Integrand size = 15, antiderivative size = 28 \[ \int \frac {(c+d x)^2}{(a+b x)^4} \, dx=-\frac {(c+d x)^3}{3 (b c-a d) (a+b x)^3} \]
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Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {37} \[ \int \frac {(c+d x)^2}{(a+b x)^4} \, dx=-\frac {(c+d x)^3}{3 (a+b x)^3 (b c-a d)} \]
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Rule 37
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^3}{3 (b c-a d) (a+b x)^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {(c+d x)^2}{(a+b x)^4} \, dx=-\frac {a^2 d^2+a b d (c+3 d x)+b^2 \left (c^2+3 c d x+3 d^2 x^2\right )}{3 b^3 (a+b x)^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(26)=52\).
Time = 0.39 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14
method | result | size |
gosper | \(-\frac {3 d^{2} x^{2} b^{2}+3 x a b \,d^{2}+3 x \,b^{2} c d +a^{2} d^{2}+a b c d +b^{2} c^{2}}{3 \left (b x +a \right )^{3} b^{3}}\) | \(60\) |
risch | \(\frac {-\frac {d^{2} x^{2}}{b}-\frac {d \left (a d +b c \right ) x}{b^{2}}-\frac {a^{2} d^{2}+a b c d +b^{2} c^{2}}{3 b^{3}}}{\left (b x +a \right )^{3}}\) | \(60\) |
parallelrisch | \(\frac {-3 d^{2} x^{2} b^{2}-3 x a b \,d^{2}-3 x \,b^{2} c d -a^{2} d^{2}-a b c d -b^{2} c^{2}}{3 b^{3} \left (b x +a \right )^{3}}\) | \(63\) |
norman | \(\frac {-\frac {d^{2} x^{2}}{b}+\frac {\left (-a \,d^{2}-b c d \right ) x}{b^{2}}+\frac {-a^{2} d^{2}-a b c d -b^{2} c^{2}}{3 b^{3}}}{\left (b x +a \right )^{3}}\) | \(66\) |
default | \(-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{3 b^{3} \left (b x +a \right )^{3}}+\frac {d \left (a d -b c \right )}{b^{3} \left (b x +a \right )^{2}}-\frac {d^{2}}{b^{3} \left (b x +a \right )}\) | \(70\) |
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (26) = 52\).
Time = 0.23 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00 \[ \int \frac {(c+d x)^2}{(a+b x)^4} \, dx=-\frac {3 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + a b c d + a^{2} d^{2} + 3 \, {\left (b^{2} c d + a b d^{2}\right )} x}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (22) = 44\).
Time = 0.35 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.14 \[ \int \frac {(c+d x)^2}{(a+b x)^4} \, dx=\frac {- a^{2} d^{2} - a b c d - b^{2} c^{2} - 3 b^{2} d^{2} x^{2} + x \left (- 3 a b d^{2} - 3 b^{2} c d\right )}{3 a^{3} b^{3} + 9 a^{2} b^{4} x + 9 a b^{5} x^{2} + 3 b^{6} x^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (26) = 52\).
Time = 0.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00 \[ \int \frac {(c+d x)^2}{(a+b x)^4} \, dx=-\frac {3 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + a b c d + a^{2} d^{2} + 3 \, {\left (b^{2} c d + a b d^{2}\right )} x}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (26) = 52\).
Time = 0.29 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {(c+d x)^2}{(a+b x)^4} \, dx=-\frac {3 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c d x + 3 \, a b d^{2} x + b^{2} c^{2} + a b c d + a^{2} d^{2}}{3 \, {\left (b x + a\right )}^{3} b^{3}} \]
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Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.86 \[ \int \frac {(c+d x)^2}{(a+b x)^4} \, dx=-\frac {\frac {a^2\,d^2+a\,b\,c\,d+b^2\,c^2}{3\,b^3}+\frac {d^2\,x^2}{b}+\frac {d\,x\,\left (a\,d+b\,c\right )}{b^2}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]
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