Integrand size = 18, antiderivative size = 193 \[ \int \frac {(c+d x)^3}{x^4 (a+b x)^3} \, dx=-\frac {c^3}{3 a^3 x^3}+\frac {3 c^2 (b c-a d)}{2 a^4 x^2}-\frac {3 c (b c-a d) (2 b c-a d)}{a^5 x}-\frac {(b c-a d)^3}{2 a^4 (a+b x)^2}-\frac {(b c-a d)^2 (4 b c-a d)}{a^5 (a+b x)}-\frac {(b c-a d) \left (10 b^2 c^2-8 a b c d+a^2 d^2\right ) \log (x)}{a^6}+\frac {(b c-a d) \left (10 b^2 c^2-8 a b c d+a^2 d^2\right ) \log (a+b x)}{a^6} \]
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Time = 0.13 (sec) , antiderivative size = 193, 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} \[ \int \frac {(c+d x)^3}{x^4 (a+b x)^3} \, dx=-\frac {3 c (b c-a d) (2 b c-a d)}{a^5 x}-\frac {(b c-a d)^2 (4 b c-a d)}{a^5 (a+b x)}+\frac {3 c^2 (b c-a d)}{2 a^4 x^2}-\frac {(b c-a d)^3}{2 a^4 (a+b x)^2}-\frac {c^3}{3 a^3 x^3}-\frac {\log (x) (b c-a d) \left (a^2 d^2-8 a b c d+10 b^2 c^2\right )}{a^6}+\frac {(b c-a d) \left (a^2 d^2-8 a b c d+10 b^2 c^2\right ) \log (a+b x)}{a^6} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^3}{a^3 x^4}+\frac {3 c^2 (-b c+a d)}{a^4 x^3}+\frac {3 c (b c-a d) (2 b c-a d)}{a^5 x^2}+\frac {(b c-a d) \left (-10 b^2 c^2+8 a b c d-a^2 d^2\right )}{a^6 x}-\frac {b (-b c+a d)^3}{a^4 (a+b x)^3}-\frac {b (-4 b c+a d) (-b c+a d)^2}{a^5 (a+b x)^2}+\frac {b (b c-a d) \left (10 b^2 c^2-8 a b c d+a^2 d^2\right )}{a^6 (a+b x)}\right ) \, dx \\ & = -\frac {c^3}{3 a^3 x^3}+\frac {3 c^2 (b c-a d)}{2 a^4 x^2}-\frac {3 c (b c-a d) (2 b c-a d)}{a^5 x}-\frac {(b c-a d)^3}{2 a^4 (a+b x)^2}-\frac {(b c-a d)^2 (4 b c-a d)}{a^5 (a+b x)}-\frac {(b c-a d) \left (10 b^2 c^2-8 a b c d+a^2 d^2\right ) \log (x)}{a^6}+\frac {(b c-a d) \left (10 b^2 c^2-8 a b c d+a^2 d^2\right ) \log (a+b x)}{a^6} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.05 \[ \int \frac {(c+d x)^3}{x^4 (a+b x)^3} \, dx=\frac {-\frac {2 a^3 c^3}{x^3}-\frac {9 a^2 c^2 (-b c+a d)}{x^2}-\frac {18 a c \left (2 b^2 c^2-3 a b c d+a^2 d^2\right )}{x}+\frac {3 a^2 (-b c+a d)^3}{(a+b x)^2}+\frac {6 a (b c-a d)^2 (-4 b c+a d)}{a+b x}+6 \left (-10 b^3 c^3+18 a b^2 c^2 d-9 a^2 b c d^2+a^3 d^3\right ) \log (x)+6 \left (10 b^3 c^3-18 a b^2 c^2 d+9 a^2 b c d^2-a^3 d^3\right ) \log (a+b x)}{6 a^6} \]
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Time = 0.47 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.28
method | result | size |
default | \(-\frac {c^{3}}{3 a^{3} x^{3}}+\frac {\left (a^{3} d^{3}-9 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -10 b^{3} c^{3}\right ) \ln \left (x \right )}{a^{6}}-\frac {3 c \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right )}{a^{5} x}-\frac {3 c^{2} \left (a d -b c \right )}{2 a^{4} x^{2}}-\frac {\left (a^{3} d^{3}-9 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -10 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{6}}+\frac {a^{3} d^{3}-6 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -4 b^{3} c^{3}}{a^{5} \left (b x +a \right )}+\frac {a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}}{2 a^{4} \left (b x +a \right )^{2}}\) | \(247\) |
norman | \(\frac {\frac {\left (a^{3} b^{2} d^{3}-9 a^{2} b^{3} c \,d^{2}+18 a \,b^{4} c^{2} d -10 c^{3} b^{5}\right ) x^{4}}{a^{5} b}-\frac {c^{3}}{3 a}-\frac {c \left (9 a^{2} d^{2}-18 a b c d +10 b^{2} c^{2}\right ) x^{2}}{3 a^{3}}-\frac {c^{2} \left (9 a d -5 b c \right ) x}{6 a^{2}}+\frac {\left (3 a^{3} b^{2} d^{3}-27 a^{2} b^{3} c \,d^{2}+54 a \,b^{4} c^{2} d -30 c^{3} b^{5}\right ) x^{3}}{2 b^{2} a^{4}}}{x^{3} \left (b x +a \right )^{2}}+\frac {\left (a^{3} d^{3}-9 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -10 b^{3} c^{3}\right ) \ln \left (x \right )}{a^{6}}-\frac {\left (a^{3} d^{3}-9 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -10 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{6}}\) | \(265\) |
risch | \(\frac {\frac {b \left (a^{3} d^{3}-9 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -10 b^{3} c^{3}\right ) x^{4}}{a^{5}}+\frac {3 \left (a^{3} d^{3}-9 a^{2} b c \,d^{2}+18 a \,b^{2} c^{2} d -10 b^{3} c^{3}\right ) x^{3}}{2 a^{4}}-\frac {c \left (9 a^{2} d^{2}-18 a b c d +10 b^{2} c^{2}\right ) x^{2}}{3 a^{3}}-\frac {c^{2} \left (9 a d -5 b c \right ) x}{6 a^{2}}-\frac {c^{3}}{3 a}}{x^{3} \left (b x +a \right )^{2}}-\frac {\ln \left (b x +a \right ) d^{3}}{a^{3}}+\frac {9 \ln \left (b x +a \right ) b c \,d^{2}}{a^{4}}-\frac {18 \ln \left (b x +a \right ) b^{2} c^{2} d}{a^{5}}+\frac {10 \ln \left (b x +a \right ) b^{3} c^{3}}{a^{6}}+\frac {\ln \left (-x \right ) d^{3}}{a^{3}}-\frac {9 \ln \left (-x \right ) b c \,d^{2}}{a^{4}}+\frac {18 \ln \left (-x \right ) b^{2} c^{2} d}{a^{5}}-\frac {10 \ln \left (-x \right ) b^{3} c^{3}}{a^{6}}\) | \(281\) |
parallelrisch | \(\frac {-54 x^{4} a^{3} b^{4} c \,d^{2}+108 x^{4} a^{2} b^{5} c^{2} d -81 x^{3} a^{4} b^{3} c \,d^{2}+162 x^{3} a^{3} b^{4} c^{2} d -18 x^{2} a^{5} b^{2} c \,d^{2}+36 x^{2} a^{4} b^{3} c^{2} d -9 x \,a^{5} b^{2} c^{2} d +6 \ln \left (x \right ) x^{5} a^{3} b^{4} d^{3}-6 \ln \left (b x +a \right ) x^{5} a^{3} b^{4} d^{3}+12 \ln \left (x \right ) x^{4} a^{4} b^{3} d^{3}-120 \ln \left (x \right ) x^{4} a \,b^{6} c^{3}-12 \ln \left (b x +a \right ) x^{4} a^{4} b^{3} d^{3}+120 \ln \left (b x +a \right ) x^{4} a \,b^{6} c^{3}+6 \ln \left (x \right ) x^{3} a^{5} b^{2} d^{3}-60 \ln \left (x \right ) x^{3} a^{2} b^{5} c^{3}-6 \ln \left (b x +a \right ) x^{3} a^{5} b^{2} d^{3}+60 \ln \left (b x +a \right ) x^{3} a^{2} b^{5} c^{3}-216 \ln \left (b x +a \right ) x^{4} a^{2} b^{5} c^{2} d -54 \ln \left (x \right ) x^{3} a^{4} b^{3} c \,d^{2}+108 \ln \left (x \right ) x^{3} a^{3} b^{4} c^{2} d +54 \ln \left (b x +a \right ) x^{3} a^{4} b^{3} c \,d^{2}-108 \ln \left (b x +a \right ) x^{3} a^{3} b^{4} c^{2} d +60 \ln \left (b x +a \right ) x^{5} b^{7} c^{3}+6 x^{4} a^{4} b^{3} d^{3}-60 x^{4} a \,b^{6} c^{3}+9 x^{3} a^{5} b^{2} d^{3}-90 x^{3} a^{2} b^{5} c^{3}-20 x^{2} a^{3} b^{4} c^{3}+5 x \,a^{4} b^{3} c^{3}-60 \ln \left (x \right ) x^{5} b^{7} c^{3}-2 a^{5} b^{2} c^{3}-54 \ln \left (x \right ) x^{5} a^{2} b^{5} c \,d^{2}+108 \ln \left (x \right ) x^{5} a \,b^{6} c^{2} d +54 \ln \left (b x +a \right ) x^{5} a^{2} b^{5} c \,d^{2}-108 \ln \left (b x +a \right ) x^{5} a \,b^{6} c^{2} d -108 \ln \left (x \right ) x^{4} a^{3} b^{4} c \,d^{2}+216 \ln \left (x \right ) x^{4} a^{2} b^{5} c^{2} d +108 \ln \left (b x +a \right ) x^{4} a^{3} b^{4} c \,d^{2}}{6 b^{2} a^{6} x^{3} \left (b x +a \right )^{2}}\) | \(644\) |
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Leaf count of result is larger than twice the leaf count of optimal. 487 vs. \(2 (187) = 374\).
Time = 0.23 (sec) , antiderivative size = 487, normalized size of antiderivative = 2.52 \[ \int \frac {(c+d x)^3}{x^4 (a+b x)^3} \, dx=-\frac {2 \, a^{5} c^{3} + 6 \, {\left (10 \, a b^{4} c^{3} - 18 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} + 9 \, {\left (10 \, a^{2} b^{3} c^{3} - 18 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 2 \, {\left (10 \, a^{3} b^{2} c^{3} - 18 \, a^{4} b c^{2} d + 9 \, a^{5} c d^{2}\right )} x^{2} - {\left (5 \, a^{4} b c^{3} - 9 \, a^{5} c^{2} d\right )} x - 6 \, {\left ({\left (10 \, b^{5} c^{3} - 18 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} + 2 \, {\left (10 \, a b^{4} c^{3} - 18 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} + {\left (10 \, a^{2} b^{3} c^{3} - 18 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3}\right )} \log \left (b x + a\right ) + 6 \, {\left ({\left (10 \, b^{5} c^{3} - 18 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} x^{5} + 2 \, {\left (10 \, a b^{4} c^{3} - 18 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} + {\left (10 \, a^{2} b^{3} c^{3} - 18 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3}\right )} \log \left (x\right )}{6 \, {\left (a^{6} b^{2} x^{5} + 2 \, a^{7} b x^{4} + a^{8} x^{3}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 505 vs. \(2 (178) = 356\).
Time = 0.99 (sec) , antiderivative size = 505, normalized size of antiderivative = 2.62 \[ \int \frac {(c+d x)^3}{x^4 (a+b x)^3} \, dx=\frac {- 2 a^{4} c^{3} + x^{4} \cdot \left (6 a^{3} b d^{3} - 54 a^{2} b^{2} c d^{2} + 108 a b^{3} c^{2} d - 60 b^{4} c^{3}\right ) + x^{3} \cdot \left (9 a^{4} d^{3} - 81 a^{3} b c d^{2} + 162 a^{2} b^{2} c^{2} d - 90 a b^{3} c^{3}\right ) + x^{2} \left (- 18 a^{4} c d^{2} + 36 a^{3} b c^{2} d - 20 a^{2} b^{2} c^{3}\right ) + x \left (- 9 a^{4} c^{2} d + 5 a^{3} b c^{3}\right )}{6 a^{7} x^{3} + 12 a^{6} b x^{4} + 6 a^{5} b^{2} x^{5}} + \frac {\left (a d - b c\right ) \left (a^{2} d^{2} - 8 a b c d + 10 b^{2} c^{2}\right ) \log {\left (x + \frac {a^{4} d^{3} - 9 a^{3} b c d^{2} + 18 a^{2} b^{2} c^{2} d - 10 a b^{3} c^{3} - a \left (a d - b c\right ) \left (a^{2} d^{2} - 8 a b c d + 10 b^{2} c^{2}\right )}{2 a^{3} b d^{3} - 18 a^{2} b^{2} c d^{2} + 36 a b^{3} c^{2} d - 20 b^{4} c^{3}} \right )}}{a^{6}} - \frac {\left (a d - b c\right ) \left (a^{2} d^{2} - 8 a b c d + 10 b^{2} c^{2}\right ) \log {\left (x + \frac {a^{4} d^{3} - 9 a^{3} b c d^{2} + 18 a^{2} b^{2} c^{2} d - 10 a b^{3} c^{3} + a \left (a d - b c\right ) \left (a^{2} d^{2} - 8 a b c d + 10 b^{2} c^{2}\right )}{2 a^{3} b d^{3} - 18 a^{2} b^{2} c d^{2} + 36 a b^{3} c^{2} d - 20 b^{4} c^{3}} \right )}}{a^{6}} \]
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Time = 0.21 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.45 \[ \int \frac {(c+d x)^3}{x^4 (a+b x)^3} \, dx=-\frac {2 \, a^{4} c^{3} + 6 \, {\left (10 \, b^{4} c^{3} - 18 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{4} + 9 \, {\left (10 \, a b^{3} c^{3} - 18 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x^{3} + 2 \, {\left (10 \, a^{2} b^{2} c^{3} - 18 \, a^{3} b c^{2} d + 9 \, a^{4} c d^{2}\right )} x^{2} - {\left (5 \, a^{3} b c^{3} - 9 \, a^{4} c^{2} d\right )} x}{6 \, {\left (a^{5} b^{2} x^{5} + 2 \, a^{6} b x^{4} + a^{7} x^{3}\right )}} + \frac {{\left (10 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x + a\right )}{a^{6}} - \frac {{\left (10 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (x\right )}{a^{6}} \]
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Time = 0.29 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.44 \[ \int \frac {(c+d x)^3}{x^4 (a+b x)^3} \, dx=-\frac {{\left (10 \, b^{3} c^{3} - 18 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{6}} + \frac {{\left (10 \, b^{4} c^{3} - 18 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{6} b} - \frac {2 \, a^{5} c^{3} + 6 \, {\left (10 \, a b^{4} c^{3} - 18 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} x^{4} + 9 \, {\left (10 \, a^{2} b^{3} c^{3} - 18 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} x^{3} + 2 \, {\left (10 \, a^{3} b^{2} c^{3} - 18 \, a^{4} b c^{2} d + 9 \, a^{5} c d^{2}\right )} x^{2} - {\left (5 \, a^{4} b c^{3} - 9 \, a^{5} c^{2} d\right )} x}{6 \, {\left (b x + a\right )}^{2} a^{6} x^{3}} \]
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Time = 0.57 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.50 \[ \int \frac {(c+d x)^3}{x^4 (a+b x)^3} \, dx=-\frac {\frac {c^3}{3\,a}-\frac {3\,x^3\,\left (a^3\,d^3-9\,a^2\,b\,c\,d^2+18\,a\,b^2\,c^2\,d-10\,b^3\,c^3\right )}{2\,a^4}+\frac {c^2\,x\,\left (9\,a\,d-5\,b\,c\right )}{6\,a^2}+\frac {c\,x^2\,\left (9\,a^2\,d^2-18\,a\,b\,c\,d+10\,b^2\,c^2\right )}{3\,a^3}-\frac {b\,x^4\,\left (a^3\,d^3-9\,a^2\,b\,c\,d^2+18\,a\,b^2\,c^2\,d-10\,b^3\,c^3\right )}{a^5}}{a^2\,x^3+2\,a\,b\,x^4+b^2\,x^5}-\frac {2\,\mathrm {atanh}\left (\frac {\left (a\,d-b\,c\right )\,\left (a+2\,b\,x\right )\,\left (a^2\,d^2-8\,a\,b\,c\,d+10\,b^2\,c^2\right )}{a\,\left (a^3\,d^3-9\,a^2\,b\,c\,d^2+18\,a\,b^2\,c^2\,d-10\,b^3\,c^3\right )}\right )\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-8\,a\,b\,c\,d+10\,b^2\,c^2\right )}{a^6} \]
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