Integrand size = 15, antiderivative size = 187 \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx=\frac {21 d^5 (b c-a d)^2 x}{b^7}-\frac {(b c-a d)^7}{4 b^8 (a+b x)^4}-\frac {7 d (b c-a d)^6}{3 b^8 (a+b x)^3}-\frac {21 d^2 (b c-a d)^5}{2 b^8 (a+b x)^2}-\frac {35 d^3 (b c-a d)^4}{b^8 (a+b x)}+\frac {7 d^6 (b c-a d) (a+b x)^2}{2 b^8}+\frac {d^7 (a+b x)^3}{3 b^8}+\frac {35 d^4 (b c-a d)^3 \log (a+b x)}{b^8} \]
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Time = 0.13 (sec) , antiderivative size = 187, 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} \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx=\frac {7 d^6 (a+b x)^2 (b c-a d)}{2 b^8}+\frac {35 d^4 (b c-a d)^3 \log (a+b x)}{b^8}-\frac {35 d^3 (b c-a d)^4}{b^8 (a+b x)}-\frac {21 d^2 (b c-a d)^5}{2 b^8 (a+b x)^2}-\frac {7 d (b c-a d)^6}{3 b^8 (a+b x)^3}-\frac {(b c-a d)^7}{4 b^8 (a+b x)^4}+\frac {d^7 (a+b x)^3}{3 b^8}+\frac {21 d^5 x (b c-a d)^2}{b^7} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {21 d^5 (b c-a d)^2}{b^7}+\frac {(b c-a d)^7}{b^7 (a+b x)^5}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^4}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^3}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^2}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)}+\frac {7 d^6 (b c-a d) (a+b x)}{b^7}+\frac {d^7 (a+b x)^2}{b^7}\right ) \, dx \\ & = \frac {21 d^5 (b c-a d)^2 x}{b^7}-\frac {(b c-a d)^7}{4 b^8 (a+b x)^4}-\frac {7 d (b c-a d)^6}{3 b^8 (a+b x)^3}-\frac {21 d^2 (b c-a d)^5}{2 b^8 (a+b x)^2}-\frac {35 d^3 (b c-a d)^4}{b^8 (a+b x)}+\frac {7 d^6 (b c-a d) (a+b x)^2}{2 b^8}+\frac {d^7 (a+b x)^3}{3 b^8}+\frac {35 d^4 (b c-a d)^3 \log (a+b x)}{b^8} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.93 \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx=\frac {12 b d^5 \left (21 b^2 c^2-35 a b c d+15 a^2 d^2\right ) x+6 b^2 d^6 (7 b c-5 a d) x^2+4 b^3 d^7 x^3-\frac {3 (b c-a d)^7}{(a+b x)^4}-\frac {28 d (b c-a d)^6}{(a+b x)^3}+\frac {126 d^2 (-b c+a d)^5}{(a+b x)^2}-\frac {420 d^3 (b c-a d)^4}{a+b x}+420 d^4 (b c-a d)^3 \log (a+b x)}{12 b^8} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(451\) vs. \(2(177)=354\).
Time = 0.22 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.42
method | result | size |
norman | \(\frac {-\frac {875 a^{7} d^{7}-2625 a^{6} b c \,d^{6}+2625 a^{5} b^{2} c^{2} d^{5}-875 a^{4} b^{3} c^{3} d^{4}+105 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +3 b^{7} c^{7}}{12 b^{8}}+\frac {d^{7} x^{7}}{3 b}-\frac {\left (140 a^{4} d^{7}-420 a^{3} b c \,d^{6}+420 a^{2} b^{2} c^{2} d^{5}-140 a \,b^{3} c^{3} d^{4}+35 b^{4} c^{4} d^{3}\right ) x^{3}}{b^{5}}-\frac {3 \left (210 a^{5} d^{7}-630 a^{4} b c \,d^{6}+630 a^{3} b^{2} c^{2} d^{5}-210 a^{2} b^{3} c^{3} d^{4}+35 a \,b^{4} c^{4} d^{3}+7 b^{5} c^{5} d^{2}\right ) x^{2}}{2 b^{6}}-\frac {\left (770 a^{6} d^{7}-2310 a^{5} b c \,d^{6}+2310 a^{4} b^{2} c^{2} d^{5}-770 a^{3} b^{3} c^{3} d^{4}+105 a^{2} b^{4} c^{4} d^{3}+21 a \,b^{5} c^{5} d^{2}+7 b^{6} c^{6} d \right ) x}{3 b^{7}}+\frac {7 d^{5} \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{5}}{b^{3}}-\frac {7 d^{6} \left (a d -3 b c \right ) x^{6}}{6 b^{2}}}{\left (b x +a \right )^{4}}-\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{8}}\) | \(452\) |
default | \(\frac {d^{5} \left (\frac {1}{3} d^{2} x^{3} b^{2}-\frac {5}{2} x^{2} a b \,d^{2}+\frac {7}{2} x^{2} b^{2} c d +15 a^{2} d^{2} x -35 a b c d x +21 b^{2} c^{2} x \right )}{b^{7}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{3 b^{8} \left (b x +a \right )^{3}}-\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{8}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{4 b^{8} \left (b x +a \right )^{4}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{2 b^{8} \left (b x +a \right )^{2}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{b^{8} \left (b x +a \right )}\) | \(453\) |
risch | \(\frac {d^{7} x^{3}}{3 b^{5}}-\frac {5 d^{7} x^{2} a}{2 b^{6}}+\frac {7 d^{6} x^{2} c}{2 b^{5}}+\frac {15 d^{7} a^{2} x}{b^{7}}-\frac {35 d^{6} a c x}{b^{6}}+\frac {21 d^{5} c^{2} x}{b^{5}}+\frac {\left (-35 a^{4} b^{2} d^{7}+140 a^{3} b^{3} c \,d^{6}-210 a^{2} b^{4} c^{2} d^{5}+140 a \,b^{5} c^{3} d^{4}-35 b^{6} c^{4} d^{3}\right ) x^{3}-\frac {21 b \,d^{2} \left (9 a^{5} d^{5}-35 a^{4} b c \,d^{4}+50 a^{3} b^{2} c^{2} d^{3}-30 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +b^{5} c^{5}\right ) x^{2}}{2}-\frac {7 d \left (37 a^{6} d^{6}-141 a^{5} b c \,d^{5}+195 a^{4} b^{2} c^{2} d^{4}-110 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}+3 a \,b^{5} c^{5} d +b^{6} c^{6}\right ) x}{3}-\frac {319 a^{7} d^{7}-1197 a^{6} b c \,d^{6}+1617 a^{5} b^{2} c^{2} d^{5}-875 a^{4} b^{3} c^{3} d^{4}+105 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +3 b^{7} c^{7}}{12 b}}{b^{7} \left (b x +a \right )^{4}}-\frac {35 d^{7} \ln \left (b x +a \right ) a^{3}}{b^{8}}+\frac {105 d^{6} \ln \left (b x +a \right ) a^{2} c}{b^{7}}-\frac {105 d^{5} \ln \left (b x +a \right ) a \,c^{2}}{b^{6}}+\frac {35 d^{4} \ln \left (b x +a \right ) c^{3}}{b^{5}}\) | \(472\) |
parallelrisch | \(-\frac {1680 \ln \left (b x +a \right ) x \,a^{6} b \,d^{7}+7 a \,b^{6} c^{6} d +21 a^{2} b^{5} c^{5} d^{2}-875 a^{4} b^{3} c^{3} d^{4}+105 a^{3} b^{4} c^{4} d^{3}-2625 a^{6} b c \,d^{6}+2625 a^{5} b^{2} c^{2} d^{5}-5040 \ln \left (b x +a \right ) x^{3} a^{3} b^{4} c \,d^{6}-7560 \ln \left (b x +a \right ) x^{2} a^{4} b^{3} c \,d^{6}+7560 \ln \left (b x +a \right ) x^{2} a^{3} b^{4} c^{2} d^{5}-1260 \ln \left (b x +a \right ) x^{4} a^{2} b^{5} c \,d^{6}+1260 \ln \left (b x +a \right ) x^{4} a \,b^{6} c^{2} d^{5}+5040 \ln \left (b x +a \right ) x \,a^{4} b^{3} c^{2} d^{5}+5040 \ln \left (b x +a \right ) x^{3} a^{2} b^{5} c^{2} d^{5}-1680 \ln \left (b x +a \right ) x^{3} a \,b^{6} c^{3} d^{4}-2520 \ln \left (b x +a \right ) x^{2} a^{2} b^{5} c^{3} d^{4}-1680 \ln \left (b x +a \right ) x \,a^{3} b^{4} c^{3} d^{4}-5040 \ln \left (b x +a \right ) x \,a^{5} b^{2} c \,d^{6}+3 b^{7} c^{7}+875 a^{7} d^{7}+2520 \ln \left (b x +a \right ) x^{2} a^{5} b^{2} d^{7}-3780 x^{2} a^{2} b^{5} c^{3} d^{4}+630 x^{2} a \,b^{6} c^{4} d^{3}-5040 x^{3} a^{3} b^{4} c \,d^{6}+5040 x^{3} a^{2} b^{5} c^{2} d^{5}-1680 x^{3} a \,b^{6} c^{3} d^{4}-1260 \ln \left (b x +a \right ) a^{6} b c \,d^{6}+1260 \ln \left (b x +a \right ) a^{5} b^{2} c^{2} d^{5}-420 \ln \left (b x +a \right ) a^{4} b^{3} c^{3} d^{4}+252 x^{5} a \,b^{6} c \,d^{6}-9240 x \,a^{5} b^{2} c \,d^{6}+9240 x \,a^{4} b^{3} c^{2} d^{5}-3080 x \,a^{3} b^{4} c^{3} d^{4}+420 x \,a^{2} b^{5} c^{4} d^{3}+84 x a \,b^{6} c^{5} d^{2}-11340 x^{2} a^{4} b^{3} c \,d^{6}+11340 x^{2} a^{3} b^{4} c^{2} d^{5}+420 \ln \left (b x +a \right ) a^{7} d^{7}-4 x^{7} d^{7} b^{7}+420 \ln \left (b x +a \right ) x^{4} a^{3} b^{4} d^{7}-420 \ln \left (b x +a \right ) x^{4} b^{7} c^{3} d^{4}-84 x^{5} a^{2} b^{5} d^{7}-252 x^{5} b^{7} c^{2} d^{5}+14 x^{6} a \,b^{6} d^{7}+3080 x \,a^{6} b \,d^{7}+28 x \,b^{7} c^{6} d +3780 x^{2} a^{5} b^{2} d^{7}+126 x^{2} b^{7} c^{5} d^{2}+1680 x^{3} a^{4} b^{3} d^{7}+420 x^{3} b^{7} c^{4} d^{3}-42 x^{6} b^{7} c \,d^{6}+1680 \ln \left (b x +a \right ) x^{3} a^{4} b^{3} d^{7}}{12 b^{8} \left (b x +a \right )^{4}}\) | \(841\) |
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Leaf count of result is larger than twice the leaf count of optimal. 754 vs. \(2 (177) = 354\).
Time = 0.22 (sec) , antiderivative size = 754, normalized size of antiderivative = 4.03 \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx=\frac {4 \, b^{7} d^{7} x^{7} - 3 \, b^{7} c^{7} - 7 \, a b^{6} c^{6} d - 21 \, a^{2} b^{5} c^{5} d^{2} - 105 \, a^{3} b^{4} c^{4} d^{3} + 875 \, a^{4} b^{3} c^{3} d^{4} - 1617 \, a^{5} b^{2} c^{2} d^{5} + 1197 \, a^{6} b c d^{6} - 319 \, a^{7} d^{7} + 14 \, {\left (3 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 84 \, {\left (3 \, b^{7} c^{2} d^{5} - 3 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 4 \, {\left (252 \, a b^{6} c^{2} d^{5} - 357 \, a^{2} b^{5} c d^{6} + 139 \, a^{3} b^{4} d^{7}\right )} x^{4} - 4 \, {\left (105 \, b^{7} c^{4} d^{3} - 420 \, a b^{6} c^{3} d^{4} + 252 \, a^{2} b^{5} c^{2} d^{5} + 168 \, a^{3} b^{4} c d^{6} - 136 \, a^{4} b^{3} d^{7}\right )} x^{3} - 6 \, {\left (21 \, b^{7} c^{5} d^{2} + 105 \, a b^{6} c^{4} d^{3} - 630 \, a^{2} b^{5} c^{3} d^{4} + 882 \, a^{3} b^{4} c^{2} d^{5} - 462 \, a^{4} b^{3} c d^{6} + 74 \, a^{5} b^{2} d^{7}\right )} x^{2} - 4 \, {\left (7 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 105 \, a^{2} b^{5} c^{4} d^{3} - 770 \, a^{3} b^{4} c^{3} d^{4} + 1302 \, a^{4} b^{3} c^{2} d^{5} - 882 \, a^{5} b^{2} c d^{6} + 214 \, a^{6} b d^{7}\right )} x + 420 \, {\left (a^{4} b^{3} c^{3} d^{4} - 3 \, a^{5} b^{2} c^{2} d^{5} + 3 \, a^{6} b c d^{6} - a^{7} d^{7} + {\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 4 \, {\left (a b^{6} c^{3} d^{4} - 3 \, a^{2} b^{5} c^{2} d^{5} + 3 \, a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x^{3} + 6 \, {\left (a^{2} b^{5} c^{3} d^{4} - 3 \, a^{3} b^{4} c^{2} d^{5} + 3 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 4 \, {\left (a^{3} b^{4} c^{3} d^{4} - 3 \, a^{4} b^{3} c^{2} d^{5} + 3 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{12} x^{4} + 4 \, a b^{11} x^{3} + 6 \, a^{2} b^{10} x^{2} + 4 \, a^{3} b^{9} x + a^{4} b^{8}\right )}} \]
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Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (177) = 354\).
Time = 0.26 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.64 \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx=-\frac {3 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} - 875 \, a^{4} b^{3} c^{3} d^{4} + 1617 \, a^{5} b^{2} c^{2} d^{5} - 1197 \, a^{6} b c d^{6} + 319 \, a^{7} d^{7} + 420 \, {\left (b^{7} c^{4} d^{3} - 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 126 \, {\left (b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} - 30 \, a^{2} b^{5} c^{3} d^{4} + 50 \, a^{3} b^{4} c^{2} d^{5} - 35 \, a^{4} b^{3} c d^{6} + 9 \, a^{5} b^{2} d^{7}\right )} x^{2} + 28 \, {\left (b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} - 110 \, a^{3} b^{4} c^{3} d^{4} + 195 \, a^{4} b^{3} c^{2} d^{5} - 141 \, a^{5} b^{2} c d^{6} + 37 \, a^{6} b d^{7}\right )} x}{12 \, {\left (b^{12} x^{4} + 4 \, a b^{11} x^{3} + 6 \, a^{2} b^{10} x^{2} + 4 \, a^{3} b^{9} x + a^{4} b^{8}\right )}} + \frac {2 \, b^{2} d^{7} x^{3} + 3 \, {\left (7 \, b^{2} c d^{6} - 5 \, a b d^{7}\right )} x^{2} + 6 \, {\left (21 \, b^{2} c^{2} d^{5} - 35 \, a b c d^{6} + 15 \, a^{2} d^{7}\right )} x}{6 \, b^{7}} + \frac {35 \, {\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (177) = 354\).
Time = 0.30 (sec) , antiderivative size = 660, normalized size of antiderivative = 3.53 \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx=\frac {{\left (2 \, d^{7} + \frac {21 \, {\left (b^{2} c d^{6} - a b d^{7}\right )}}{{\left (b x + a\right )} b} + \frac {126 \, {\left (b^{4} c^{2} d^{5} - 2 \, a b^{3} c d^{6} + a^{2} b^{2} d^{7}\right )}}{{\left (b x + a\right )}^{2} b^{2}}\right )} {\left (b x + a\right )}^{3}}{6 \, b^{8}} - \frac {35 \, {\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{8}} - \frac {\frac {3 \, b^{43} c^{7}}{{\left (b x + a\right )}^{4}} + \frac {28 \, b^{42} c^{6} d}{{\left (b x + a\right )}^{3}} - \frac {21 \, a b^{42} c^{6} d}{{\left (b x + a\right )}^{4}} + \frac {126 \, b^{41} c^{5} d^{2}}{{\left (b x + a\right )}^{2}} - \frac {168 \, a b^{41} c^{5} d^{2}}{{\left (b x + a\right )}^{3}} + \frac {63 \, a^{2} b^{41} c^{5} d^{2}}{{\left (b x + a\right )}^{4}} + \frac {420 \, b^{40} c^{4} d^{3}}{b x + a} - \frac {630 \, a b^{40} c^{4} d^{3}}{{\left (b x + a\right )}^{2}} + \frac {420 \, a^{2} b^{40} c^{4} d^{3}}{{\left (b x + a\right )}^{3}} - \frac {105 \, a^{3} b^{40} c^{4} d^{3}}{{\left (b x + a\right )}^{4}} - \frac {1680 \, a b^{39} c^{3} d^{4}}{b x + a} + \frac {1260 \, a^{2} b^{39} c^{3} d^{4}}{{\left (b x + a\right )}^{2}} - \frac {560 \, a^{3} b^{39} c^{3} d^{4}}{{\left (b x + a\right )}^{3}} + \frac {105 \, a^{4} b^{39} c^{3} d^{4}}{{\left (b x + a\right )}^{4}} + \frac {2520 \, a^{2} b^{38} c^{2} d^{5}}{b x + a} - \frac {1260 \, a^{3} b^{38} c^{2} d^{5}}{{\left (b x + a\right )}^{2}} + \frac {420 \, a^{4} b^{38} c^{2} d^{5}}{{\left (b x + a\right )}^{3}} - \frac {63 \, a^{5} b^{38} c^{2} d^{5}}{{\left (b x + a\right )}^{4}} - \frac {1680 \, a^{3} b^{37} c d^{6}}{b x + a} + \frac {630 \, a^{4} b^{37} c d^{6}}{{\left (b x + a\right )}^{2}} - \frac {168 \, a^{5} b^{37} c d^{6}}{{\left (b x + a\right )}^{3}} + \frac {21 \, a^{6} b^{37} c d^{6}}{{\left (b x + a\right )}^{4}} + \frac {420 \, a^{4} b^{36} d^{7}}{b x + a} - \frac {126 \, a^{5} b^{36} d^{7}}{{\left (b x + a\right )}^{2}} + \frac {28 \, a^{6} b^{36} d^{7}}{{\left (b x + a\right )}^{3}} - \frac {3 \, a^{7} b^{36} d^{7}}{{\left (b x + a\right )}^{4}}}{12 \, b^{44}} \]
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Time = 0.17 (sec) , antiderivative size = 512, normalized size of antiderivative = 2.74 \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx=x\,\left (\frac {5\,a\,\left (\frac {5\,a\,d^7}{b^6}-\frac {7\,c\,d^6}{b^5}\right )}{b}-\frac {10\,a^2\,d^7}{b^7}+\frac {21\,c^2\,d^5}{b^5}\right )-x^2\,\left (\frac {5\,a\,d^7}{2\,b^6}-\frac {7\,c\,d^6}{2\,b^5}\right )-\frac {\frac {319\,a^7\,d^7-1197\,a^6\,b\,c\,d^6+1617\,a^5\,b^2\,c^2\,d^5-875\,a^4\,b^3\,c^3\,d^4+105\,a^3\,b^4\,c^4\,d^3+21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d+3\,b^7\,c^7}{12\,b}+x\,\left (\frac {259\,a^6\,d^7}{3}-329\,a^5\,b\,c\,d^6+455\,a^4\,b^2\,c^2\,d^5-\frac {770\,a^3\,b^3\,c^3\,d^4}{3}+35\,a^2\,b^4\,c^4\,d^3+7\,a\,b^5\,c^5\,d^2+\frac {7\,b^6\,c^6\,d}{3}\right )+x^3\,\left (35\,a^4\,b^2\,d^7-140\,a^3\,b^3\,c\,d^6+210\,a^2\,b^4\,c^2\,d^5-140\,a\,b^5\,c^3\,d^4+35\,b^6\,c^4\,d^3\right )+x^2\,\left (\frac {189\,a^5\,b\,d^7}{2}-\frac {735\,a^4\,b^2\,c\,d^6}{2}+525\,a^3\,b^3\,c^2\,d^5-315\,a^2\,b^4\,c^3\,d^4+\frac {105\,a\,b^5\,c^4\,d^3}{2}+\frac {21\,b^6\,c^5\,d^2}{2}\right )}{a^4\,b^7+4\,a^3\,b^8\,x+6\,a^2\,b^9\,x^2+4\,a\,b^{10}\,x^3+b^{11}\,x^4}-\frac {\ln \left (a+b\,x\right )\,\left (35\,a^3\,d^7-105\,a^2\,b\,c\,d^6+105\,a\,b^2\,c^2\,d^5-35\,b^3\,c^3\,d^4\right )}{b^8}+\frac {d^7\,x^3}{3\,b^5} \]
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