Integrand size = 15, antiderivative size = 78 \[ \int \frac {(c+d x)^3}{(a+b x)^3} \, dx=\frac {d^3 x}{b^3}-\frac {(b c-a d)^3}{2 b^4 (a+b x)^2}-\frac {3 d (b c-a d)^2}{b^4 (a+b x)}+\frac {3 d^2 (b c-a d) \log (a+b x)}{b^4} \]
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Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {(c+d x)^3}{(a+b x)^3} \, dx=\frac {3 d^2 (b c-a d) \log (a+b x)}{b^4}-\frac {3 d (b c-a d)^2}{b^4 (a+b x)}-\frac {(b c-a d)^3}{2 b^4 (a+b x)^2}+\frac {d^3 x}{b^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^3}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)^3}+\frac {3 d (b c-a d)^2}{b^3 (a+b x)^2}+\frac {3 d^2 (b c-a d)}{b^3 (a+b x)}\right ) \, dx \\ & = \frac {d^3 x}{b^3}-\frac {(b c-a d)^3}{2 b^4 (a+b x)^2}-\frac {3 d (b c-a d)^2}{b^4 (a+b x)}+\frac {3 d^2 (b c-a d) \log (a+b x)}{b^4} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.46 \[ \int \frac {(c+d x)^3}{(a+b x)^3} \, dx=\frac {-5 a^3 d^3+a^2 b d^2 (9 c-4 d x)+a b^2 d \left (-3 c^2+12 c d x+4 d^2 x^2\right )-b^3 \left (c^3+6 c^2 d x-2 d^3 x^3\right )-6 d^2 (-b c+a d) (a+b x)^2 \log (a+b x)}{2 b^4 (a+b x)^2} \]
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Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.46
method | result | size |
default | \(\frac {d^{3} x}{b^{3}}-\frac {3 d^{2} \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{4}}-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{2 b^{4} \left (b x +a \right )^{2}}-\frac {3 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{4} \left (b x +a \right )}\) | \(114\) |
norman | \(\frac {\frac {d^{3} x^{3}}{b}-\frac {9 a^{3} d^{3}-9 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}}{2 b^{4}}-\frac {\left (6 a^{2} d^{3}-6 a b c \,d^{2}+3 b^{2} c^{2} d \right ) x}{b^{3}}}{\left (b x +a \right )^{2}}-\frac {3 d^{2} \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{4}}\) | \(116\) |
risch | \(\frac {d^{3} x}{b^{3}}+\frac {\left (-3 a^{2} d^{3}+6 a b c \,d^{2}-3 b^{2} c^{2} d \right ) x -\frac {5 a^{3} d^{3}-9 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}}{2 b}}{b^{3} \left (b x +a \right )^{2}}-\frac {3 d^{3} \ln \left (b x +a \right ) a}{b^{4}}+\frac {3 d^{2} \ln \left (b x +a \right ) c}{b^{3}}\) | \(121\) |
parallelrisch | \(-\frac {6 \ln \left (b x +a \right ) x^{2} a \,b^{2} d^{3}-6 \ln \left (b x +a \right ) x^{2} b^{3} c \,d^{2}-2 d^{3} x^{3} b^{3}+12 \ln \left (b x +a \right ) x \,a^{2} b \,d^{3}-12 \ln \left (b x +a \right ) x a \,b^{2} c \,d^{2}+6 \ln \left (b x +a \right ) a^{3} d^{3}-6 \ln \left (b x +a \right ) a^{2} b c \,d^{2}+12 x \,a^{2} b \,d^{3}-12 x a \,b^{2} c \,d^{2}+6 x \,b^{3} c^{2} d +9 a^{3} d^{3}-9 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}}{2 b^{4} \left (b x +a \right )^{2}}\) | \(190\) |
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Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (76) = 152\).
Time = 0.23 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.41 \[ \int \frac {(c+d x)^3}{(a+b x)^3} \, dx=\frac {2 \, b^{3} d^{3} x^{3} + 4 \, a b^{2} d^{3} x^{2} - b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3} - 2 \, {\left (3 \, b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x + 6 \, {\left (a^{2} b c d^{2} - a^{3} d^{3} + {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b^{2} c d^{2} - a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} \]
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Time = 0.51 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.64 \[ \int \frac {(c+d x)^3}{(a+b x)^3} \, dx=\frac {- 5 a^{3} d^{3} + 9 a^{2} b c d^{2} - 3 a b^{2} c^{2} d - b^{3} c^{3} + x \left (- 6 a^{2} b d^{3} + 12 a b^{2} c d^{2} - 6 b^{3} c^{2} d\right )}{2 a^{2} b^{4} + 4 a b^{5} x + 2 b^{6} x^{2}} + \frac {d^{3} x}{b^{3}} - \frac {3 d^{2} \left (a d - b c\right ) \log {\left (a + b x \right )}}{b^{4}} \]
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none
Time = 0.20 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.60 \[ \int \frac {(c+d x)^3}{(a+b x)^3} \, dx=\frac {d^{3} x}{b^{3}} - \frac {b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3} + 6 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \, {\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}} + \frac {3 \, {\left (b c d^{2} - a d^{3}\right )} \log \left (b x + a\right )}{b^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.44 \[ \int \frac {(c+d x)^3}{(a+b x)^3} \, dx=\frac {d^{3} x}{b^{3}} + \frac {3 \, {\left (b c d^{2} - a d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4}} - \frac {b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3} + 6 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{4}} \]
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Time = 0.38 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.67 \[ \int \frac {(c+d x)^3}{(a+b x)^3} \, dx=\frac {d^3\,x}{b^3}-\frac {\ln \left (a+b\,x\right )\,\left (3\,a\,d^3-3\,b\,c\,d^2\right )}{b^4}-\frac {\frac {5\,a^3\,d^3-9\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d+b^3\,c^3}{2\,b}+x\,\left (3\,a^2\,d^3-6\,a\,b\,c\,d^2+3\,b^2\,c^2\,d\right )}{a^2\,b^3+2\,a\,b^4\,x+b^5\,x^2} \]
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