Integrand size = 15, antiderivative size = 158 \[ \int \frac {(a+b x)^6}{(c+d x)^3} \, dx=-\frac {20 b^3 (b c-a d)^3 x}{d^6}-\frac {(b c-a d)^6}{2 d^7 (c+d x)^2}+\frac {6 b (b c-a d)^5}{d^7 (c+d x)}+\frac {15 b^4 (b c-a d)^2 (c+d x)^2}{2 d^7}-\frac {2 b^5 (b c-a d) (c+d x)^3}{d^7}+\frac {b^6 (c+d x)^4}{4 d^7}+\frac {15 b^2 (b c-a d)^4 \log (c+d x)}{d^7} \]
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Time = 0.14 (sec) , antiderivative size = 158, 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} \[ \int \frac {(a+b x)^6}{(c+d x)^3} \, dx=-\frac {2 b^5 (c+d x)^3 (b c-a d)}{d^7}+\frac {15 b^4 (c+d x)^2 (b c-a d)^2}{2 d^7}-\frac {20 b^3 x (b c-a d)^3}{d^6}+\frac {15 b^2 (b c-a d)^4 \log (c+d x)}{d^7}+\frac {6 b (b c-a d)^5}{d^7 (c+d x)}-\frac {(b c-a d)^6}{2 d^7 (c+d x)^2}+\frac {b^6 (c+d x)^4}{4 d^7} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {20 b^3 (b c-a d)^3}{d^6}+\frac {(-b c+a d)^6}{d^6 (c+d x)^3}-\frac {6 b (b c-a d)^5}{d^6 (c+d x)^2}+\frac {15 b^2 (b c-a d)^4}{d^6 (c+d x)}+\frac {15 b^4 (b c-a d)^2 (c+d x)}{d^6}-\frac {6 b^5 (b c-a d) (c+d x)^2}{d^6}+\frac {b^6 (c+d x)^3}{d^6}\right ) \, dx \\ & = -\frac {20 b^3 (b c-a d)^3 x}{d^6}-\frac {(b c-a d)^6}{2 d^7 (c+d x)^2}+\frac {6 b (b c-a d)^5}{d^7 (c+d x)}+\frac {15 b^4 (b c-a d)^2 (c+d x)^2}{2 d^7}-\frac {2 b^5 (b c-a d) (c+d x)^3}{d^7}+\frac {b^6 (c+d x)^4}{4 d^7}+\frac {15 b^2 (b c-a d)^4 \log (c+d x)}{d^7} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.92 \[ \int \frac {(a+b x)^6}{(c+d x)^3} \, dx=\frac {-2 a^6 d^6-12 a^5 b d^5 (c+2 d x)+30 a^4 b^2 c d^4 (3 c+4 d x)+40 a^3 b^3 d^3 \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )+30 a^2 b^4 d^2 \left (7 c^4+2 c^3 d x-11 c^2 d^2 x^2-4 c d^3 x^3+d^4 x^4\right )+4 a b^5 d \left (-27 c^5+6 c^4 d x+63 c^3 d^2 x^2+20 c^2 d^3 x^3-5 c d^4 x^4+2 d^5 x^5\right )+b^6 \left (22 c^6-16 c^5 d x-68 c^4 d^2 x^2-20 c^3 d^3 x^3+5 c^2 d^4 x^4-2 c d^5 x^5+d^6 x^6\right )+60 b^2 (b c-a d)^4 (c+d x)^2 \log (c+d x)}{4 d^7 (c+d x)^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(342\) vs. \(2(152)=304\).
Time = 0.23 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.17
method | result | size |
norman | \(\frac {-\frac {a^{6} d^{6}+6 a^{5} b c \,d^{5}-45 a^{4} b^{2} c^{2} d^{4}+180 a^{3} b^{3} c^{3} d^{3}-270 a^{2} b^{4} c^{4} d^{2}+180 a \,b^{5} c^{5} d -45 b^{6} c^{6}}{2 d^{7}}+\frac {b^{6} x^{6}}{4 d}-\frac {2 \left (3 a^{5} b \,d^{5}-15 a^{4} b^{2} c \,d^{4}+60 a^{3} b^{3} c^{2} d^{3}-90 a^{2} b^{4} c^{3} d^{2}+60 a \,b^{5} c^{4} d -15 b^{6} c^{5}\right ) x}{d^{6}}+\frac {5 b^{3} \left (4 a^{3} d^{3}-6 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) x^{3}}{d^{4}}+\frac {5 b^{4} \left (6 a^{2} d^{2}-4 a b c d +b^{2} c^{2}\right ) x^{4}}{4 d^{3}}+\frac {b^{5} \left (4 a d -b c \right ) x^{5}}{2 d^{2}}}{\left (d x +c \right )^{2}}+\frac {15 b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (d x +c \right )}{d^{7}}\) | \(343\) |
default | \(\frac {b^{3} \left (\frac {1}{4} d^{3} x^{4} b^{3}+2 x^{3} a \,b^{2} d^{3}-x^{3} b^{3} c \,d^{2}+\frac {15}{2} x^{2} a^{2} b \,d^{3}-9 x^{2} a \,b^{2} c \,d^{2}+3 x^{2} b^{3} c^{2} d +20 a^{3} d^{3} x -45 a^{2} b c \,d^{2} x +36 a \,b^{2} c^{2} d x -10 b^{3} c^{3} x \right )}{d^{6}}-\frac {a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}}{2 d^{7} \left (d x +c \right )^{2}}+\frac {15 b^{2} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (d x +c \right )}{d^{7}}-\frac {6 b \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{d^{7} \left (d x +c \right )}\) | \(351\) |
risch | \(\frac {b^{6} x^{4}}{4 d^{3}}+\frac {2 b^{5} x^{3} a}{d^{3}}-\frac {b^{6} x^{3} c}{d^{4}}+\frac {15 b^{4} x^{2} a^{2}}{2 d^{3}}-\frac {9 b^{5} x^{2} a c}{d^{4}}+\frac {3 b^{6} x^{2} c^{2}}{d^{5}}+\frac {20 b^{3} a^{3} x}{d^{3}}-\frac {45 b^{4} a^{2} c x}{d^{4}}+\frac {36 b^{5} a \,c^{2} x}{d^{5}}-\frac {10 b^{6} c^{3} x}{d^{6}}+\frac {\left (-6 a^{5} b \,d^{5}+30 a^{4} b^{2} c \,d^{4}-60 a^{3} b^{3} c^{2} d^{3}+60 a^{2} b^{4} c^{3} d^{2}-30 a \,b^{5} c^{4} d +6 b^{6} c^{5}\right ) x -\frac {a^{6} d^{6}+6 a^{5} b c \,d^{5}-45 a^{4} b^{2} c^{2} d^{4}+100 a^{3} b^{3} c^{3} d^{3}-105 a^{2} b^{4} c^{4} d^{2}+54 a \,b^{5} c^{5} d -11 b^{6} c^{6}}{2 d}}{d^{6} \left (d x +c \right )^{2}}+\frac {15 b^{2} \ln \left (d x +c \right ) a^{4}}{d^{3}}-\frac {60 b^{3} \ln \left (d x +c \right ) a^{3} c}{d^{4}}+\frac {90 b^{4} \ln \left (d x +c \right ) a^{2} c^{2}}{d^{5}}-\frac {60 b^{5} \ln \left (d x +c \right ) a \,c^{3}}{d^{6}}+\frac {15 b^{6} \ln \left (d x +c \right ) c^{4}}{d^{7}}\) | \(383\) |
parallelrisch | \(\frac {-360 a^{3} b^{3} c^{3} d^{3}+540 a^{2} b^{4} c^{4} d^{2}-360 a \,b^{5} c^{5} d -2 a^{6} d^{6}+120 b^{6} c^{5} d x -24 a^{5} b \,d^{6} x +720 \ln \left (d x +c \right ) x \,a^{2} b^{4} c^{3} d^{3}-480 \ln \left (d x +c \right ) x a \,b^{5} c^{4} d^{2}-240 \ln \left (d x +c \right ) x^{2} a^{3} b^{3} c \,d^{5}+360 \ln \left (d x +c \right ) x^{2} a^{2} b^{4} c^{2} d^{4}-240 \ln \left (d x +c \right ) x^{2} a \,b^{5} c^{3} d^{3}+120 \ln \left (d x +c \right ) x \,a^{4} b^{2} c \,d^{5}-480 \ln \left (d x +c \right ) x \,a^{3} b^{3} c^{2} d^{4}+8 x^{5} a \,b^{5} d^{6}-2 x^{5} b^{6} c \,d^{5}+30 x^{4} a^{2} b^{4} d^{6}+5 x^{4} b^{6} c^{2} d^{4}+80 x^{3} a^{3} b^{3} d^{6}-20 x^{3} b^{6} c^{3} d^{3}-480 a^{3} b^{3} c^{2} d^{4} x +720 a^{2} b^{4} c^{3} d^{3} x -480 a \,b^{5} c^{4} d^{2} x +60 \ln \left (d x +c \right ) b^{6} c^{6}+60 \ln \left (d x +c \right ) x^{2} a^{4} b^{2} d^{6}+60 \ln \left (d x +c \right ) x^{2} b^{6} c^{4} d^{2}+120 \ln \left (d x +c \right ) x \,b^{6} c^{5} d +60 \ln \left (d x +c \right ) a^{4} b^{2} c^{2} d^{4}-240 \ln \left (d x +c \right ) a^{3} b^{3} c^{3} d^{3}+360 \ln \left (d x +c \right ) a^{2} b^{4} c^{4} d^{2}-240 \ln \left (d x +c \right ) a \,b^{5} c^{5} d -20 x^{4} a \,b^{5} c \,d^{5}-120 x^{3} a^{2} b^{4} c \,d^{5}+80 x^{3} a \,b^{5} c^{2} d^{4}+120 x \,a^{4} b^{2} c \,d^{5}+x^{6} b^{6} d^{6}-12 a^{5} b c \,d^{5}+90 a^{4} b^{2} c^{2} d^{4}+90 b^{6} c^{6}}{4 d^{7} \left (d x +c \right )^{2}}\) | \(592\) |
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Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (152) = 304\).
Time = 0.22 (sec) , antiderivative size = 548, normalized size of antiderivative = 3.47 \[ \int \frac {(a+b x)^6}{(c+d x)^3} \, dx=\frac {b^{6} d^{6} x^{6} + 22 \, b^{6} c^{6} - 108 \, a b^{5} c^{5} d + 210 \, a^{2} b^{4} c^{4} d^{2} - 200 \, a^{3} b^{3} c^{3} d^{3} + 90 \, a^{4} b^{2} c^{2} d^{4} - 12 \, a^{5} b c d^{5} - 2 \, a^{6} d^{6} - 2 \, {\left (b^{6} c d^{5} - 4 \, a b^{5} d^{6}\right )} x^{5} + 5 \, {\left (b^{6} c^{2} d^{4} - 4 \, a b^{5} c d^{5} + 6 \, a^{2} b^{4} d^{6}\right )} x^{4} - 20 \, {\left (b^{6} c^{3} d^{3} - 4 \, a b^{5} c^{2} d^{4} + 6 \, a^{2} b^{4} c d^{5} - 4 \, a^{3} b^{3} d^{6}\right )} x^{3} - 2 \, {\left (34 \, b^{6} c^{4} d^{2} - 126 \, a b^{5} c^{3} d^{3} + 165 \, a^{2} b^{4} c^{2} d^{4} - 80 \, a^{3} b^{3} c d^{5}\right )} x^{2} - 4 \, {\left (4 \, b^{6} c^{5} d - 6 \, a b^{5} c^{4} d^{2} - 15 \, a^{2} b^{4} c^{3} d^{3} + 40 \, a^{3} b^{3} c^{2} d^{4} - 30 \, a^{4} b^{2} c d^{5} + 6 \, a^{5} b d^{6}\right )} x + 60 \, {\left (b^{6} c^{6} - 4 \, a b^{5} c^{5} d + 6 \, a^{2} b^{4} c^{4} d^{2} - 4 \, a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{2} + 2 \, {\left (b^{6} c^{5} d - 4 \, a b^{5} c^{4} d^{2} + 6 \, a^{2} b^{4} c^{3} d^{3} - 4 \, a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5}\right )} x\right )} \log \left (d x + c\right )}{4 \, {\left (d^{9} x^{2} + 2 \, c d^{8} x + c^{2} d^{7}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (144) = 288\).
Time = 1.28 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.15 \[ \int \frac {(a+b x)^6}{(c+d x)^3} \, dx=\frac {b^{6} x^{4}}{4 d^{3}} + \frac {15 b^{2} \left (a d - b c\right )^{4} \log {\left (c + d x \right )}}{d^{7}} + x^{3} \cdot \left (\frac {2 a b^{5}}{d^{3}} - \frac {b^{6} c}{d^{4}}\right ) + x^{2} \cdot \left (\frac {15 a^{2} b^{4}}{2 d^{3}} - \frac {9 a b^{5} c}{d^{4}} + \frac {3 b^{6} c^{2}}{d^{5}}\right ) + x \left (\frac {20 a^{3} b^{3}}{d^{3}} - \frac {45 a^{2} b^{4} c}{d^{4}} + \frac {36 a b^{5} c^{2}}{d^{5}} - \frac {10 b^{6} c^{3}}{d^{6}}\right ) + \frac {- a^{6} d^{6} - 6 a^{5} b c d^{5} + 45 a^{4} b^{2} c^{2} d^{4} - 100 a^{3} b^{3} c^{3} d^{3} + 105 a^{2} b^{4} c^{4} d^{2} - 54 a b^{5} c^{5} d + 11 b^{6} c^{6} + x \left (- 12 a^{5} b d^{6} + 60 a^{4} b^{2} c d^{5} - 120 a^{3} b^{3} c^{2} d^{4} + 120 a^{2} b^{4} c^{3} d^{3} - 60 a b^{5} c^{4} d^{2} + 12 b^{6} c^{5} d\right )}{2 c^{2} d^{7} + 4 c d^{8} x + 2 d^{9} x^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (152) = 304\).
Time = 0.24 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.30 \[ \int \frac {(a+b x)^6}{(c+d x)^3} \, dx=\frac {11 \, b^{6} c^{6} - 54 \, a b^{5} c^{5} d + 105 \, a^{2} b^{4} c^{4} d^{2} - 100 \, a^{3} b^{3} c^{3} d^{3} + 45 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} - a^{6} d^{6} + 12 \, {\left (b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} + 10 \, a^{2} b^{4} c^{3} d^{3} - 10 \, a^{3} b^{3} c^{2} d^{4} + 5 \, a^{4} b^{2} c d^{5} - a^{5} b d^{6}\right )} x}{2 \, {\left (d^{9} x^{2} + 2 \, c d^{8} x + c^{2} d^{7}\right )}} + \frac {b^{6} d^{3} x^{4} - 4 \, {\left (b^{6} c d^{2} - 2 \, a b^{5} d^{3}\right )} x^{3} + 6 \, {\left (2 \, b^{6} c^{2} d - 6 \, a b^{5} c d^{2} + 5 \, a^{2} b^{4} d^{3}\right )} x^{2} - 4 \, {\left (10 \, b^{6} c^{3} - 36 \, a b^{5} c^{2} d + 45 \, a^{2} b^{4} c d^{2} - 20 \, a^{3} b^{3} d^{3}\right )} x}{4 \, d^{6}} + \frac {15 \, {\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} \log \left (d x + c\right )}{d^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (152) = 304\).
Time = 0.32 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.29 \[ \int \frac {(a+b x)^6}{(c+d x)^3} \, dx=\frac {15 \, {\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{7}} + \frac {11 \, b^{6} c^{6} - 54 \, a b^{5} c^{5} d + 105 \, a^{2} b^{4} c^{4} d^{2} - 100 \, a^{3} b^{3} c^{3} d^{3} + 45 \, a^{4} b^{2} c^{2} d^{4} - 6 \, a^{5} b c d^{5} - a^{6} d^{6} + 12 \, {\left (b^{6} c^{5} d - 5 \, a b^{5} c^{4} d^{2} + 10 \, a^{2} b^{4} c^{3} d^{3} - 10 \, a^{3} b^{3} c^{2} d^{4} + 5 \, a^{4} b^{2} c d^{5} - a^{5} b d^{6}\right )} x}{2 \, {\left (d x + c\right )}^{2} d^{7}} + \frac {b^{6} d^{9} x^{4} - 4 \, b^{6} c d^{8} x^{3} + 8 \, a b^{5} d^{9} x^{3} + 12 \, b^{6} c^{2} d^{7} x^{2} - 36 \, a b^{5} c d^{8} x^{2} + 30 \, a^{2} b^{4} d^{9} x^{2} - 40 \, b^{6} c^{3} d^{6} x + 144 \, a b^{5} c^{2} d^{7} x - 180 \, a^{2} b^{4} c d^{8} x + 80 \, a^{3} b^{3} d^{9} x}{4 \, d^{12}} \]
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Time = 0.31 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.79 \[ \int \frac {(a+b x)^6}{(c+d x)^3} \, dx=x^3\,\left (\frac {2\,a\,b^5}{d^3}-\frac {b^6\,c}{d^4}\right )-\frac {\frac {a^6\,d^6+6\,a^5\,b\,c\,d^5-45\,a^4\,b^2\,c^2\,d^4+100\,a^3\,b^3\,c^3\,d^3-105\,a^2\,b^4\,c^4\,d^2+54\,a\,b^5\,c^5\,d-11\,b^6\,c^6}{2\,d}-x\,\left (-6\,a^5\,b\,d^5+30\,a^4\,b^2\,c\,d^4-60\,a^3\,b^3\,c^2\,d^3+60\,a^2\,b^4\,c^3\,d^2-30\,a\,b^5\,c^4\,d+6\,b^6\,c^5\right )}{c^2\,d^6+2\,c\,d^7\,x+d^8\,x^2}-x^2\,\left (\frac {3\,c\,\left (\frac {6\,a\,b^5}{d^3}-\frac {3\,b^6\,c}{d^4}\right )}{2\,d}-\frac {15\,a^2\,b^4}{2\,d^3}+\frac {3\,b^6\,c^2}{2\,d^5}\right )+x\,\left (\frac {3\,c\,\left (\frac {3\,c\,\left (\frac {6\,a\,b^5}{d^3}-\frac {3\,b^6\,c}{d^4}\right )}{d}-\frac {15\,a^2\,b^4}{d^3}+\frac {3\,b^6\,c^2}{d^5}\right )}{d}+\frac {20\,a^3\,b^3}{d^3}-\frac {b^6\,c^3}{d^6}-\frac {3\,c^2\,\left (\frac {6\,a\,b^5}{d^3}-\frac {3\,b^6\,c}{d^4}\right )}{d^2}\right )+\frac {\ln \left (c+d\,x\right )\,\left (15\,a^4\,b^2\,d^4-60\,a^3\,b^3\,c\,d^3+90\,a^2\,b^4\,c^2\,d^2-60\,a\,b^5\,c^3\,d+15\,b^6\,c^4\right )}{d^7}+\frac {b^6\,x^4}{4\,d^3} \]
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