Integrand size = 16, antiderivative size = 114 \[ \int \frac {x (c+d x)^3}{(a+b x)^3} \, dx=\frac {3 d^2 (b c-a d) x}{b^4}+\frac {d^3 x^2}{2 b^3}+\frac {a (b c-a d)^3}{2 b^5 (a+b x)^2}-\frac {(b c-4 a d) (b c-a d)^2}{b^5 (a+b x)}+\frac {3 d (b c-2 a d) (b c-a d) \log (a+b x)}{b^5} \]
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Time = 0.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int \frac {x (c+d x)^3}{(a+b x)^3} \, dx=-\frac {(b c-4 a d) (b c-a d)^2}{b^5 (a+b x)}+\frac {a (b c-a d)^3}{2 b^5 (a+b x)^2}+\frac {3 d (b c-2 a d) (b c-a d) \log (a+b x)}{b^5}+\frac {3 d^2 x (b c-a d)}{b^4}+\frac {d^3 x^2}{2 b^3} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 d^2 (b c-a d)}{b^4}+\frac {d^3 x}{b^3}+\frac {a (-b c+a d)^3}{b^4 (a+b x)^3}+\frac {(b c-4 a d) (b c-a d)^2}{b^4 (a+b x)^2}+\frac {3 d (b c-2 a d) (b c-a d)}{b^4 (a+b x)}\right ) \, dx \\ & = \frac {3 d^2 (b c-a d) x}{b^4}+\frac {d^3 x^2}{2 b^3}+\frac {a (b c-a d)^3}{2 b^5 (a+b x)^2}-\frac {(b c-4 a d) (b c-a d)^2}{b^5 (a+b x)}+\frac {3 d (b c-2 a d) (b c-a d) \log (a+b x)}{b^5} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.45 \[ \int \frac {x (c+d x)^3}{(a+b x)^3} \, dx=\frac {7 a^4 d^3+a^3 b d^2 (-15 c+2 d x)+a^2 b^2 d \left (9 c^2-12 c d x-11 d^2 x^2\right )+b^4 x \left (-2 c^3+6 c d^2 x^2+d^3 x^3\right )-a b^3 \left (c^3-12 c^2 d x-12 c d^2 x^2+4 d^3 x^3\right )+6 d \left (b^2 c^2-3 a b c d+2 a^2 d^2\right ) (a+b x)^2 \log (a+b x)}{2 b^5 (a+b x)^2} \]
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Time = 0.46 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.39
method | result | size |
default | \(-\frac {d^{2} \left (-\frac {1}{2} b d \,x^{2}+3 a d x -3 b c x \right )}{b^{4}}+\frac {3 d \left (2 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{5}}-\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 b^{5} \left (b x +a \right )^{2}}-\frac {-4 a^{3} d^{3}+9 a^{2} b c \,d^{2}-6 a \,b^{2} c^{2} d +b^{3} c^{3}}{b^{5} \left (b x +a \right )}\) | \(159\) |
norman | \(\frac {\frac {\left (12 a^{3} d^{3}-18 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{b^{4}}+\frac {a \left (18 a^{3} d^{3}-27 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 b^{5}}+\frac {d^{3} x^{4}}{2 b}-\frac {d^{2} \left (2 a d -3 b c \right ) x^{3}}{b^{2}}}{\left (b x +a \right )^{2}}+\frac {3 d \left (2 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) \ln \left (b x +a \right )}{b^{5}}\) | \(161\) |
risch | \(\frac {d^{3} x^{2}}{2 b^{3}}-\frac {3 d^{3} a x}{b^{4}}+\frac {3 d^{2} c x}{b^{3}}+\frac {\left (4 a^{3} d^{3}-9 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x +\frac {a \left (7 a^{3} d^{3}-15 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 b}}{b^{4} \left (b x +a \right )^{2}}+\frac {6 d^{3} \ln \left (b x +a \right ) a^{2}}{b^{5}}-\frac {9 d^{2} \ln \left (b x +a \right ) a c}{b^{4}}+\frac {3 d \ln \left (b x +a \right ) c^{2}}{b^{3}}\) | \(175\) |
parallelrisch | \(\frac {d^{3} x^{4} b^{4}+12 \ln \left (b x +a \right ) x^{2} a^{2} b^{2} d^{3}-18 \ln \left (b x +a \right ) x^{2} a \,b^{3} c \,d^{2}+6 \ln \left (b x +a \right ) x^{2} b^{4} c^{2} d -4 x^{3} a \,b^{3} d^{3}+6 x^{3} b^{4} c \,d^{2}+24 \ln \left (b x +a \right ) x \,a^{3} b \,d^{3}-36 \ln \left (b x +a \right ) x \,a^{2} b^{2} c \,d^{2}+12 \ln \left (b x +a \right ) x a \,b^{3} c^{2} d +12 \ln \left (b x +a \right ) a^{4} d^{3}-18 \ln \left (b x +a \right ) a^{3} b c \,d^{2}+6 \ln \left (b x +a \right ) a^{2} b^{2} c^{2} d +24 x \,a^{3} b \,d^{3}-36 x \,a^{2} b^{2} c \,d^{2}+12 x a \,b^{3} c^{2} d -2 b^{4} c^{3} x +18 a^{4} d^{3}-27 a^{3} b c \,d^{2}+9 a^{2} b^{2} c^{2} d -a \,b^{3} c^{3}}{2 b^{5} \left (b x +a \right )^{2}}\) | \(287\) |
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Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (110) = 220\).
Time = 0.23 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.40 \[ \int \frac {x (c+d x)^3}{(a+b x)^3} \, dx=\frac {b^{4} d^{3} x^{4} - a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 15 \, a^{3} b c d^{2} + 7 \, a^{4} d^{3} + 2 \, {\left (3 \, b^{4} c d^{2} - 2 \, a b^{3} d^{3}\right )} x^{3} + {\left (12 \, a b^{3} c d^{2} - 11 \, a^{2} b^{2} d^{3}\right )} x^{2} - 2 \, {\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 6 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x + 6 \, {\left (a^{2} b^{2} c^{2} d - 3 \, a^{3} b c d^{2} + 2 \, a^{4} d^{3} + {\left (b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} d - 3 \, a^{2} b^{2} c d^{2} + 2 \, a^{3} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} \]
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Time = 0.57 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.54 \[ \int \frac {x (c+d x)^3}{(a+b x)^3} \, dx=x \left (- \frac {3 a d^{3}}{b^{4}} + \frac {3 c d^{2}}{b^{3}}\right ) + \frac {7 a^{4} d^{3} - 15 a^{3} b c d^{2} + 9 a^{2} b^{2} c^{2} d - a b^{3} c^{3} + x \left (8 a^{3} b d^{3} - 18 a^{2} b^{2} c d^{2} + 12 a b^{3} c^{2} d - 2 b^{4} c^{3}\right )}{2 a^{2} b^{5} + 4 a b^{6} x + 2 b^{7} x^{2}} + \frac {d^{3} x^{2}}{2 b^{3}} + \frac {3 d \left (a d - b c\right ) \left (2 a d - b c\right ) \log {\left (a + b x \right )}}{b^{5}} \]
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Time = 0.22 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.53 \[ \int \frac {x (c+d x)^3}{(a+b x)^3} \, dx=-\frac {a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} + 2 \, {\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x}{2 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}} + \frac {b d^{3} x^{2} + 6 \, {\left (b c d^{2} - a d^{3}\right )} x}{2 \, b^{4}} + \frac {3 \, {\left (b^{2} c^{2} d - 3 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} \log \left (b x + a\right )}{b^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.46 \[ \int \frac {x (c+d x)^3}{(a+b x)^3} \, dx=\frac {3 \, {\left (b^{2} c^{2} d - 3 \, a b c d^{2} + 2 \, a^{2} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{5}} + \frac {b^{3} d^{3} x^{2} + 6 \, b^{3} c d^{2} x - 6 \, a b^{2} d^{3} x}{2 \, b^{6}} - \frac {a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 15 \, a^{3} b c d^{2} - 7 \, a^{4} d^{3} + 2 \, {\left (b^{4} c^{3} - 6 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 4 \, a^{3} b d^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{5}} \]
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Time = 0.46 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.58 \[ \int \frac {x (c+d x)^3}{(a+b x)^3} \, dx=\frac {x\,\left (4\,a^3\,d^3-9\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d-b^3\,c^3\right )+\frac {7\,a^4\,d^3-15\,a^3\,b\,c\,d^2+9\,a^2\,b^2\,c^2\,d-a\,b^3\,c^3}{2\,b}}{a^2\,b^4+2\,a\,b^5\,x+b^6\,x^2}-x\,\left (\frac {3\,a\,d^3}{b^4}-\frac {3\,c\,d^2}{b^3}\right )+\frac {\ln \left (a+b\,x\right )\,\left (6\,a^2\,d^3-9\,a\,b\,c\,d^2+3\,b^2\,c^2\,d\right )}{b^5}+\frac {d^3\,x^2}{2\,b^3} \]
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