Integrand size = 29, antiderivative size = 92 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^{12}} \, dx=-\frac {(b c-a d)^3}{8 b^4 (a+b x)^8}-\frac {3 d (b c-a d)^2}{7 b^4 (a+b x)^7}-\frac {d^2 (b c-a d)}{2 b^4 (a+b x)^6}-\frac {d^3}{5 b^4 (a+b x)^5} \]
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Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {640, 45} \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^{12}} \, dx=-\frac {d^2 (b c-a d)}{2 b^4 (a+b x)^6}-\frac {3 d (b c-a d)^2}{7 b^4 (a+b x)^7}-\frac {(b c-a d)^3}{8 b^4 (a+b x)^8}-\frac {d^3}{5 b^4 (a+b x)^5} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {(c+d x)^3}{(a+b x)^9} \, dx \\ & = \int \left (\frac {(b c-a d)^3}{b^3 (a+b x)^9}+\frac {3 d (b c-a d)^2}{b^3 (a+b x)^8}+\frac {3 d^2 (b c-a d)}{b^3 (a+b x)^7}+\frac {d^3}{b^3 (a+b x)^6}\right ) \, dx \\ & = -\frac {(b c-a d)^3}{8 b^4 (a+b x)^8}-\frac {3 d (b c-a d)^2}{7 b^4 (a+b x)^7}-\frac {d^2 (b c-a d)}{2 b^4 (a+b x)^6}-\frac {d^3}{5 b^4 (a+b x)^5} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^{12}} \, dx=-\frac {a^3 d^3+a^2 b d^2 (5 c+8 d x)+a b^2 d \left (15 c^2+40 c d x+28 d^2 x^2\right )+b^3 \left (35 c^3+120 c^2 d x+140 c d^2 x^2+56 d^3 x^3\right )}{280 b^4 (a+b x)^8} \]
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Time = 2.36 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.20
method | result | size |
risch | \(\frac {-\frac {d^{3} x^{3}}{5 b}-\frac {d^{2} \left (a d +5 b c \right ) x^{2}}{10 b^{2}}-\frac {d \left (a^{2} d^{2}+5 a b c d +15 b^{2} c^{2}\right ) x}{35 b^{3}}-\frac {a^{3} d^{3}+5 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d +35 b^{3} c^{3}}{280 b^{4}}}{\left (b x +a \right )^{8}}\) | \(110\) |
gosper | \(-\frac {56 d^{3} x^{3} b^{3}+28 x^{2} a \,b^{2} d^{3}+140 x^{2} b^{3} c \,d^{2}+8 x \,a^{2} b \,d^{3}+40 x a \,b^{2} c \,d^{2}+120 x \,b^{3} c^{2} d +a^{3} d^{3}+5 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d +35 b^{3} c^{3}}{280 b^{4} \left (b x +a \right )^{8}}\) | \(115\) |
default | \(-\frac {d^{3}}{5 b^{4} \left (b x +a \right )^{5}}+\frac {d^{2} \left (a d -b c \right )}{2 b^{4} \left (b x +a \right )^{6}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{8 b^{4} \left (b x +a \right )^{8}}-\frac {3 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{7 b^{4} \left (b x +a \right )^{7}}\) | \(122\) |
parallelrisch | \(\frac {-56 d^{3} x^{3} b^{7}-28 a \,b^{6} d^{3} x^{2}-140 b^{7} c \,d^{2} x^{2}-8 a^{2} b^{5} d^{3} x -40 a \,b^{6} c \,d^{2} x -120 b^{7} c^{2} d x -a^{3} b^{4} d^{3}-5 a^{2} b^{5} c \,d^{2}-15 a \,c^{2} d \,b^{6}-35 c^{3} b^{7}}{280 b^{8} \left (b x +a \right )^{8}}\) | \(123\) |
norman | \(\frac {\frac {a^{3} \left (-a^{3} b^{7} d^{3}-5 a^{2} b^{8} c \,d^{2}-15 a \,b^{9} c^{2} d -35 b^{10} c^{3}\right )}{280 b^{11}}-\frac {b^{2} d^{3} x^{6}}{5}+\frac {\left (-7 a \,d^{3} b^{7}-5 c \,d^{2} b^{8}\right ) x^{5}}{10 b^{6}}+\frac {\left (-13 a^{2} b^{7} d^{3}-23 a c \,d^{2} b^{8}-6 c^{2} d \,b^{9}\right ) x^{4}}{14 b^{7}}+\frac {\left (-33 a^{3} b^{7} d^{3}-109 a^{2} b^{8} c \,d^{2}-75 a \,b^{9} c^{2} d -7 b^{10} c^{3}\right ) x^{3}}{56 b^{8}}+\frac {a \left (-11 a^{3} b^{7} d^{3}-55 a^{2} b^{8} c \,d^{2}-81 a \,b^{9} c^{2} d -21 b^{10} c^{3}\right ) x^{2}}{56 b^{9}}+\frac {a^{2} \left (-11 a^{3} b^{7} d^{3}-55 a^{2} b^{8} c \,d^{2}-165 a \,b^{9} c^{2} d -105 b^{10} c^{3}\right ) x}{280 b^{10}}}{\left (b x +a \right )^{11}}\) | \(289\) |
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Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (84) = 168\).
Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.10 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^{12}} \, dx=-\frac {56 \, b^{3} d^{3} x^{3} + 35 \, b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3} + 28 \, {\left (5 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 8 \, {\left (15 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{280 \, {\left (b^{12} x^{8} + 8 \, a b^{11} x^{7} + 28 \, a^{2} b^{10} x^{6} + 56 \, a^{3} b^{9} x^{5} + 70 \, a^{4} b^{8} x^{4} + 56 \, a^{5} b^{7} x^{3} + 28 \, a^{6} b^{6} x^{2} + 8 \, a^{7} b^{5} x + a^{8} b^{4}\right )}} \]
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Timed out. \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^{12}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (84) = 168\).
Time = 0.21 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.10 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^{12}} \, dx=-\frac {56 \, b^{3} d^{3} x^{3} + 35 \, b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3} + 28 \, {\left (5 \, b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{2} + 8 \, {\left (15 \, b^{3} c^{2} d + 5 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{280 \, {\left (b^{12} x^{8} + 8 \, a b^{11} x^{7} + 28 \, a^{2} b^{10} x^{6} + 56 \, a^{3} b^{9} x^{5} + 70 \, a^{4} b^{8} x^{4} + 56 \, a^{5} b^{7} x^{3} + 28 \, a^{6} b^{6} x^{2} + 8 \, a^{7} b^{5} x + a^{8} b^{4}\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^{12}} \, dx=-\frac {56 \, b^{3} d^{3} x^{3} + 140 \, b^{3} c d^{2} x^{2} + 28 \, a b^{2} d^{3} x^{2} + 120 \, b^{3} c^{2} d x + 40 \, a b^{2} c d^{2} x + 8 \, a^{2} b d^{3} x + 35 \, b^{3} c^{3} + 15 \, a b^{2} c^{2} d + 5 \, a^{2} b c d^{2} + a^{3} d^{3}}{280 \, {\left (b x + a\right )}^{8} b^{4}} \]
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Time = 0.10 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.03 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^3}{(a+b x)^{12}} \, dx=-\frac {\frac {a^3\,d^3+5\,a^2\,b\,c\,d^2+15\,a\,b^2\,c^2\,d+35\,b^3\,c^3}{280\,b^4}+\frac {d^3\,x^3}{5\,b}+\frac {d\,x\,\left (a^2\,d^2+5\,a\,b\,c\,d+15\,b^2\,c^2\right )}{35\,b^3}+\frac {d^2\,x^2\,\left (a\,d+5\,b\,c\right )}{10\,b^2}}{a^8+8\,a^7\,b\,x+28\,a^6\,b^2\,x^2+56\,a^5\,b^3\,x^3+70\,a^4\,b^4\,x^4+56\,a^3\,b^5\,x^5+28\,a^2\,b^6\,x^6+8\,a\,b^7\,x^7+b^8\,x^8} \]
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