Integrand size = 29, antiderivative size = 65 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^8} \, dx=-\frac {(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac {d (b c-a d)}{2 b^3 (a+b x)^4}-\frac {d^2}{3 b^3 (a+b x)^3} \]
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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} \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^8} \, dx=-\frac {d (b c-a d)}{2 b^3 (a+b x)^4}-\frac {(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac {d^2}{3 b^3 (a+b x)^3} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {(c+d x)^2}{(a+b x)^6} \, dx \\ & = \int \left (\frac {(b c-a d)^2}{b^2 (a+b x)^6}+\frac {2 d (b c-a d)}{b^2 (a+b x)^5}+\frac {d^2}{b^2 (a+b x)^4}\right ) \, dx \\ & = -\frac {(b c-a d)^2}{5 b^3 (a+b x)^5}-\frac {d (b c-a d)}{2 b^3 (a+b x)^4}-\frac {d^2}{3 b^3 (a+b x)^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^8} \, dx=-\frac {a^2 d^2+a b d (3 c+5 d x)+b^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )}{30 b^3 (a+b x)^5} \]
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Time = 2.35 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95
method | result | size |
gosper | \(-\frac {10 d^{2} x^{2} b^{2}+5 x a b \,d^{2}+15 x \,b^{2} c d +a^{2} d^{2}+3 a b c d +6 b^{2} c^{2}}{30 b^{3} \left (b x +a \right )^{5}}\) | \(62\) |
risch | \(\frac {-\frac {d^{2} x^{2}}{3 b}-\frac {d \left (a d +3 b c \right ) x}{6 b^{2}}-\frac {a^{2} d^{2}+3 a b c d +6 b^{2} c^{2}}{30 b^{3}}}{\left (b x +a \right )^{5}}\) | \(63\) |
parallelrisch | \(\frac {-10 d^{2} x^{2} b^{4}-5 a \,b^{3} d^{2} x -15 b^{4} c d x -b^{2} d^{2} a^{2}-3 a \,b^{3} c d -6 b^{4} c^{2}}{30 b^{5} \left (b x +a \right )^{5}}\) | \(70\) |
default | \(-\frac {d^{2}}{3 b^{3} \left (b x +a \right )^{3}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{5 b^{3} \left (b x +a \right )^{5}}+\frac {\left (a d -b c \right ) d}{2 b^{3} \left (b x +a \right )^{4}}\) | \(71\) |
norman | \(\frac {\frac {a^{2} \left (-a^{2} b^{4} d^{2}-3 a c d \,b^{5}-6 c^{2} b^{6}\right )}{30 b^{7}}-\frac {b \,d^{2} x^{4}}{3}+\frac {\left (-5 a \,b^{4} d^{2}-3 b^{5} c d \right ) x^{3}}{6 b^{4}}+\frac {\left (-7 a^{2} b^{4} d^{2}-11 a c d \,b^{5}-2 c^{2} b^{6}\right ) x^{2}}{10 b^{5}}+\frac {a \left (-7 a^{2} b^{4} d^{2}-21 a c d \,b^{5}-12 c^{2} b^{6}\right ) x}{30 b^{6}}}{\left (b x +a \right )^{7}}\) | \(151\) |
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Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^8} \, dx=-\frac {10 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2} + 5 \, {\left (3 \, b^{2} c d + a b d^{2}\right )} x}{30 \, {\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (56) = 112\).
Time = 0.63 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.78 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^8} \, dx=\frac {- a^{2} d^{2} - 3 a b c d - 6 b^{2} c^{2} - 10 b^{2} d^{2} x^{2} + x \left (- 5 a b d^{2} - 15 b^{2} c d\right )}{30 a^{5} b^{3} + 150 a^{4} b^{4} x + 300 a^{3} b^{5} x^{2} + 300 a^{2} b^{6} x^{3} + 150 a b^{7} x^{4} + 30 b^{8} x^{5}} \]
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Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^8} \, dx=-\frac {10 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2} + 5 \, {\left (3 \, b^{2} c d + a b d^{2}\right )} x}{30 \, {\left (b^{8} x^{5} + 5 \, a b^{7} x^{4} + 10 \, a^{2} b^{6} x^{3} + 10 \, a^{3} b^{5} x^{2} + 5 \, a^{4} b^{4} x + a^{5} b^{3}\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^8} \, dx=-\frac {10 \, b^{2} d^{2} x^{2} + 15 \, b^{2} c d x + 5 \, a b d^{2} x + 6 \, b^{2} c^{2} + 3 \, a b c d + a^{2} d^{2}}{30 \, {\left (b x + a\right )}^{5} b^{3}} \]
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Time = 9.79 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.65 \[ \int \frac {\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^8} \, dx=-\frac {\frac {a^2\,d^2+3\,a\,b\,c\,d+6\,b^2\,c^2}{30\,b^3}+\frac {d^2\,x^2}{3\,b}+\frac {d\,x\,\left (a\,d+3\,b\,c\right )}{6\,b^2}}{a^5+5\,a^4\,b\,x+10\,a^3\,b^2\,x^2+10\,a^2\,b^3\,x^3+5\,a\,b^4\,x^4+b^5\,x^5} \]
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