Integrand size = 15, antiderivative size = 130 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=-\frac {10 b^2 (b c-a d)^3 x}{d^5}+\frac {(b c-a d)^5}{d^6 (c+d x)}+\frac {5 b^3 (b c-a d)^2 (c+d x)^2}{d^6}-\frac {5 b^4 (b c-a d) (c+d x)^3}{3 d^6}+\frac {b^5 (c+d x)^4}{4 d^6}+\frac {5 b (b c-a d)^4 \log (c+d x)}{d^6} \]
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Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=-\frac {5 b^4 (c+d x)^3 (b c-a d)}{3 d^6}+\frac {5 b^3 (c+d x)^2 (b c-a d)^2}{d^6}-\frac {10 b^2 x (b c-a d)^3}{d^5}+\frac {(b c-a d)^5}{d^6 (c+d x)}+\frac {5 b (b c-a d)^4 \log (c+d x)}{d^6}+\frac {b^5 (c+d x)^4}{4 d^6} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {10 b^2 (b c-a d)^3}{d^5}+\frac {(-b c+a d)^5}{d^5 (c+d x)^2}+\frac {5 b (b c-a d)^4}{d^5 (c+d x)}+\frac {10 b^3 (b c-a d)^2 (c+d x)}{d^5}-\frac {5 b^4 (b c-a d) (c+d x)^2}{d^5}+\frac {b^5 (c+d x)^3}{d^5}\right ) \, dx \\ & = -\frac {10 b^2 (b c-a d)^3 x}{d^5}+\frac {(b c-a d)^5}{d^6 (c+d x)}+\frac {5 b^3 (b c-a d)^2 (c+d x)^2}{d^6}-\frac {5 b^4 (b c-a d) (c+d x)^3}{3 d^6}+\frac {b^5 (c+d x)^4}{4 d^6}+\frac {5 b (b c-a d)^4 \log (c+d x)}{d^6} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.75 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=\frac {60 a^4 b c d^4-12 a^5 d^5+120 a^3 b^2 d^3 \left (-c^2+c d x+d^2 x^2\right )+60 a^2 b^3 d^2 \left (2 c^3-4 c^2 d x-3 c d^2 x^2+d^3 x^3\right )+20 a b^4 d \left (-3 c^4+9 c^3 d x+6 c^2 d^2 x^2-2 c d^3 x^3+d^4 x^4\right )+b^5 \left (12 c^5-48 c^4 d x-30 c^3 d^2 x^2+10 c^2 d^3 x^3-5 c d^4 x^4+3 d^5 x^5\right )+60 b (b c-a d)^4 (c+d x) \log (c+d x)}{12 d^6 (c+d x)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(255\) vs. \(2(126)=252\).
Time = 0.65 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.97
method | result | size |
norman | \(\frac {\frac {\left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+20 a^{3} b^{2} c^{2} d^{3}-30 a^{2} b^{3} c^{3} d^{2}+20 a \,b^{4} c^{4} d -5 b^{5} c^{5}\right ) x}{d^{5} c}+\frac {b^{5} x^{5}}{4 d}+\frac {5 b^{2} \left (4 a^{3} d^{3}-6 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{2}}{2 d^{4}}+\frac {5 b^{3} \left (6 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) x^{3}}{6 d^{3}}+\frac {5 b^{4} \left (4 a d -b c \right ) x^{4}}{12 d^{2}}}{d x +c}+\frac {5 b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (d x +c \right )}{d^{6}}\) | \(256\) |
default | \(\frac {b^{2} \left (\frac {1}{4} d^{3} x^{4} b^{3}+\frac {5}{3} x^{3} a \,b^{2} d^{3}-\frac {2}{3} x^{3} b^{3} c \,d^{2}+5 x^{2} a^{2} b \,d^{3}-5 x^{2} a \,b^{2} c \,d^{2}+\frac {3}{2} x^{2} b^{3} c^{2} d +10 a^{3} d^{3} x -20 a^{2} b c \,d^{2} x +15 a \,b^{2} c^{2} d x -4 b^{3} c^{3} x \right )}{d^{5}}+\frac {5 b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (d x +c \right )}{d^{6}}-\frac {a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}{d^{6} \left (d x +c \right )}\) | \(259\) |
risch | \(\frac {b^{5} x^{4}}{4 d^{2}}+\frac {5 b^{4} x^{3} a}{3 d^{2}}-\frac {2 b^{5} x^{3} c}{3 d^{3}}+\frac {5 b^{3} x^{2} a^{2}}{d^{2}}-\frac {5 b^{4} x^{2} a c}{d^{3}}+\frac {3 b^{5} x^{2} c^{2}}{2 d^{4}}+\frac {10 b^{2} a^{3} x}{d^{2}}-\frac {20 b^{3} a^{2} c x}{d^{3}}+\frac {15 b^{4} a \,c^{2} x}{d^{4}}-\frac {4 b^{5} c^{3} x}{d^{5}}+\frac {5 b \ln \left (d x +c \right ) a^{4}}{d^{2}}-\frac {20 b^{2} \ln \left (d x +c \right ) a^{3} c}{d^{3}}+\frac {30 b^{3} \ln \left (d x +c \right ) a^{2} c^{2}}{d^{4}}-\frac {20 b^{4} \ln \left (d x +c \right ) a \,c^{3}}{d^{5}}+\frac {5 b^{5} \ln \left (d x +c \right ) c^{4}}{d^{6}}-\frac {a^{5}}{d \left (d x +c \right )}+\frac {5 a^{4} b c}{d^{2} \left (d x +c \right )}-\frac {10 a^{3} b^{2} c^{2}}{d^{3} \left (d x +c \right )}+\frac {10 a^{2} b^{3} c^{3}}{d^{4} \left (d x +c \right )}-\frac {5 a \,b^{4} c^{4}}{d^{5} \left (d x +c \right )}+\frac {b^{5} c^{5}}{d^{6} \left (d x +c \right )}\) | \(326\) |
parallelrisch | \(\frac {60 a^{4} b c \,d^{4}-240 a^{3} b^{2} c^{2} d^{3}+360 a^{2} b^{3} c^{3} d^{2}-240 a \,b^{4} c^{4} d -12 a^{5} d^{5}-240 \ln \left (d x +c \right ) x \,a^{3} b^{2} c \,d^{4}+360 \ln \left (d x +c \right ) x \,a^{2} b^{3} c^{2} d^{3}-240 \ln \left (d x +c \right ) x a \,b^{4} c^{3} d^{2}+3 x^{5} b^{5} d^{5}+60 \ln \left (d x +c \right ) b^{5} c^{5}+20 x^{4} a \,b^{4} d^{5}-5 x^{4} b^{5} c \,d^{4}+60 x^{3} a^{2} b^{3} d^{5}+10 x^{3} b^{5} c^{2} d^{3}+120 x^{2} a^{3} b^{2} d^{5}-30 x^{2} b^{5} c^{3} d^{2}+360 \ln \left (d x +c \right ) a^{2} b^{3} c^{3} d^{2}-240 \ln \left (d x +c \right ) a \,b^{4} c^{4} d -40 x^{3} a \,b^{4} c \,d^{4}-180 x^{2} a^{2} b^{3} c \,d^{4}+120 x^{2} a \,b^{4} c^{2} d^{3}+60 \ln \left (d x +c \right ) a^{4} b c \,d^{4}-240 \ln \left (d x +c \right ) a^{3} b^{2} c^{2} d^{3}+60 b^{5} c^{5}+60 \ln \left (d x +c \right ) x \,a^{4} b \,d^{5}+60 \ln \left (d x +c \right ) x \,b^{5} c^{4} d}{12 d^{6} \left (d x +c \right )}\) | \(389\) |
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Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (126) = 252\).
Time = 0.23 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.87 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=\frac {3 \, b^{5} d^{5} x^{5} + 12 \, b^{5} c^{5} - 60 \, a b^{4} c^{4} d + 120 \, a^{2} b^{3} c^{3} d^{2} - 120 \, a^{3} b^{2} c^{2} d^{3} + 60 \, a^{4} b c d^{4} - 12 \, a^{5} d^{5} - 5 \, {\left (b^{5} c d^{4} - 4 \, a b^{4} d^{5}\right )} x^{4} + 10 \, {\left (b^{5} c^{2} d^{3} - 4 \, a b^{4} c d^{4} + 6 \, a^{2} b^{3} d^{5}\right )} x^{3} - 30 \, {\left (b^{5} c^{3} d^{2} - 4 \, a b^{4} c^{2} d^{3} + 6 \, a^{2} b^{3} c d^{4} - 4 \, a^{3} b^{2} d^{5}\right )} x^{2} - 12 \, {\left (4 \, b^{5} c^{4} d - 15 \, a b^{4} c^{3} d^{2} + 20 \, a^{2} b^{3} c^{2} d^{3} - 10 \, a^{3} b^{2} c d^{4}\right )} x + 60 \, {\left (b^{5} c^{5} - 4 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} - 4 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} + {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x\right )} \log \left (d x + c\right )}{12 \, {\left (d^{7} x + c d^{6}\right )}} \]
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Time = 0.53 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.78 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=\frac {b^{5} x^{4}}{4 d^{2}} + \frac {5 b \left (a d - b c\right )^{4} \log {\left (c + d x \right )}}{d^{6}} + x^{3} \cdot \left (\frac {5 a b^{4}}{3 d^{2}} - \frac {2 b^{5} c}{3 d^{3}}\right ) + x^{2} \cdot \left (\frac {5 a^{2} b^{3}}{d^{2}} - \frac {5 a b^{4} c}{d^{3}} + \frac {3 b^{5} c^{2}}{2 d^{4}}\right ) + x \left (\frac {10 a^{3} b^{2}}{d^{2}} - \frac {20 a^{2} b^{3} c}{d^{3}} + \frac {15 a b^{4} c^{2}}{d^{4}} - \frac {4 b^{5} c^{3}}{d^{5}}\right ) + \frac {- a^{5} d^{5} + 5 a^{4} b c d^{4} - 10 a^{3} b^{2} c^{2} d^{3} + 10 a^{2} b^{3} c^{3} d^{2} - 5 a b^{4} c^{4} d + b^{5} c^{5}}{c d^{6} + d^{7} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (126) = 252\).
Time = 0.21 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.03 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=\frac {b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}}{d^{7} x + c d^{6}} + \frac {3 \, b^{5} d^{3} x^{4} - 4 \, {\left (2 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{3} + 6 \, {\left (3 \, b^{5} c^{2} d - 10 \, a b^{4} c d^{2} + 10 \, a^{2} b^{3} d^{3}\right )} x^{2} - 12 \, {\left (4 \, b^{5} c^{3} - 15 \, a b^{4} c^{2} d + 20 \, a^{2} b^{3} c d^{2} - 10 \, a^{3} b^{2} d^{3}\right )} x}{12 \, d^{5}} + \frac {5 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} \log \left (d x + c\right )}{d^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (126) = 252\).
Time = 0.30 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.61 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=\frac {{\left (3 \, b^{5} - \frac {20 \, {\left (b^{5} c d - a b^{4} d^{2}\right )}}{{\left (d x + c\right )} d} + \frac {60 \, {\left (b^{5} c^{2} d^{2} - 2 \, a b^{4} c d^{3} + a^{2} b^{3} d^{4}\right )}}{{\left (d x + c\right )}^{2} d^{2}} - \frac {120 \, {\left (b^{5} c^{3} d^{3} - 3 \, a b^{4} c^{2} d^{4} + 3 \, a^{2} b^{3} c d^{5} - a^{3} b^{2} d^{6}\right )}}{{\left (d x + c\right )}^{3} d^{3}}\right )} {\left (d x + c\right )}^{4}}{12 \, d^{6}} - \frac {5 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} \log \left (\frac {{\left | d x + c \right |}}{{\left (d x + c\right )}^{2} {\left | d \right |}}\right )}{d^{6}} + \frac {\frac {b^{5} c^{5} d^{4}}{d x + c} - \frac {5 \, a b^{4} c^{4} d^{5}}{d x + c} + \frac {10 \, a^{2} b^{3} c^{3} d^{6}}{d x + c} - \frac {10 \, a^{3} b^{2} c^{2} d^{7}}{d x + c} + \frac {5 \, a^{4} b c d^{8}}{d x + c} - \frac {a^{5} d^{9}}{d x + c}}{d^{10}} \]
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Time = 0.27 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.52 \[ \int \frac {(a+b x)^5}{(c+d x)^2} \, dx=x^3\,\left (\frac {5\,a\,b^4}{3\,d^2}-\frac {2\,b^5\,c}{3\,d^3}\right )+x\,\left (\frac {2\,c\,\left (\frac {2\,c\,\left (\frac {5\,a\,b^4}{d^2}-\frac {2\,b^5\,c}{d^3}\right )}{d}-\frac {10\,a^2\,b^3}{d^2}+\frac {b^5\,c^2}{d^4}\right )}{d}+\frac {10\,a^3\,b^2}{d^2}-\frac {c^2\,\left (\frac {5\,a\,b^4}{d^2}-\frac {2\,b^5\,c}{d^3}\right )}{d^2}\right )-x^2\,\left (\frac {c\,\left (\frac {5\,a\,b^4}{d^2}-\frac {2\,b^5\,c}{d^3}\right )}{d}-\frac {5\,a^2\,b^3}{d^2}+\frac {b^5\,c^2}{2\,d^4}\right )+\frac {\ln \left (c+d\,x\right )\,\left (5\,a^4\,b\,d^4-20\,a^3\,b^2\,c\,d^3+30\,a^2\,b^3\,c^2\,d^2-20\,a\,b^4\,c^3\,d+5\,b^5\,c^4\right )}{d^6}-\frac {a^5\,d^5-5\,a^4\,b\,c\,d^4+10\,a^3\,b^2\,c^2\,d^3-10\,a^2\,b^3\,c^3\,d^2+5\,a\,b^4\,c^4\,d-b^5\,c^5}{d\,\left (x\,d^6+c\,d^5\right )}+\frac {b^5\,x^4}{4\,d^2} \]
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