Integrand size = 18, antiderivative size = 152 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=\frac {a^2 (b c-a d)^3 x}{b^6}-\frac {a (b c-a d)^3 x^2}{2 b^5}+\frac {(b c-a d)^3 x^3}{3 b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac {d^2 (3 b c-a d) x^5}{5 b^2}+\frac {d^3 x^6}{6 b}-\frac {a^3 (b c-a d)^3 \log (a+b x)}{b^7} \]
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Time = 0.11 (sec) , antiderivative size = 152, 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} \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=-\frac {a^3 (b c-a d)^3 \log (a+b x)}{b^7}+\frac {a^2 x (b c-a d)^3}{b^6}+\frac {d x^4 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{4 b^3}-\frac {a x^2 (b c-a d)^3}{2 b^5}+\frac {x^3 (b c-a d)^3}{3 b^4}+\frac {d^2 x^5 (3 b c-a d)}{5 b^2}+\frac {d^3 x^6}{6 b} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^2 (-b c+a d)^3}{b^6}+\frac {a (-b c+a d)^3 x}{b^5}+\frac {(b c-a d)^3 x^2}{b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^3}{b^3}+\frac {d^2 (3 b c-a d) x^4}{b^2}+\frac {d^3 x^5}{b}+\frac {a^3 (-b c+a d)^3}{b^6 (a+b x)}\right ) \, dx \\ & = \frac {a^2 (b c-a d)^3 x}{b^6}-\frac {a (b c-a d)^3 x^2}{2 b^5}+\frac {(b c-a d)^3 x^3}{3 b^4}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^4}{4 b^3}+\frac {d^2 (3 b c-a d) x^5}{5 b^2}+\frac {d^3 x^6}{6 b}-\frac {a^3 (b c-a d)^3 \log (a+b x)}{b^7} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.95 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=\frac {-60 a^2 b (-b c+a d)^3 x+30 a b^2 (-b c+a d)^3 x^2+20 b^3 (b c-a d)^3 x^3+15 b^4 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^4+12 b^5 d^2 (3 b c-a d) x^5+10 b^6 d^3 x^6+60 a^3 (-b c+a d)^3 \log (a+b x)}{60 b^7} \]
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Time = 1.20 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.62
method | result | size |
norman | \(-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{3}}{3 b^{4}}+\frac {d^{3} x^{6}}{6 b}+\frac {a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{2}}{2 b^{5}}-\frac {a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x}{b^{6}}-\frac {d^{2} \left (a d -3 b c \right ) x^{5}}{5 b^{2}}+\frac {d \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{4}}{4 b^{3}}+\frac {a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{7}}\) | \(246\) |
risch | \(\frac {d^{3} x^{6}}{6 b}-\frac {x^{5} a \,d^{3}}{5 b^{2}}+\frac {3 x^{5} d^{2} c}{5 b}-\frac {3 x^{4} a c \,d^{2}}{4 b^{2}}+\frac {3 x^{4} c^{2} d}{4 b}+\frac {x^{4} a^{2} d^{3}}{4 b^{3}}-\frac {x^{3} a^{3} d^{3}}{3 b^{4}}+\frac {x^{3} a^{2} c \,d^{2}}{b^{3}}-\frac {x^{3} a \,c^{2} d}{b^{2}}+\frac {x^{3} c^{3}}{3 b}-\frac {3 x^{2} a^{3} c \,d^{2}}{2 b^{4}}+\frac {3 x^{2} a^{2} c^{2} d}{2 b^{3}}-\frac {x^{2} a \,c^{3}}{2 b^{2}}+\frac {x^{2} a^{4} d^{3}}{2 b^{5}}-\frac {a^{5} d^{3} x}{b^{6}}+\frac {3 a^{4} c \,d^{2} x}{b^{5}}-\frac {3 a^{3} c^{2} d x}{b^{4}}+\frac {a^{2} c^{3} x}{b^{3}}+\frac {a^{6} \ln \left (b x +a \right ) d^{3}}{b^{7}}-\frac {3 a^{5} \ln \left (b x +a \right ) c \,d^{2}}{b^{6}}+\frac {3 a^{4} \ln \left (b x +a \right ) c^{2} d}{b^{5}}-\frac {a^{3} \ln \left (b x +a \right ) c^{3}}{b^{4}}\) | \(302\) |
parallelrisch | \(\frac {10 x^{6} d^{3} b^{6}-12 x^{5} a \,b^{5} d^{3}+36 x^{5} b^{6} c \,d^{2}+15 x^{4} a^{2} b^{4} d^{3}-45 x^{4} a \,b^{5} c \,d^{2}+45 x^{4} b^{6} c^{2} d -20 x^{3} a^{3} b^{3} d^{3}+60 x^{3} a^{2} b^{4} c \,d^{2}-60 x^{3} a \,b^{5} c^{2} d +20 x^{3} b^{6} c^{3}+30 x^{2} a^{4} b^{2} d^{3}-90 x^{2} a^{3} b^{3} c \,d^{2}+90 x^{2} a^{2} b^{4} c^{2} d -30 x^{2} a \,b^{5} c^{3}+60 \ln \left (b x +a \right ) a^{6} d^{3}-180 \ln \left (b x +a \right ) a^{5} b c \,d^{2}+180 \ln \left (b x +a \right ) a^{4} b^{2} c^{2} d -60 \ln \left (b x +a \right ) a^{3} b^{3} c^{3}-60 x \,a^{5} b \,d^{3}+180 x \,a^{4} b^{2} c \,d^{2}-180 x \,a^{3} b^{3} c^{2} d +60 x \,a^{2} b^{4} c^{3}}{60 b^{7}}\) | \(303\) |
default | \(-\frac {-\frac {d^{3} x^{6} b^{5}}{6}+\frac {\left (\left (a d -b c \right ) b^{4} d^{2}-2 b^{5} d^{2} c \right ) x^{5}}{5}+\frac {\left (2 \left (a d -b c \right ) b^{4} c d -b d \left (a^{2} b^{2} d^{2}-a \,b^{3} c d +b^{4} c^{2}\right )\right ) x^{4}}{4}+\frac {\left (\left (a d -b c \right ) \left (a^{2} b^{2} d^{2}-a \,b^{3} c d +b^{4} c^{2}\right )-b d \left (a^{2} b^{2} c d -a \,b^{3} c^{2}\right )\right ) x^{3}}{3}+\frac {\left (\left (a d -b c \right ) \left (a^{2} b^{2} c d -a \,b^{3} c^{2}\right )-b d \left (a^{4} d^{2}-2 a^{3} b c d +a^{2} b^{2} c^{2}\right )\right ) x^{2}}{2}+\left (a d -b c \right ) \left (a^{4} d^{2}-2 a^{3} b c d +a^{2} b^{2} c^{2}\right ) x}{b^{6}}+\frac {a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{7}}\) | \(313\) |
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Time = 0.22 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.76 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=\frac {10 \, b^{6} d^{3} x^{6} + 12 \, {\left (3 \, b^{6} c d^{2} - a b^{5} d^{3}\right )} x^{5} + 15 \, {\left (3 \, b^{6} c^{2} d - 3 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}\right )} x^{4} + 20 \, {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} - 30 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 60 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x - 60 \, {\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \log \left (b x + a\right )}{60 \, b^{7}} \]
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Time = 0.25 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.60 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=\frac {a^{3} \left (a d - b c\right )^{3} \log {\left (a + b x \right )}}{b^{7}} + x^{5} \left (- \frac {a d^{3}}{5 b^{2}} + \frac {3 c d^{2}}{5 b}\right ) + x^{4} \left (\frac {a^{2} d^{3}}{4 b^{3}} - \frac {3 a c d^{2}}{4 b^{2}} + \frac {3 c^{2} d}{4 b}\right ) + x^{3} \left (- \frac {a^{3} d^{3}}{3 b^{4}} + \frac {a^{2} c d^{2}}{b^{3}} - \frac {a c^{2} d}{b^{2}} + \frac {c^{3}}{3 b}\right ) + x^{2} \left (\frac {a^{4} d^{3}}{2 b^{5}} - \frac {3 a^{3} c d^{2}}{2 b^{4}} + \frac {3 a^{2} c^{2} d}{2 b^{3}} - \frac {a c^{3}}{2 b^{2}}\right ) + x \left (- \frac {a^{5} d^{3}}{b^{6}} + \frac {3 a^{4} c d^{2}}{b^{5}} - \frac {3 a^{3} c^{2} d}{b^{4}} + \frac {a^{2} c^{3}}{b^{3}}\right ) + \frac {d^{3} x^{6}}{6 b} \]
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Time = 0.20 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.75 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=\frac {10 \, b^{5} d^{3} x^{6} + 12 \, {\left (3 \, b^{5} c d^{2} - a b^{4} d^{3}\right )} x^{5} + 15 \, {\left (3 \, b^{5} c^{2} d - 3 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{4} + 20 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{3} - 30 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{2} + 60 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x}{60 \, b^{6}} - \frac {{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \log \left (b x + a\right )}{b^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (142) = 284\).
Time = 0.27 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.88 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=\frac {10 \, b^{5} d^{3} x^{6} + 36 \, b^{5} c d^{2} x^{5} - 12 \, a b^{4} d^{3} x^{5} + 45 \, b^{5} c^{2} d x^{4} - 45 \, a b^{4} c d^{2} x^{4} + 15 \, a^{2} b^{3} d^{3} x^{4} + 20 \, b^{5} c^{3} x^{3} - 60 \, a b^{4} c^{2} d x^{3} + 60 \, a^{2} b^{3} c d^{2} x^{3} - 20 \, a^{3} b^{2} d^{3} x^{3} - 30 \, a b^{4} c^{3} x^{2} + 90 \, a^{2} b^{3} c^{2} d x^{2} - 90 \, a^{3} b^{2} c d^{2} x^{2} + 30 \, a^{4} b d^{3} x^{2} + 60 \, a^{2} b^{3} c^{3} x - 180 \, a^{3} b^{2} c^{2} d x + 180 \, a^{4} b c d^{2} x - 60 \, a^{5} d^{3} x}{60 \, b^{6}} - \frac {{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} \]
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Time = 0.07 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.89 \[ \int \frac {x^3 (c+d x)^3}{a+b x} \, dx=x^3\,\left (\frac {c^3}{3\,b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{3\,b}\right )-x^5\,\left (\frac {a\,d^3}{5\,b^2}-\frac {3\,c\,d^2}{5\,b}\right )+x^4\,\left (\frac {3\,c^2\,d}{4\,b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{4\,b}\right )+\frac {\ln \left (a+b\,x\right )\,\left (a^6\,d^3-3\,a^5\,b\,c\,d^2+3\,a^4\,b^2\,c^2\,d-a^3\,b^3\,c^3\right )}{b^7}+\frac {d^3\,x^6}{6\,b}-\frac {a\,x^2\,\left (\frac {c^3}{b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{b}\right )}{2\,b}+\frac {a^2\,x\,\left (\frac {c^3}{b}-\frac {a\,\left (\frac {3\,c^2\,d}{b}+\frac {a\,\left (\frac {a\,d^3}{b^2}-\frac {3\,c\,d^2}{b}\right )}{b}\right )}{b}\right )}{b^2} \]
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