Integrand size = 18, antiderivative size = 162 \[ \int \frac {(c+d x)^3}{x^5 (a+b x)^2} \, dx=-\frac {c^3}{4 a^2 x^4}+\frac {c^2 (2 b c-3 a d)}{3 a^3 x^3}-\frac {3 c (b c-a d)^2}{2 a^4 x^2}+\frac {(b c-a d)^2 (4 b c-a d)}{a^5 x}+\frac {b (b c-a d)^3}{a^5 (a+b x)}+\frac {b (5 b c-2 a d) (b c-a d)^2 \log (x)}{a^6}-\frac {b (5 b c-2 a d) (b c-a d)^2 \log (a+b x)}{a^6} \]
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Time = 0.10 (sec) , antiderivative size = 162, 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} \[ \int \frac {(c+d x)^3}{x^5 (a+b x)^2} \, dx=\frac {b \log (x) (5 b c-2 a d) (b c-a d)^2}{a^6}-\frac {b (5 b c-2 a d) (b c-a d)^2 \log (a+b x)}{a^6}+\frac {(b c-a d)^2 (4 b c-a d)}{a^5 x}+\frac {b (b c-a d)^3}{a^5 (a+b x)}-\frac {3 c (b c-a d)^2}{2 a^4 x^2}+\frac {c^2 (2 b c-3 a d)}{3 a^3 x^3}-\frac {c^3}{4 a^2 x^4} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^3}{a^2 x^5}+\frac {c^2 (-2 b c+3 a d)}{a^3 x^4}+\frac {3 c (-b c+a d)^2}{a^4 x^3}+\frac {(-4 b c+a d) (-b c+a d)^2}{a^5 x^2}-\frac {b (-b c+a d)^2 (-5 b c+2 a d)}{a^6 x}+\frac {b^2 (-b c+a d)^3}{a^5 (a+b x)^2}+\frac {b^2 (-b c+a d)^2 (-5 b c+2 a d)}{a^6 (a+b x)}\right ) \, dx \\ & = -\frac {c^3}{4 a^2 x^4}+\frac {c^2 (2 b c-3 a d)}{3 a^3 x^3}-\frac {3 c (b c-a d)^2}{2 a^4 x^2}+\frac {(b c-a d)^2 (4 b c-a d)}{a^5 x}+\frac {b (b c-a d)^3}{a^5 (a+b x)}+\frac {b (5 b c-2 a d) (b c-a d)^2 \log (x)}{a^6}-\frac {b (5 b c-2 a d) (b c-a d)^2 \log (a+b x)}{a^6} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.96 \[ \int \frac {(c+d x)^3}{x^5 (a+b x)^2} \, dx=-\frac {\frac {3 a^4 c^3}{x^4}+\frac {4 a^3 c^2 (-2 b c+3 a d)}{x^3}+\frac {18 a^2 c (b c-a d)^2}{x^2}+\frac {12 a (b c-a d)^2 (-4 b c+a d)}{x}+\frac {12 a b (-b c+a d)^3}{a+b x}-12 b (5 b c-2 a d) (b c-a d)^2 \log (x)+12 b (5 b c-2 a d) (b c-a d)^2 \log (a+b x)}{12 a^6} \]
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Time = 0.47 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.54
method | result | size |
default | \(-\frac {c^{3}}{4 a^{2} x^{4}}-\frac {a^{3} d^{3}-6 a^{2} b c \,d^{2}+9 a \,b^{2} c^{2} d -4 b^{3} c^{3}}{a^{5} x}-\frac {c^{2} \left (3 a d -2 b c \right )}{3 a^{3} x^{3}}-\frac {b \left (2 a^{3} d^{3}-9 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right ) \ln \left (x \right )}{a^{6}}-\frac {3 c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 a^{4} x^{2}}+\frac {b \left (2 a^{3} d^{3}-9 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{6}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b}{a^{5} \left (b x +a \right )}\) | \(249\) |
norman | \(\frac {\frac {b \left (2 b \,d^{3} a^{3}-9 c \,d^{2} b^{2} a^{2}+12 b^{3} c^{2} d a -5 b^{4} c^{3}\right ) x^{5}}{a^{6}}-\frac {c^{3}}{4 a}-\frac {\left (2 a^{3} d^{3}-9 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right ) x^{3}}{2 a^{4}}-\frac {c \left (9 a^{2} d^{2}-12 a b c d +5 b^{2} c^{2}\right ) x^{2}}{6 a^{3}}-\frac {c^{2} \left (12 a d -5 b c \right ) x}{12 a^{2}}}{x^{4} \left (b x +a \right )}+\frac {b \left (2 a^{3} d^{3}-9 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right ) \ln \left (b x +a \right )}{a^{6}}-\frac {b \left (2 a^{3} d^{3}-9 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right ) \ln \left (x \right )}{a^{6}}\) | \(258\) |
risch | \(\frac {-\frac {b \left (2 a^{3} d^{3}-9 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right ) x^{4}}{a^{5}}-\frac {\left (2 a^{3} d^{3}-9 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right ) x^{3}}{2 a^{4}}-\frac {c \left (9 a^{2} d^{2}-12 a b c d +5 b^{2} c^{2}\right ) x^{2}}{6 a^{3}}-\frac {c^{2} \left (12 a d -5 b c \right ) x}{12 a^{2}}-\frac {c^{3}}{4 a}}{x^{4} \left (b x +a \right )}-\frac {2 b \ln \left (x \right ) d^{3}}{a^{3}}+\frac {9 b^{2} \ln \left (x \right ) c \,d^{2}}{a^{4}}-\frac {12 b^{3} \ln \left (x \right ) c^{2} d}{a^{5}}+\frac {5 b^{4} \ln \left (x \right ) c^{3}}{a^{6}}+\frac {2 b \ln \left (-b x -a \right ) d^{3}}{a^{3}}-\frac {9 b^{2} \ln \left (-b x -a \right ) c \,d^{2}}{a^{4}}+\frac {12 b^{3} \ln \left (-b x -a \right ) c^{2} d}{a^{5}}-\frac {5 b^{4} \ln \left (-b x -a \right ) c^{3}}{a^{6}}\) | \(295\) |
parallelrisch | \(-\frac {60 \ln \left (b x +a \right ) x^{4} a \,b^{4} c^{3}+24 \ln \left (x \right ) x^{4} a^{4} b \,d^{3}-60 \ln \left (x \right ) x^{4} a \,b^{4} c^{3}-24 \ln \left (b x +a \right ) x^{4} a^{4} b \,d^{3}+108 x^{5} a^{2} b^{3} c \,d^{2}-144 x^{5} a \,b^{4} c^{2} d +12 a^{5} d^{3} x^{3}+3 c^{3} a^{5}-144 \ln \left (b x +a \right ) x^{4} a^{2} b^{3} c^{2} d -108 \ln \left (x \right ) x^{4} a^{3} b^{2} c \,d^{2}+108 \ln \left (b x +a \right ) x^{4} a^{3} b^{2} c \,d^{2}+144 \ln \left (x \right ) x^{4} a^{2} b^{3} c^{2} d +108 \ln \left (b x +a \right ) x^{5} a^{2} b^{3} c \,d^{2}-144 \ln \left (b x +a \right ) x^{5} a \,b^{4} c^{2} d -108 \ln \left (x \right ) x^{5} a^{2} b^{3} c \,d^{2}+144 \ln \left (x \right ) x^{5} a \,b^{4} c^{2} d +12 a^{5} c^{2} d x -5 a^{4} b \,c^{3} x +10 a^{3} b^{2} c^{3} x^{2}+18 a^{5} c \,d^{2} x^{2}-30 a^{2} b^{3} c^{3} x^{3}-24 a^{4} b \,c^{2} d \,x^{2}-54 a^{4} b c \,d^{2} x^{3}+72 a^{3} b^{2} c^{2} d \,x^{3}+24 \ln \left (x \right ) x^{5} a^{3} b^{2} d^{3}-24 \ln \left (b x +a \right ) x^{5} a^{3} b^{2} d^{3}-24 x^{5} a^{3} b^{2} d^{3}-60 \ln \left (x \right ) x^{5} b^{5} c^{3}+60 \ln \left (b x +a \right ) x^{5} b^{5} c^{3}+60 x^{5} b^{5} c^{3}}{12 a^{6} x^{4} \left (b x +a \right )}\) | \(468\) |
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Leaf count of result is larger than twice the leaf count of optimal. 382 vs. \(2 (156) = 312\).
Time = 0.22 (sec) , antiderivative size = 382, normalized size of antiderivative = 2.36 \[ \int \frac {(c+d x)^3}{x^5 (a+b x)^2} \, dx=-\frac {3 \, a^{5} c^{3} - 12 \, {\left (5 \, a b^{4} c^{3} - 12 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 2 \, a^{4} b d^{3}\right )} x^{4} - 6 \, {\left (5 \, a^{2} b^{3} c^{3} - 12 \, a^{3} b^{2} c^{2} d + 9 \, a^{4} b c d^{2} - 2 \, a^{5} d^{3}\right )} x^{3} + 2 \, {\left (5 \, a^{3} b^{2} c^{3} - 12 \, a^{4} b c^{2} d + 9 \, a^{5} c d^{2}\right )} x^{2} - {\left (5 \, a^{4} b c^{3} - 12 \, a^{5} c^{2} d\right )} x + 12 \, {\left ({\left (5 \, b^{5} c^{3} - 12 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - 2 \, a^{3} b^{2} d^{3}\right )} x^{5} + {\left (5 \, a b^{4} c^{3} - 12 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 2 \, a^{4} b d^{3}\right )} x^{4}\right )} \log \left (b x + a\right ) - 12 \, {\left ({\left (5 \, b^{5} c^{3} - 12 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - 2 \, a^{3} b^{2} d^{3}\right )} x^{5} + {\left (5 \, a b^{4} c^{3} - 12 \, a^{2} b^{3} c^{2} d + 9 \, a^{3} b^{2} c d^{2} - 2 \, a^{4} b d^{3}\right )} x^{4}\right )} \log \left (x\right )}{12 \, {\left (a^{6} b x^{5} + a^{7} x^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (151) = 302\).
Time = 0.88 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.88 \[ \int \frac {(c+d x)^3}{x^5 (a+b x)^2} \, dx=\frac {- 3 a^{4} c^{3} + x^{4} \left (- 24 a^{3} b d^{3} + 108 a^{2} b^{2} c d^{2} - 144 a b^{3} c^{2} d + 60 b^{4} c^{3}\right ) + x^{3} \left (- 12 a^{4} d^{3} + 54 a^{3} b c d^{2} - 72 a^{2} b^{2} c^{2} d + 30 a b^{3} c^{3}\right ) + x^{2} \left (- 18 a^{4} c d^{2} + 24 a^{3} b c^{2} d - 10 a^{2} b^{2} c^{3}\right ) + x \left (- 12 a^{4} c^{2} d + 5 a^{3} b c^{3}\right )}{12 a^{6} x^{4} + 12 a^{5} b x^{5}} - \frac {b \left (a d - b c\right )^{2} \cdot \left (2 a d - 5 b c\right ) \log {\left (x + \frac {2 a^{4} b d^{3} - 9 a^{3} b^{2} c d^{2} + 12 a^{2} b^{3} c^{2} d - 5 a b^{4} c^{3} - a b \left (a d - b c\right )^{2} \cdot \left (2 a d - 5 b c\right )}{4 a^{3} b^{2} d^{3} - 18 a^{2} b^{3} c d^{2} + 24 a b^{4} c^{2} d - 10 b^{5} c^{3}} \right )}}{a^{6}} + \frac {b \left (a d - b c\right )^{2} \cdot \left (2 a d - 5 b c\right ) \log {\left (x + \frac {2 a^{4} b d^{3} - 9 a^{3} b^{2} c d^{2} + 12 a^{2} b^{3} c^{2} d - 5 a b^{4} c^{3} + a b \left (a d - b c\right )^{2} \cdot \left (2 a d - 5 b c\right )}{4 a^{3} b^{2} d^{3} - 18 a^{2} b^{3} c d^{2} + 24 a b^{4} c^{2} d - 10 b^{5} c^{3}} \right )}}{a^{6}} \]
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Time = 0.21 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.70 \[ \int \frac {(c+d x)^3}{x^5 (a+b x)^2} \, dx=-\frac {3 \, a^{4} c^{3} - 12 \, {\left (5 \, b^{4} c^{3} - 12 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x^{4} - 6 \, {\left (5 \, a b^{3} c^{3} - 12 \, a^{2} b^{2} c^{2} d + 9 \, a^{3} b c d^{2} - 2 \, a^{4} d^{3}\right )} x^{3} + 2 \, {\left (5 \, a^{2} b^{2} c^{3} - 12 \, a^{3} b c^{2} d + 9 \, a^{4} c d^{2}\right )} x^{2} - {\left (5 \, a^{3} b c^{3} - 12 \, a^{4} c^{2} d\right )} x}{12 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}} - \frac {{\left (5 \, b^{4} c^{3} - 12 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} \log \left (b x + a\right )}{a^{6}} + \frac {{\left (5 \, b^{4} c^{3} - 12 \, a b^{3} c^{2} d + 9 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} \log \left (x\right )}{a^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (156) = 312\).
Time = 0.27 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.30 \[ \int \frac {(c+d x)^3}{x^5 (a+b x)^2} \, dx=\frac {{\left (5 \, b^{5} c^{3} - 12 \, a b^{4} c^{2} d + 9 \, a^{2} b^{3} c d^{2} - 2 \, a^{3} b^{2} d^{3}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{6} b} + \frac {\frac {b^{9} c^{3}}{b x + a} - \frac {3 \, a b^{8} c^{2} d}{b x + a} + \frac {3 \, a^{2} b^{7} c d^{2}}{b x + a} - \frac {a^{3} b^{6} d^{3}}{b x + a}}{a^{5} b^{5}} + \frac {77 \, b^{4} c^{3} - 156 \, a b^{3} c^{2} d + 90 \, a^{2} b^{2} c d^{2} - 12 \, a^{3} b d^{3} - \frac {4 \, {\left (65 \, a b^{5} c^{3} - 129 \, a^{2} b^{4} c^{2} d + 72 \, a^{3} b^{3} c d^{2} - 9 \, a^{4} b^{2} d^{3}\right )}}{{\left (b x + a\right )} b} + \frac {6 \, {\left (50 \, a^{2} b^{6} c^{3} - 96 \, a^{3} b^{5} c^{2} d + 51 \, a^{4} b^{4} c d^{2} - 6 \, a^{5} b^{3} d^{3}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {12 \, {\left (10 \, a^{3} b^{7} c^{3} - 18 \, a^{4} b^{6} c^{2} d + 9 \, a^{5} b^{5} c d^{2} - a^{6} b^{4} d^{3}\right )}}{{\left (b x + a\right )}^{3} b^{3}}}{12 \, a^{6} {\left (\frac {a}{b x + a} - 1\right )}^{4}} \]
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Time = 0.65 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.62 \[ \int \frac {(c+d x)^3}{x^5 (a+b x)^2} \, dx=-\frac {\frac {c^3}{4\,a}+\frac {x^3\,\left (2\,a^3\,d^3-9\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d-5\,b^3\,c^3\right )}{2\,a^4}+\frac {c^2\,x\,\left (12\,a\,d-5\,b\,c\right )}{12\,a^2}+\frac {c\,x^2\,\left (9\,a^2\,d^2-12\,a\,b\,c\,d+5\,b^2\,c^2\right )}{6\,a^3}+\frac {b\,x^4\,\left (2\,a^3\,d^3-9\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d-5\,b^3\,c^3\right )}{a^5}}{b\,x^5+a\,x^4}-\frac {2\,b\,\mathrm {atanh}\left (\frac {b\,{\left (a\,d-b\,c\right )}^2\,\left (2\,a\,d-5\,b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (-2\,a^3\,b\,d^3+9\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+5\,b^4\,c^3\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (2\,a\,d-5\,b\,c\right )}{a^6} \]
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