Integrand size = 18, antiderivative size = 196 \[ \int \frac {x^3 (c+d x)^3}{(a+b x)^3} \, dx=\frac {(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right ) x}{b^6}+\frac {3 d (b c-2 a d) (b c-a d) x^2}{2 b^5}+\frac {d^2 (b c-a d) x^3}{b^4}+\frac {d^3 x^4}{4 b^3}+\frac {a^3 (b c-a d)^3}{2 b^7 (a+b x)^2}-\frac {3 a^2 (b c-2 a d) (b c-a d)^2}{b^7 (a+b x)}-\frac {3 a (b c-a d) \left (b^2 c^2-5 a b c d+5 a^2 d^2\right ) \log (a+b x)}{b^7} \]
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Time = 0.16 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {90} \[ \int \frac {x^3 (c+d x)^3}{(a+b x)^3} \, dx=\frac {a^3 (b c-a d)^3}{2 b^7 (a+b x)^2}-\frac {3 a^2 (b c-2 a d) (b c-a d)^2}{b^7 (a+b x)}-\frac {3 a (b c-a d) \left (5 a^2 d^2-5 a b c d+b^2 c^2\right ) \log (a+b x)}{b^7}+\frac {x (b c-a d) \left (10 a^2 d^2-8 a b c d+b^2 c^2\right )}{b^6}+\frac {3 d x^2 (b c-2 a d) (b c-a d)}{2 b^5}+\frac {d^2 x^3 (b c-a d)}{b^4}+\frac {d^3 x^4}{4 b^3} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right )}{b^6}+\frac {3 d (b c-2 a d) (b c-a d) x}{b^5}+\frac {3 d^2 (b c-a d) x^2}{b^4}+\frac {d^3 x^3}{b^3}+\frac {a^3 (-b c+a d)^3}{b^6 (a+b x)^3}-\frac {3 a^2 (-b c+a d)^2 (-b c+2 a d)}{b^6 (a+b x)^2}+\frac {3 a (b c-a d) \left (-b^2 c^2+5 a b c d-5 a^2 d^2\right )}{b^6 (a+b x)}\right ) \, dx \\ & = \frac {(b c-a d) \left (b^2 c^2-8 a b c d+10 a^2 d^2\right ) x}{b^6}+\frac {3 d (b c-2 a d) (b c-a d) x^2}{2 b^5}+\frac {d^2 (b c-a d) x^3}{b^4}+\frac {d^3 x^4}{4 b^3}+\frac {a^3 (b c-a d)^3}{2 b^7 (a+b x)^2}-\frac {3 a^2 (b c-2 a d) (b c-a d)^2}{b^7 (a+b x)}-\frac {3 a (b c-a d) \left (b^2 c^2-5 a b c d+5 a^2 d^2\right ) \log (a+b x)}{b^7} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 (c+d x)^3}{(a+b x)^3} \, dx=\frac {4 b \left (b^3 c^3-9 a b^2 c^2 d+18 a^2 b c d^2-10 a^3 d^3\right ) x+6 b^2 d \left (b^2 c^2-3 a b c d+2 a^2 d^2\right ) x^2+4 b^3 d^2 (b c-a d) x^3+b^4 d^3 x^4+\frac {2 a^3 (b c-a d)^3}{(a+b x)^2}+\frac {12 a^2 (b c-a d)^2 (-b c+2 a d)}{a+b x}+12 a \left (-b^3 c^3+6 a b^2 c^2 d-10 a^2 b c d^2+5 a^3 d^3\right ) \log (a+b x)}{4 b^7} \]
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Time = 0.46 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.34
method | result | size |
norman | \(\frac {\frac {a^{2} \left (45 a^{4} d^{3}-90 a^{3} b c \,d^{2}+54 a^{2} b^{2} c^{2} d -9 a \,b^{3} c^{3}\right )}{2 b^{7}}-\frac {\left (5 a^{3} d^{3}-10 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{3}}{b^{4}}+\frac {d^{3} x^{6}}{4 b}+\frac {2 a \left (15 a^{4} d^{3}-30 a^{3} b c \,d^{2}+18 a^{2} b^{2} c^{2} d -3 a \,b^{3} c^{3}\right ) x}{b^{6}}+\frac {d \left (5 a^{2} d^{2}-10 a b c d +6 b^{2} c^{2}\right ) x^{4}}{4 b^{3}}-\frac {d^{2} \left (a d -2 b c \right ) x^{5}}{2 b^{2}}}{\left (b x +a \right )^{2}}+\frac {3 a \left (5 a^{3} d^{3}-10 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{7}}\) | \(262\) |
default | \(-\frac {-\frac {1}{4} d^{3} x^{4} b^{3}+x^{3} a \,b^{2} d^{3}-x^{3} b^{3} c \,d^{2}-3 x^{2} a^{2} b \,d^{3}+\frac {9}{2} x^{2} a \,b^{2} c \,d^{2}-\frac {3}{2} x^{2} b^{3} c^{2} d +10 a^{3} d^{3} x -18 a^{2} b c \,d^{2} x +9 a \,b^{2} c^{2} d x -b^{3} c^{3} x}{b^{6}}+\frac {3 a \left (5 a^{3} d^{3}-10 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{7}}-\frac {a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{2 b^{7} \left (b x +a \right )^{2}}+\frac {3 a^{2} \left (2 a^{3} d^{3}-5 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{7} \left (b x +a \right )}\) | \(271\) |
risch | \(\frac {d^{3} x^{4}}{4 b^{3}}-\frac {x^{3} a \,d^{3}}{b^{4}}+\frac {x^{3} c \,d^{2}}{b^{3}}+\frac {3 x^{2} a^{2} d^{3}}{b^{5}}-\frac {9 x^{2} a c \,d^{2}}{2 b^{4}}+\frac {3 x^{2} c^{2} d}{2 b^{3}}-\frac {10 a^{3} d^{3} x}{b^{6}}+\frac {18 a^{2} c \,d^{2} x}{b^{5}}-\frac {9 a \,c^{2} d x}{b^{4}}+\frac {c^{3} x}{b^{3}}+\frac {\left (6 a^{5} d^{3}-15 a^{4} b c \,d^{2}+12 a^{3} b^{2} c^{2} d -3 a^{2} b^{3} c^{3}\right ) x +\frac {a^{3} \left (11 a^{3} d^{3}-27 a^{2} b c \,d^{2}+21 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right )}{2 b}}{b^{6} \left (b x +a \right )^{2}}+\frac {15 a^{4} \ln \left (b x +a \right ) d^{3}}{b^{7}}-\frac {30 a^{3} \ln \left (b x +a \right ) c \,d^{2}}{b^{6}}+\frac {18 a^{2} \ln \left (b x +a \right ) c^{2} d}{b^{5}}-\frac {3 a \ln \left (b x +a \right ) c^{3}}{b^{4}}\) | \(288\) |
parallelrisch | \(\frac {60 \ln \left (b x +a \right ) x^{2} a^{4} b^{2} d^{3}-12 \ln \left (b x +a \right ) x^{2} a \,b^{5} c^{3}+90 a^{6} d^{3}-180 a^{5} b c \,d^{2}+108 a^{4} b^{2} c^{2} d +72 \ln \left (b x +a \right ) x^{2} a^{2} b^{4} c^{2} d -120 \ln \left (b x +a \right ) x^{2} a^{3} b^{3} c \,d^{2}+144 \ln \left (b x +a \right ) x \,a^{3} b^{3} c^{2} d -240 \ln \left (b x +a \right ) x \,a^{4} b^{2} c \,d^{2}-10 x^{4} a \,b^{5} c \,d^{2}+40 x^{3} a^{2} b^{4} c \,d^{2}-24 x^{3} a \,b^{5} c^{2} d -120 \ln \left (b x +a \right ) a^{5} b c \,d^{2}+72 \ln \left (b x +a \right ) a^{4} b^{2} c^{2} d -240 x \,a^{4} b^{2} c \,d^{2}+144 x \,a^{3} b^{3} c^{2} d -18 a^{3} b^{3} c^{3}+120 \ln \left (b x +a \right ) x \,a^{5} b \,d^{3}-24 \ln \left (b x +a \right ) x \,a^{2} b^{4} c^{3}+4 x^{3} b^{6} c^{3}+60 \ln \left (b x +a \right ) a^{6} d^{3}+x^{6} d^{3} b^{6}-2 x^{5} a \,b^{5} d^{3}+4 x^{5} b^{6} c \,d^{2}+5 x^{4} a^{2} b^{4} d^{3}+6 x^{4} b^{6} c^{2} d -20 x^{3} a^{3} b^{3} d^{3}-12 \ln \left (b x +a \right ) a^{3} b^{3} c^{3}+120 x \,a^{5} b \,d^{3}-24 x \,a^{2} b^{4} c^{3}}{4 b^{7} \left (b x +a \right )^{2}}\) | \(446\) |
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Leaf count of result is larger than twice the leaf count of optimal. 425 vs. \(2 (190) = 380\).
Time = 0.23 (sec) , antiderivative size = 425, normalized size of antiderivative = 2.17 \[ \int \frac {x^3 (c+d x)^3}{(a+b x)^3} \, dx=\frac {b^{6} d^{3} x^{6} - 10 \, a^{3} b^{3} c^{3} + 42 \, a^{4} b^{2} c^{2} d - 54 \, a^{5} b c d^{2} + 22 \, a^{6} d^{3} + 2 \, {\left (2 \, b^{6} c d^{2} - a b^{5} d^{3}\right )} x^{5} + {\left (6 \, b^{6} c^{2} d - 10 \, a b^{5} c d^{2} + 5 \, a^{2} b^{4} d^{3}\right )} x^{4} + 4 \, {\left (b^{6} c^{3} - 6 \, a b^{5} c^{2} d + 10 \, a^{2} b^{4} c d^{2} - 5 \, a^{3} b^{3} d^{3}\right )} x^{3} + 2 \, {\left (4 \, a b^{5} c^{3} - 33 \, a^{2} b^{4} c^{2} d + 63 \, a^{3} b^{3} c d^{2} - 34 \, a^{4} b^{2} d^{3}\right )} x^{2} - 4 \, {\left (2 \, a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d - 3 \, a^{4} b^{2} c d^{2} + 4 \, a^{5} b d^{3}\right )} x - 12 \, {\left (a^{3} b^{3} c^{3} - 6 \, a^{4} b^{2} c^{2} d + 10 \, a^{5} b c d^{2} - 5 \, a^{6} d^{3} + {\left (a b^{5} c^{3} - 6 \, a^{2} b^{4} c^{2} d + 10 \, a^{3} b^{3} c d^{2} - 5 \, a^{4} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a^{2} b^{4} c^{3} - 6 \, a^{3} b^{3} c^{2} d + 10 \, a^{4} b^{2} c d^{2} - 5 \, a^{5} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{4 \, {\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} \]
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Time = 0.80 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.45 \[ \int \frac {x^3 (c+d x)^3}{(a+b x)^3} \, dx=\frac {3 a \left (a d - b c\right ) \left (5 a^{2} d^{2} - 5 a b c d + b^{2} c^{2}\right ) \log {\left (a + b x \right )}}{b^{7}} + x^{3} \left (- \frac {a d^{3}}{b^{4}} + \frac {c d^{2}}{b^{3}}\right ) + x^{2} \cdot \left (\frac {3 a^{2} d^{3}}{b^{5}} - \frac {9 a c d^{2}}{2 b^{4}} + \frac {3 c^{2} d}{2 b^{3}}\right ) + x \left (- \frac {10 a^{3} d^{3}}{b^{6}} + \frac {18 a^{2} c d^{2}}{b^{5}} - \frac {9 a c^{2} d}{b^{4}} + \frac {c^{3}}{b^{3}}\right ) + \frac {11 a^{6} d^{3} - 27 a^{5} b c d^{2} + 21 a^{4} b^{2} c^{2} d - 5 a^{3} b^{3} c^{3} + x \left (12 a^{5} b d^{3} - 30 a^{4} b^{2} c d^{2} + 24 a^{3} b^{3} c^{2} d - 6 a^{2} b^{4} c^{3}\right )}{2 a^{2} b^{7} + 4 a b^{8} x + 2 b^{9} x^{2}} + \frac {d^{3} x^{4}}{4 b^{3}} \]
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Time = 0.24 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.41 \[ \int \frac {x^3 (c+d x)^3}{(a+b x)^3} \, dx=-\frac {5 \, a^{3} b^{3} c^{3} - 21 \, a^{4} b^{2} c^{2} d + 27 \, a^{5} b c d^{2} - 11 \, a^{6} d^{3} + 6 \, {\left (a^{2} b^{4} c^{3} - 4 \, a^{3} b^{3} c^{2} d + 5 \, a^{4} b^{2} c d^{2} - 2 \, a^{5} b d^{3}\right )} x}{2 \, {\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} + \frac {b^{3} d^{3} x^{4} + 4 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} + 6 \, {\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + 2 \, a^{2} b d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 10 \, a^{3} d^{3}\right )} x}{4 \, b^{6}} - \frac {3 \, {\left (a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 10 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left (b x + a\right )}{b^{7}} \]
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Time = 0.29 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.41 \[ \int \frac {x^3 (c+d x)^3}{(a+b x)^3} \, dx=-\frac {3 \, {\left (a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 10 \, a^{3} b c d^{2} - 5 \, a^{4} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac {5 \, a^{3} b^{3} c^{3} - 21 \, a^{4} b^{2} c^{2} d + 27 \, a^{5} b c d^{2} - 11 \, a^{6} d^{3} + 6 \, {\left (a^{2} b^{4} c^{3} - 4 \, a^{3} b^{3} c^{2} d + 5 \, a^{4} b^{2} c d^{2} - 2 \, a^{5} b d^{3}\right )} x}{2 \, {\left (b x + a\right )}^{2} b^{7}} + \frac {b^{9} d^{3} x^{4} + 4 \, b^{9} c d^{2} x^{3} - 4 \, a b^{8} d^{3} x^{3} + 6 \, b^{9} c^{2} d x^{2} - 18 \, a b^{8} c d^{2} x^{2} + 12 \, a^{2} b^{7} d^{3} x^{2} + 4 \, b^{9} c^{3} x - 36 \, a b^{8} c^{2} d x + 72 \, a^{2} b^{7} c d^{2} x - 40 \, a^{3} b^{6} d^{3} x}{4 \, b^{12}} \]
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Time = 0.11 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.80 \[ \int \frac {x^3 (c+d x)^3}{(a+b x)^3} \, dx=\frac {x\,\left (6\,a^5\,d^3-15\,a^4\,b\,c\,d^2+12\,a^3\,b^2\,c^2\,d-3\,a^2\,b^3\,c^3\right )+\frac {11\,a^6\,d^3-27\,a^5\,b\,c\,d^2+21\,a^4\,b^2\,c^2\,d-5\,a^3\,b^3\,c^3}{2\,b}}{a^2\,b^6+2\,a\,b^7\,x+b^8\,x^2}-x^3\,\left (\frac {a\,d^3}{b^4}-\frac {c\,d^2}{b^3}\right )+x\,\left (\frac {c^3}{b^3}-\frac {3\,a\,\left (\frac {3\,c^2\,d}{b^3}+\frac {3\,a\,\left (\frac {3\,a\,d^3}{b^4}-\frac {3\,c\,d^2}{b^3}\right )}{b}-\frac {3\,a^2\,d^3}{b^5}\right )}{b}-\frac {a^3\,d^3}{b^6}+\frac {3\,a^2\,\left (\frac {3\,a\,d^3}{b^4}-\frac {3\,c\,d^2}{b^3}\right )}{b^2}\right )+x^2\,\left (\frac {3\,c^2\,d}{2\,b^3}+\frac {3\,a\,\left (\frac {3\,a\,d^3}{b^4}-\frac {3\,c\,d^2}{b^3}\right )}{2\,b}-\frac {3\,a^2\,d^3}{2\,b^5}\right )+\frac {d^3\,x^4}{4\,b^3}+\frac {\ln \left (a+b\,x\right )\,\left (15\,a^4\,d^3-30\,a^3\,b\,c\,d^2+18\,a^2\,b^2\,c^2\,d-3\,a\,b^3\,c^3\right )}{b^7} \]
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