Integrand size = 29, antiderivative size = 65 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^9} \, dx=-\frac {(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac {2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac {d^2}{4 b^3 (a+b x)^4} \]
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Time = 0.03 (sec) , antiderivative size = 65, 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} \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^9} \, dx=-\frac {2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac {(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac {d^2}{4 b^3 (a+b x)^4} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {(c+d x)^2}{(a+b x)^7} \, dx \\ & = \int \left (\frac {(b c-a d)^2}{b^2 (a+b x)^7}+\frac {2 d (b c-a d)}{b^2 (a+b x)^6}+\frac {d^2}{b^2 (a+b x)^5}\right ) \, dx \\ & = -\frac {(b c-a d)^2}{6 b^3 (a+b x)^6}-\frac {2 d (b c-a d)}{5 b^3 (a+b x)^5}-\frac {d^2}{4 b^3 (a+b x)^4} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^9} \, dx=-\frac {a^2 d^2+2 a b d (2 c+3 d x)+b^2 \left (10 c^2+24 c d x+15 d^2 x^2\right )}{60 b^3 (a+b x)^6} \]
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Time = 2.52 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95
method | result | size |
gosper | \(-\frac {15 d^{2} x^{2} b^{2}+6 x a b \,d^{2}+24 x \,b^{2} c d +a^{2} d^{2}+4 a b c d +10 b^{2} c^{2}}{60 b^{3} \left (b x +a \right )^{6}}\) | \(62\) |
risch | \(\frac {-\frac {d^{2} x^{2}}{4 b}-\frac {d \left (a d +4 b c \right ) x}{10 b^{2}}-\frac {a^{2} d^{2}+4 a b c d +10 b^{2} c^{2}}{60 b^{3}}}{\left (b x +a \right )^{6}}\) | \(63\) |
parallelrisch | \(\frac {-15 d^{2} x^{2} b^{5}-6 a \,b^{4} d^{2} x -24 b^{5} c d x -a^{2} b^{3} d^{2}-4 a \,b^{4} c d -10 b^{5} c^{2}}{60 b^{6} \left (b x +a \right )^{6}}\) | \(70\) |
default | \(\frac {2 \left (a d -b c \right ) d}{5 b^{3} \left (b x +a \right )^{5}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{6 b^{3} \left (b x +a \right )^{6}}-\frac {d^{2}}{4 b^{3} \left (b x +a \right )^{4}}\) | \(71\) |
norman | \(\frac {\frac {a^{2} \left (-a^{2} b^{5} d^{2}-4 a \,b^{6} c d -10 b^{7} c^{2}\right )}{60 b^{8}}-\frac {b \,d^{2} x^{4}}{4}+\frac {\left (-3 a \,b^{5} d^{2}-2 c d \,b^{6}\right ) x^{3}}{5 b^{5}}+\frac {\left (-14 a^{2} b^{5} d^{2}-26 a \,b^{6} c d -5 b^{7} c^{2}\right ) x^{2}}{30 b^{6}}+\frac {a \left (-2 a^{2} b^{5} d^{2}-8 a \,b^{6} c d -5 b^{7} c^{2}\right ) x}{15 b^{7}}}{\left (b x +a \right )^{8}}\) | \(151\) |
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Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (59) = 118\).
Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^9} \, dx=-\frac {15 \, b^{2} d^{2} x^{2} + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} + 6 \, {\left (4 \, b^{2} c d + a b d^{2}\right )} x}{60 \, {\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (58) = 116\).
Time = 1.05 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.97 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^9} \, dx=\frac {- a^{2} d^{2} - 4 a b c d - 10 b^{2} c^{2} - 15 b^{2} d^{2} x^{2} + x \left (- 6 a b d^{2} - 24 b^{2} c d\right )}{60 a^{6} b^{3} + 360 a^{5} b^{4} x + 900 a^{4} b^{5} x^{2} + 1200 a^{3} b^{6} x^{3} + 900 a^{2} b^{7} x^{4} + 360 a b^{8} x^{5} + 60 b^{9} x^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (59) = 118\).
Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.85 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^9} \, dx=-\frac {15 \, b^{2} d^{2} x^{2} + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2} + 6 \, {\left (4 \, b^{2} c d + a b d^{2}\right )} x}{60 \, {\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^9} \, dx=-\frac {15 \, b^{2} d^{2} x^{2} + 24 \, b^{2} c d x + 6 \, a b d^{2} x + 10 \, b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}}{60 \, {\left (b x + a\right )}^{6} b^{3}} \]
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Time = 9.86 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.82 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^9} \, dx=-\frac {\frac {a^2\,d^2+4\,a\,b\,c\,d+10\,b^2\,c^2}{60\,b^3}+\frac {d^2\,x^2}{4\,b}+\frac {d\,x\,\left (a\,d+4\,b\,c\right )}{10\,b^2}}{a^6+6\,a^5\,b\,x+15\,a^4\,b^2\,x^2+20\,a^3\,b^3\,x^3+15\,a^2\,b^4\,x^4+6\,a\,b^5\,x^5+b^6\,x^6} \]
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