Integrand size = 18, antiderivative size = 167 \[ \int \frac {(c+d x)^2}{x^5 (a+b x)^2} \, dx=-\frac {c^2}{4 a^2 x^4}+\frac {2 c (b c-a d)}{3 a^3 x^3}-\frac {(b c-a d) (3 b c-a d)}{2 a^4 x^2}+\frac {2 b (b c-a d) (2 b c-a d)}{a^5 x}+\frac {b^2 (b c-a d)^2}{a^5 (a+b x)}+\frac {b^2 (5 b c-3 a d) (b c-a d) \log (x)}{a^6}-\frac {b^2 (5 b c-3 a d) (b c-a d) \log (a+b x)}{a^6} \]
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Time = 0.10 (sec) , antiderivative size = 167, 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} \[ \int \frac {(c+d x)^2}{x^5 (a+b x)^2} \, dx=\frac {b^2 \log (x) (5 b c-3 a d) (b c-a d)}{a^6}-\frac {b^2 (5 b c-3 a d) (b c-a d) \log (a+b x)}{a^6}+\frac {b^2 (b c-a d)^2}{a^5 (a+b x)}+\frac {2 b (b c-a d) (2 b c-a d)}{a^5 x}-\frac {(b c-a d) (3 b c-a d)}{2 a^4 x^2}+\frac {2 c (b c-a d)}{3 a^3 x^3}-\frac {c^2}{4 a^2 x^4} \]
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Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^2}{a^2 x^5}+\frac {2 c (-b c+a d)}{a^3 x^4}+\frac {(b c-a d) (3 b c-a d)}{a^4 x^3}+\frac {2 b (b c-a d) (-2 b c+a d)}{a^5 x^2}+\frac {b^2 (5 b c-3 a d) (b c-a d)}{a^6 x}-\frac {b^3 (-b c+a d)^2}{a^5 (a+b x)^2}+\frac {b^3 (5 b c-3 a d) (-b c+a d)}{a^6 (a+b x)}\right ) \, dx \\ & = -\frac {c^2}{4 a^2 x^4}+\frac {2 c (b c-a d)}{3 a^3 x^3}-\frac {(b c-a d) (3 b c-a d)}{2 a^4 x^2}+\frac {2 b (b c-a d) (2 b c-a d)}{a^5 x}+\frac {b^2 (b c-a d)^2}{a^5 (a+b x)}+\frac {b^2 (5 b c-3 a d) (b c-a d) \log (x)}{a^6}-\frac {b^2 (5 b c-3 a d) (b c-a d) \log (a+b x)}{a^6} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.09 \[ \int \frac {(c+d x)^2}{x^5 (a+b x)^2} \, dx=\frac {-\frac {3 a^4 c^2}{x^4}-\frac {8 a^3 c (-b c+a d)}{x^3}-\frac {6 a^2 \left (3 b^2 c^2-4 a b c d+a^2 d^2\right )}{x^2}+\frac {24 a b \left (2 b^2 c^2-3 a b c d+a^2 d^2\right )}{x}+\frac {12 a b^2 (b c-a d)^2}{a+b x}+12 b^2 \left (5 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \log (x)-12 b^2 \left (5 b^2 c^2-8 a b c d+3 a^2 d^2\right ) \log (a+b x)}{12 a^6} \]
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Time = 0.47 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.17
method | result | size |
default | \(-\frac {c^{2}}{4 a^{2} x^{4}}-\frac {a^{2} d^{2}-4 a b c d +3 b^{2} c^{2}}{2 a^{4} x^{2}}+\frac {b^{2} \left (3 a^{2} d^{2}-8 a b c d +5 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{6}}+\frac {2 b \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right )}{a^{5} x}-\frac {2 c \left (a d -b c \right )}{3 a^{3} x^{3}}-\frac {b^{2} \left (3 a^{2} d^{2}-8 a b c d +5 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{a^{6}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2}}{a^{5} \left (b x +a \right )}\) | \(195\) |
norman | \(\frac {\frac {b \left (-3 a^{2} b^{2} d^{2}+8 a \,b^{3} c d -5 b^{4} c^{2}\right ) x^{5}}{a^{6}}-\frac {c^{2}}{4 a}-\frac {\left (3 a^{2} d^{2}-8 a b c d +5 b^{2} c^{2}\right ) x^{2}}{6 a^{3}}+\frac {b \left (3 a^{2} d^{2}-8 a b c d +5 b^{2} c^{2}\right ) x^{3}}{2 a^{4}}-\frac {c \left (8 a d -5 b c \right ) x}{12 a^{2}}}{x^{4} \left (b x +a \right )}+\frac {b^{2} \left (3 a^{2} d^{2}-8 a b c d +5 b^{2} c^{2}\right ) \ln \left (x \right )}{a^{6}}-\frac {b^{2} \left (3 a^{2} d^{2}-8 a b c d +5 b^{2} c^{2}\right ) \ln \left (b x +a \right )}{a^{6}}\) | \(206\) |
risch | \(\frac {\frac {b^{2} \left (3 a^{2} d^{2}-8 a b c d +5 b^{2} c^{2}\right ) x^{4}}{a^{5}}+\frac {b \left (3 a^{2} d^{2}-8 a b c d +5 b^{2} c^{2}\right ) x^{3}}{2 a^{4}}-\frac {\left (3 a^{2} d^{2}-8 a b c d +5 b^{2} c^{2}\right ) x^{2}}{6 a^{3}}-\frac {c \left (8 a d -5 b c \right ) x}{12 a^{2}}-\frac {c^{2}}{4 a}}{x^{4} \left (b x +a \right )}+\frac {3 b^{2} \ln \left (-x \right ) d^{2}}{a^{4}}-\frac {8 b^{3} \ln \left (-x \right ) c d}{a^{5}}+\frac {5 b^{4} \ln \left (-x \right ) c^{2}}{a^{6}}-\frac {3 b^{2} \ln \left (b x +a \right ) d^{2}}{a^{4}}+\frac {8 b^{3} \ln \left (b x +a \right ) c d}{a^{5}}-\frac {5 b^{4} \ln \left (b x +a \right ) c^{2}}{a^{6}}\) | \(228\) |
parallelrisch | \(\frac {60 \ln \left (x \right ) x^{5} b^{5} c^{2}-60 \ln \left (b x +a \right ) x^{5} b^{5} c^{2}+96 \ln \left (b x +a \right ) x^{4} a^{2} b^{3} c d -96 \ln \left (x \right ) x^{5} a \,b^{4} c d +96 \ln \left (b x +a \right ) x^{5} a \,b^{4} c d -96 \ln \left (x \right ) x^{4} a^{2} b^{3} c d +96 x^{5} a \,b^{4} c d -48 x^{3} a^{3} b^{2} c d +16 x^{2} a^{4} b c d +36 \ln \left (x \right ) x^{5} a^{2} b^{3} d^{2}-36 \ln \left (b x +a \right ) x^{5} a^{2} b^{3} d^{2}+36 \ln \left (x \right ) x^{4} a^{3} b^{2} d^{2}+60 \ln \left (x \right ) x^{4} a \,b^{4} c^{2}-36 \ln \left (b x +a \right ) x^{4} a^{3} b^{2} d^{2}-60 \ln \left (b x +a \right ) x^{4} a \,b^{4} c^{2}-3 c^{2} a^{5}-60 x^{5} b^{5} c^{2}-6 x^{2} a^{5} d^{2}-8 x \,a^{5} c d +5 x \,a^{4} b \,c^{2}+18 x^{3} a^{4} b \,d^{2}+30 x^{3} a^{2} b^{3} c^{2}-10 x^{2} a^{3} b^{2} c^{2}-36 x^{5} a^{2} b^{3} d^{2}}{12 a^{6} x^{4} \left (b x +a \right )}\) | \(352\) |
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Time = 0.23 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.81 \[ \int \frac {(c+d x)^2}{x^5 (a+b x)^2} \, dx=-\frac {3 \, a^{5} c^{2} - 12 \, {\left (5 \, a b^{4} c^{2} - 8 \, a^{2} b^{3} c d + 3 \, a^{3} b^{2} d^{2}\right )} x^{4} - 6 \, {\left (5 \, a^{2} b^{3} c^{2} - 8 \, a^{3} b^{2} c d + 3 \, a^{4} b d^{2}\right )} x^{3} + 2 \, {\left (5 \, a^{3} b^{2} c^{2} - 8 \, a^{4} b c d + 3 \, a^{5} d^{2}\right )} x^{2} - {\left (5 \, a^{4} b c^{2} - 8 \, a^{5} c d\right )} x + 12 \, {\left ({\left (5 \, b^{5} c^{2} - 8 \, a b^{4} c d + 3 \, a^{2} b^{3} d^{2}\right )} x^{5} + {\left (5 \, a b^{4} c^{2} - 8 \, a^{2} b^{3} c d + 3 \, a^{3} b^{2} d^{2}\right )} x^{4}\right )} \log \left (b x + a\right ) - 12 \, {\left ({\left (5 \, b^{5} c^{2} - 8 \, a b^{4} c d + 3 \, a^{2} b^{3} d^{2}\right )} x^{5} + {\left (5 \, a b^{4} c^{2} - 8 \, a^{2} b^{3} c d + 3 \, a^{3} b^{2} d^{2}\right )} x^{4}\right )} \log \left (x\right )}{12 \, {\left (a^{6} b x^{5} + a^{7} x^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (156) = 312\).
Time = 0.58 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.26 \[ \int \frac {(c+d x)^2}{x^5 (a+b x)^2} \, dx=\frac {- 3 a^{4} c^{2} + x^{4} \cdot \left (36 a^{2} b^{2} d^{2} - 96 a b^{3} c d + 60 b^{4} c^{2}\right ) + x^{3} \cdot \left (18 a^{3} b d^{2} - 48 a^{2} b^{2} c d + 30 a b^{3} c^{2}\right ) + x^{2} \left (- 6 a^{4} d^{2} + 16 a^{3} b c d - 10 a^{2} b^{2} c^{2}\right ) + x \left (- 8 a^{4} c d + 5 a^{3} b c^{2}\right )}{12 a^{6} x^{4} + 12 a^{5} b x^{5}} + \frac {b^{2} \left (a d - b c\right ) \left (3 a d - 5 b c\right ) \log {\left (x + \frac {3 a^{3} b^{2} d^{2} - 8 a^{2} b^{3} c d + 5 a b^{4} c^{2} - a b^{2} \left (a d - b c\right ) \left (3 a d - 5 b c\right )}{6 a^{2} b^{3} d^{2} - 16 a b^{4} c d + 10 b^{5} c^{2}} \right )}}{a^{6}} - \frac {b^{2} \left (a d - b c\right ) \left (3 a d - 5 b c\right ) \log {\left (x + \frac {3 a^{3} b^{2} d^{2} - 8 a^{2} b^{3} c d + 5 a b^{4} c^{2} + a b^{2} \left (a d - b c\right ) \left (3 a d - 5 b c\right )}{6 a^{2} b^{3} d^{2} - 16 a b^{4} c d + 10 b^{5} c^{2}} \right )}}{a^{6}} \]
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Time = 0.21 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.34 \[ \int \frac {(c+d x)^2}{x^5 (a+b x)^2} \, dx=-\frac {3 \, a^{4} c^{2} - 12 \, {\left (5 \, b^{4} c^{2} - 8 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} x^{4} - 6 \, {\left (5 \, a b^{3} c^{2} - 8 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2}\right )} x^{3} + 2 \, {\left (5 \, a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + 3 \, a^{4} d^{2}\right )} x^{2} - {\left (5 \, a^{3} b c^{2} - 8 \, a^{4} c d\right )} x}{12 \, {\left (a^{5} b x^{5} + a^{6} x^{4}\right )}} - \frac {{\left (5 \, b^{4} c^{2} - 8 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} \log \left (b x + a\right )}{a^{6}} + \frac {{\left (5 \, b^{4} c^{2} - 8 \, a b^{3} c d + 3 \, a^{2} b^{2} d^{2}\right )} \log \left (x\right )}{a^{6}} \]
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Time = 0.28 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.69 \[ \int \frac {(c+d x)^2}{x^5 (a+b x)^2} \, dx=\frac {{\left (5 \, b^{5} c^{2} - 8 \, a b^{4} c d + 3 \, a^{2} b^{3} d^{2}\right )} \log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{6} b} + \frac {\frac {b^{9} c^{2}}{b x + a} - \frac {2 \, a b^{8} c d}{b x + a} + \frac {a^{2} b^{7} d^{2}}{b x + a}}{a^{5} b^{5}} + \frac {77 \, b^{4} c^{2} - 104 \, a b^{3} c d + 30 \, a^{2} b^{2} d^{2} - \frac {4 \, {\left (65 \, a b^{5} c^{2} - 86 \, a^{2} b^{4} c d + 24 \, a^{3} b^{3} d^{2}\right )}}{{\left (b x + a\right )} b} + \frac {6 \, {\left (50 \, a^{2} b^{6} c^{2} - 64 \, a^{3} b^{5} c d + 17 \, a^{4} b^{4} d^{2}\right )}}{{\left (b x + a\right )}^{2} b^{2}} - \frac {12 \, {\left (10 \, a^{3} b^{7} c^{2} - 12 \, a^{4} b^{6} c d + 3 \, a^{5} b^{5} d^{2}\right )}}{{\left (b x + a\right )}^{3} b^{3}}}{12 \, a^{6} {\left (\frac {a}{b x + a} - 1\right )}^{4}} \]
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Time = 0.59 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.34 \[ \int \frac {(c+d x)^2}{x^5 (a+b x)^2} \, dx=-\frac {\frac {c^2}{4\,a}+\frac {x^2\,\left (3\,a^2\,d^2-8\,a\,b\,c\,d+5\,b^2\,c^2\right )}{6\,a^3}-\frac {b\,x^3\,\left (3\,a^2\,d^2-8\,a\,b\,c\,d+5\,b^2\,c^2\right )}{2\,a^4}+\frac {c\,x\,\left (8\,a\,d-5\,b\,c\right )}{12\,a^2}-\frac {b^2\,x^4\,\left (3\,a^2\,d^2-8\,a\,b\,c\,d+5\,b^2\,c^2\right )}{a^5}}{b\,x^5+a\,x^4}-\frac {2\,b^2\,\mathrm {atanh}\left (\frac {b^2\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-5\,b\,c\right )\,\left (a+2\,b\,x\right )}{a\,\left (3\,a^2\,b^2\,d^2-8\,a\,b^3\,c\,d+5\,b^4\,c^2\right )}\right )\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-5\,b\,c\right )}{a^6} \]
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