Integrand size = 24, antiderivative size = 73 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=-\frac {\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac {b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac {1}{32 c^3 d^6 (b+2 c x)} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=-\frac {\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac {b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac {1}{32 c^3 d^6 (b+2 c x)} \]
[In]
[Out]
Rule 697
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^2}{16 c^2 d^6 (b+2 c x)^6}+\frac {-b^2+4 a c}{8 c^2 d^6 (b+2 c x)^4}+\frac {1}{16 c^2 d^6 (b+2 c x)^2}\right ) \, dx \\ & = -\frac {\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac {b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac {1}{32 c^3 d^6 (b+2 c x)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=\frac {-3 \left (b^2-4 a c\right )^2+10 \left (b^2-4 a c\right ) (b+2 c x)^2-15 (b+2 c x)^4}{480 c^3 d^6 (b+2 c x)^5} \]
[In]
[Out]
Time = 2.93 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01
method | result | size |
default | \(\frac {-\frac {16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}{160 c^{3} \left (2 c x +b \right )^{5}}-\frac {4 a c -b^{2}}{48 c^{3} \left (2 c x +b \right )^{3}}-\frac {1}{32 c^{3} \left (2 c x +b \right )}}{d^{6}}\) | \(74\) |
risch | \(\frac {-\frac {x^{4} c}{2}-b \,x^{3}-\frac {\left (a c +2 b^{2}\right ) x^{2}}{3 c}-\frac {\left (2 a c +b^{2}\right ) b x}{6 c^{2}}-\frac {6 a^{2} c^{2}+2 a \,b^{2} c +b^{4}}{60 c^{3}}}{d^{6} \left (2 c x +b \right )^{5}}\) | \(82\) |
gosper | \(-\frac {30 c^{4} x^{4}+60 b \,c^{3} x^{3}+20 x^{2} a \,c^{3}+40 b^{2} c^{2} x^{2}+20 a b \,c^{2} x +10 b^{3} c x +6 a^{2} c^{2}+2 a \,b^{2} c +b^{4}}{60 \left (2 c x +b \right )^{5} c^{3} d^{6}}\) | \(88\) |
norman | \(\frac {\frac {a^{2} x}{b d}+\frac {\left (4 a^{2} c +a \,b^{2}\right ) x^{2}}{b^{2} d}+\frac {\left (24 a^{2} c^{2}+8 a \,b^{2} c +b^{4}\right ) x^{3}}{3 b^{3} d}+\frac {c \left (48 a^{2} c^{2}+16 a \,b^{2} c +5 b^{4}\right ) x^{4}}{6 b^{4} d}+\frac {8 c^{2} \left (6 a^{2} c^{2}+2 a \,b^{2} c +b^{4}\right ) x^{5}}{15 b^{5} d}}{d^{5} \left (2 c x +b \right )^{5}}\) | \(143\) |
parallelrisch | \(\frac {96 x^{5} a^{2} c^{4}+32 x^{5} a \,b^{2} c^{3}+16 b^{4} c^{2} x^{5}+240 a^{2} b \,c^{3} x^{4}+80 x^{4} a \,b^{3} c^{2}+25 x^{4} b^{5} c +240 x^{3} a^{2} b^{2} c^{2}+80 a \,b^{4} c \,x^{3}+10 x^{3} b^{6}+120 x^{2} a^{2} b^{3} c +30 a \,b^{5} x^{2}+30 a^{2} b^{4} x}{30 b^{5} d^{6} \left (2 c x +b \right )^{5}}\) | \(147\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (67) = 134\).
Time = 0.34 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.04 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=-\frac {30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2} + 20 \, {\left (2 \, b^{2} c^{2} + a c^{3}\right )} x^{2} + 10 \, {\left (b^{3} c + 2 \, a b c^{2}\right )} x}{60 \, {\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (68) = 136\).
Time = 0.73 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.16 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=\frac {- 6 a^{2} c^{2} - 2 a b^{2} c - b^{4} - 60 b c^{3} x^{3} - 30 c^{4} x^{4} + x^{2} \left (- 20 a c^{3} - 40 b^{2} c^{2}\right ) + x \left (- 20 a b c^{2} - 10 b^{3} c\right )}{60 b^{5} c^{3} d^{6} + 600 b^{4} c^{4} d^{6} x + 2400 b^{3} c^{5} d^{6} x^{2} + 4800 b^{2} c^{6} d^{6} x^{3} + 4800 b c^{7} d^{6} x^{4} + 1920 c^{8} d^{6} x^{5}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (67) = 134\).
Time = 0.21 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.04 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=-\frac {30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2} + 20 \, {\left (2 \, b^{2} c^{2} + a c^{3}\right )} x^{2} + 10 \, {\left (b^{3} c + 2 \, a b c^{2}\right )} x}{60 \, {\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )}} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=-\frac {30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + 40 \, b^{2} c^{2} x^{2} + 20 \, a c^{3} x^{2} + 10 \, b^{3} c x + 20 \, a b c^{2} x + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2}}{60 \, {\left (2 \, c x + b\right )}^{5} c^{3} d^{6}} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=-\frac {\frac {6\,a^2\,c^2+2\,a\,b^2\,c+b^4}{60\,c^3}+b\,x^3+\frac {c\,x^4}{2}+\frac {x^2\,\left (2\,b^2+a\,c\right )}{3\,c}+\frac {b\,x\,\left (b^2+2\,a\,c\right )}{6\,c^2}}{b^5\,d^6+10\,b^4\,c\,d^6\,x+40\,b^3\,c^2\,d^6\,x^2+80\,b^2\,c^3\,d^6\,x^3+80\,b\,c^4\,d^6\,x^4+32\,c^5\,d^6\,x^5} \]
[In]
[Out]