-d^2 x^2}}}{\sqrt {b^2-4 a c}}}{\left (b^2 d^2-\left (a d^2+c\right )^2\right ) d^2}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c \left (2 c^2+2 a d^2 c-b^2 d^2\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (c x^2+b x+a\right ) \sqrt {1-d^2 x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {\left (b \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right )-\left (a^2 b^2 d^6-4 a^3 c d^6+2 b^4 d^4-6 a^2 c^2 d^4-6 a b^2 c d^4-b^2 c^2 d^2+2 c^4\right ) x\right ) d^2}{\left (b^2 d^2-\left (a d^2+c\right )^2\right ) \sqrt {1-d^2 x^2}}-\frac {\frac {c \left (6 a b^4 d^8+4 b^2 c \left (b^2-7 a^2 d^2\right ) d^6+24 a c^4 d^4-b \left (b+\sqrt {b^2-4 a c}\right ) \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right ) d^4+4 c^5 d^2-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right ) \text {arctanh}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) x d^2+2 c}{\sqrt {2} \sqrt {2 c^2+2 a d^2 c-b \left (b+\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2+2 a d^2 c-b \left (b+\sqrt {b^2-4 a c}\right ) d^2}}-\frac {c \left (6 a b^4 d^8+4 b^2 c \left (b^2-7 a^2 d^2\right ) d^6+24 a c^4 d^4-b \left (b-\sqrt {b^2-4 a c}\right ) \left (3 a b^2 d^4-11 a^2 c d^4-10 a c^2 d^2+2 b^2 c d^2+c^3\right ) d^4+4 c^5 d^2-4 c^3 \left (2 b^2 d^4-9 a^2 d^6\right )-8 c^2 \left (3 a b^2 d^6-2 a^3 d^8\right )\right ) \text {arctanh}\left (\frac {\left (b-\sqrt {b^2-4 a c}\right ) x d^2+2 c}{\sqrt {2} \sqrt {2 c^2+2 a d^2 c-b \left (b-\sqrt {b^2-4 a c}\right ) d^2} \sqrt {1-d^2 x^2}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {2 c^2+2 a d^2 c-b \left (b-\sqrt {b^2-4 a c}\right ) d^2}}}{\left (b^2 d^2-\left (a d^2+c\right )^2\right ) d^2}}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right )}-\frac {b \left (b^2 d^2-c \left (3 a d^2+c\right )\right )-c \left (2 c^2+2 a d^2 c-b^2 d^2\right ) x}{\left (b^2-4 a c\right ) \left (b^2 d^2-\left (a d^2+c\right )^2\right ) \left (c x^2+b x+a\right ) \sqrt {1-d^2 x^2}}\) |
-((b*(b^2*d^2 - c*(c + 3*a*d^2)) - c*(2*c^2 - b^2*d^2 + 2*a*c*d^2)*x)/((b^ 2 - 4*a*c)*(b^2*d^2 - (c + a*d^2)^2)*(a + b*x + c*x^2)*Sqrt[1 - d^2*x^2])) + (-((d^2*(b*(c^3 + 2*b^2*c*d^2 - 10*a*c^2*d^2 + 3*a*b^2*d^4 - 11*a^2*c*d ^4) - (2*c^4 - b^2*c^2*d^2 + 2*b^4*d^4 - 6*a*b^2*c*d^4 - 6*a^2*c^2*d^4 + a ^2*b^2*d^6 - 4*a^3*c*d^6)*x))/((b^2*d^2 - (c + a*d^2)^2)*Sqrt[1 - d^2*x^2] )) - (-((c*(4*c^5*d^2 + 24*a*c^4*d^4 + 6*a*b^4*d^8 + 4*b^2*c*d^6*(b^2 - 7* a^2*d^2) - b*(b - Sqrt[b^2 - 4*a*c])*d^4*(c^3 + 2*b^2*c*d^2 - 10*a*c^2*d^2 + 3*a*b^2*d^4 - 11*a^2*c*d^4) - 4*c^3*(2*b^2*d^4 - 9*a^2*d^6) - 8*c^2*(3* a*b^2*d^6 - 2*a^3*d^8))*ArcTanh[(2*c + (b - Sqrt[b^2 - 4*a*c])*d^2*x)/(Sqr t[2]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b - Sqrt[b^2 - 4*a*c])*d^2]*Sqrt[1 - d^2* x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b - Sqrt[b^ 2 - 4*a*c])*d^2])) + (c*(4*c^5*d^2 + 24*a*c^4*d^4 + 6*a*b^4*d^8 + 4*b^2*c* d^6*(b^2 - 7*a^2*d^2) - b*(b + Sqrt[b^2 - 4*a*c])*d^4*(c^3 + 2*b^2*c*d^2 - 10*a*c^2*d^2 + 3*a*b^2*d^4 - 11*a^2*c*d^4) - 4*c^3*(2*b^2*d^4 - 9*a^2*d^6 ) - 8*c^2*(3*a*b^2*d^6 - 2*a^3*d^8))*ArcTanh[(2*c + (b + Sqrt[b^2 - 4*a*c] )*d^2*x)/(Sqrt[2]*Sqrt[2*c^2 + 2*a*c*d^2 - b*(b + Sqrt[b^2 - 4*a*c])*d^2]* Sqrt[1 - d^2*x^2])])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[2*c^2 + 2*a*c*d^2 - b *(b + Sqrt[b^2 - 4*a*c])*d^2]))/(d^2*(b^2*d^2 - (c + a*d^2)^2)))/((b^2 - 4 *a*c)*(b^2*d^2 - (c + a*d^2)^2))
3.9.1.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_.) + (f_.)*(x_)^2)^(q_), x _Symbol] :> Simp[(b^3*f + b*c*(c*d - 3*a*f) + c*(2*c^2*d + b^2*f - c*(2*a*f ))*x)*(a + b*x + c*x^2)^(p + 1)*((d + f*x^2)^(q + 1)/((b^2 - 4*a*c)*(b^2*d* f + (c*d - a*f)^2)*(p + 1))), x] - Simp[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)) Int[(a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^q*Simp[2*c*(b^ 2*d*f + (c*d - a*f)^2)*(p + 1) - (2*c^2*d + b^2*f - c*(2*a*f))*(a*f*(p + 1) - c*d*(p + 2)) + (2*f*(b^3*f + b*c*(c*d - 3*a*f))*(p + q + 2) - (2*c^2*d + b^2*f - c*(2*a*f))*(b*f*(p + 1)))*x + c*f*(2*c^2*d + b^2*f - c*(2*a*f))*(2 *p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, f, q}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0] && !( !IntegerQ [p] && ILtQ[q, -1]) && !IGtQ[q, 0]
Int[((g_.) + (h_.)*(x_))/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (f _.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(2*c*g - h*( b - q))/q Int[1/((b - q + 2*c*x)*Sqrt[d + f*x^2]), x], x] - Simp[(2*c*g - h*(b + q))/q Int[1/((b + q + 2*c*x)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{ a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c]
Int[(Px_)*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_ ), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[P x, x, 2]}, Simp[(a + c*x^2)^(p + 1)*((d + e*x + f*x^2)^(q + 1)/((-4*a*c)*(a *c*e^2 + (c*d - a*f)^2)*(p + 1)))*((A*c - a*C)*(2*a*c*e) + ((-a)*B)*(2*c^2* d - c*(2*a*f)) + c*(A*(2*c^2*d - c*(2*a*f)) - B*(-2*a*c*e) + C*(-2*a*(c*d - a*f)))*x), x] + Simp[1/((-4*a*c)*(a*c*e^2 + (c*d - a*f)^2)*(p + 1)) Int[ (a + c*x^2)^(p + 1)*(d + e*x + f*x^2)^q*Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^ 2 - ((-a)*e)*(c*e))*(p + 1) + (2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C* f)))*(a*f*(p + 1) - c*d*(p + 2)) - e*((A*c - a*C)*(2*a*c*e) + ((-a)*B)*(2*c ^2*d - c*((Plus[2])*a*f)))*(p + q + 2) - (2*f*((A*c - a*C)*(2*a*c*e) + ((-a )*B)*(2*c^2*d + (-c)*((Plus[2])*a*f)))*(p + q + 2) - (2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*((-c)*e*(2*p + q + 4)))*x - c*f*(2*(A*c*(c*d - a*f) - a*(c*C*d - B*c*e - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]] /; Free Q[{a, c, d, e, f, q}, x] && PolyQ[Px, x, 2] && LtQ[p, -1] && NeQ[a*c*e^2 + (c*d - a*f)^2, 0] && !( !IntegerQ[p] && ILtQ[q, -1]) && !IGtQ[q, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.58 (sec) , antiderivative size = 108969, normalized size of antiderivative = 116.05
Timed out. \[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{2} {\left (d x + 1\right )}^{\frac {3}{2}} {\left (-d x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(1-d x)^{3/2} (1+d x)^{3/2} \left (a+b x+c x^2\right )^2} \, dx=\int \frac {1}{{\left (1-d\,x\right )}^{3/2}\,{\left (d\,x+1\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^2} \,d x \]