Integrand size = 15, antiderivative size = 133 \[ \int \frac {(a+b x)^5}{(c+d x)^3} \, dx=\frac {10 b^3 (b c-a d)^2 x}{d^5}+\frac {(b c-a d)^5}{2 d^6 (c+d x)^2}-\frac {5 b (b c-a d)^4}{d^6 (c+d x)}-\frac {5 b^4 (b c-a d) (c+d x)^2}{2 d^6}+\frac {b^5 (c+d x)^3}{3 d^6}-\frac {10 b^2 (b c-a d)^3 \log (c+d x)}{d^6} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 133, 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} \[ \int \frac {(a+b x)^5}{(c+d x)^3} \, dx=-\frac {5 b^4 (c+d x)^2 (b c-a d)}{2 d^6}+\frac {10 b^3 x (b c-a d)^2}{d^5}-\frac {10 b^2 (b c-a d)^3 \log (c+d x)}{d^6}-\frac {5 b (b c-a d)^4}{d^6 (c+d x)}+\frac {(b c-a d)^5}{2 d^6 (c+d x)^2}+\frac {b^5 (c+d x)^3}{3 d^6} \]
[In]
[Out]
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {10 b^3 (b c-a d)^2}{d^5}+\frac {(-b c+a d)^5}{d^5 (c+d x)^3}+\frac {5 b (b c-a d)^4}{d^5 (c+d x)^2}-\frac {10 b^2 (b c-a d)^3}{d^5 (c+d x)}-\frac {5 b^4 (b c-a d) (c+d x)}{d^5}+\frac {b^5 (c+d x)^2}{d^5}\right ) \, dx \\ & = \frac {10 b^3 (b c-a d)^2 x}{d^5}+\frac {(b c-a d)^5}{2 d^6 (c+d x)^2}-\frac {5 b (b c-a d)^4}{d^6 (c+d x)}-\frac {5 b^4 (b c-a d) (c+d x)^2}{2 d^6}+\frac {b^5 (c+d x)^3}{3 d^6}-\frac {10 b^2 (b c-a d)^3 \log (c+d x)}{d^6} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.73 \[ \int \frac {(a+b x)^5}{(c+d x)^3} \, dx=\frac {-3 a^5 d^5-15 a^4 b d^4 (c+2 d x)+30 a^3 b^2 c d^3 (3 c+4 d x)+30 a^2 b^3 d^2 \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )+15 a b^4 d \left (7 c^4+2 c^3 d x-11 c^2 d^2 x^2-4 c d^3 x^3+d^4 x^4\right )+b^5 \left (-27 c^5+6 c^4 d x+63 c^3 d^2 x^2+20 c^2 d^3 x^3-5 c d^4 x^4+2 d^5 x^5\right )-60 b^2 (b c-a d)^3 (c+d x)^2 \log (c+d x)}{6 d^6 (c+d x)^2} \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.91
method | result | size |
default | \(\frac {b^{3} \left (\frac {1}{3} d^{2} x^{3} b^{2}+\frac {5}{2} x^{2} a b \,d^{2}-\frac {3}{2} x^{2} b^{2} c d +10 a^{2} d^{2} x -15 a b c d x +6 b^{2} c^{2} x \right )}{d^{5}}-\frac {a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}}{2 d^{6} \left (d x +c \right )^{2}}+\frac {10 b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (d x +c \right )}{d^{6}}-\frac {5 b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{d^{6} \left (d x +c \right )}\) | \(254\) |
norman | \(\frac {-\frac {a^{5} d^{5}+5 a^{4} b c \,d^{4}-30 a^{3} b^{2} c^{2} d^{3}+90 a^{2} b^{3} c^{3} d^{2}-90 a \,b^{4} c^{4} d +30 b^{5} c^{5}}{2 d^{6}}+\frac {b^{5} x^{5}}{3 d}-\frac {\left (5 a^{4} b \,d^{4}-20 a^{3} b^{2} c \,d^{3}+60 a^{2} b^{3} c^{2} d^{2}-60 a \,b^{4} c^{3} d +20 b^{5} c^{4}\right ) x}{d^{5}}+\frac {10 b^{3} \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) x^{3}}{3 d^{3}}+\frac {5 b^{4} \left (3 a d -b c \right ) x^{4}}{6 d^{2}}}{\left (d x +c \right )^{2}}+\frac {10 b^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (d x +c \right )}{d^{6}}\) | \(254\) |
risch | \(\frac {b^{5} x^{3}}{3 d^{3}}+\frac {5 b^{4} x^{2} a}{2 d^{3}}-\frac {3 b^{5} x^{2} c}{2 d^{4}}+\frac {10 b^{3} a^{2} x}{d^{3}}-\frac {15 b^{4} a c x}{d^{4}}+\frac {6 b^{5} c^{2} x}{d^{5}}+\frac {\left (-5 a^{4} b \,d^{4}+20 a^{3} b^{2} c \,d^{3}-30 a^{2} b^{3} c^{2} d^{2}+20 a \,b^{4} c^{3} d -5 b^{5} c^{4}\right ) x -\frac {a^{5} d^{5}+5 a^{4} b c \,d^{4}-30 a^{3} b^{2} c^{2} d^{3}+50 a^{2} b^{3} c^{3} d^{2}-35 a \,b^{4} c^{4} d +9 b^{5} c^{5}}{2 d}}{d^{5} \left (d x +c \right )^{2}}+\frac {10 b^{2} \ln \left (d x +c \right ) a^{3}}{d^{3}}-\frac {30 b^{3} \ln \left (d x +c \right ) a^{2} c}{d^{4}}+\frac {30 b^{4} \ln \left (d x +c \right ) a \,c^{2}}{d^{5}}-\frac {10 b^{5} \ln \left (d x +c \right ) c^{3}}{d^{6}}\) | \(279\) |
parallelrisch | \(\frac {-15 a^{4} b c \,d^{4}+90 a^{3} b^{2} c^{2} d^{3}-270 a^{2} b^{3} c^{3} d^{2}+270 a \,b^{4} c^{4} d -3 a^{5} d^{5}+180 \ln \left (d x +c \right ) x^{2} a \,b^{4} c^{2} d^{3}-180 \ln \left (d x +c \right ) x^{2} a^{2} b^{3} c \,d^{4}+120 \ln \left (d x +c \right ) x \,a^{3} b^{2} c \,d^{4}-360 \ln \left (d x +c \right ) x \,a^{2} b^{3} c^{2} d^{3}+360 \ln \left (d x +c \right ) x a \,b^{4} c^{3} d^{2}+2 x^{5} b^{5} d^{5}-60 \ln \left (d x +c \right ) b^{5} c^{5}+15 x^{4} a \,b^{4} d^{5}-5 x^{4} b^{5} c \,d^{4}+60 x^{3} a^{2} b^{3} d^{5}+20 x^{3} b^{5} c^{2} d^{3}-30 x \,a^{4} b \,d^{5}-120 x \,b^{5} c^{4} d -180 \ln \left (d x +c \right ) a^{2} b^{3} c^{3} d^{2}+180 \ln \left (d x +c \right ) a \,b^{4} c^{4} d +120 x \,a^{3} b^{2} c \,d^{4}-360 x \,a^{2} b^{3} c^{2} d^{3}+360 x a \,b^{4} c^{3} d^{2}-60 x^{3} a \,b^{4} c \,d^{4}+60 \ln \left (d x +c \right ) a^{3} b^{2} c^{2} d^{3}+60 \ln \left (d x +c \right ) x^{2} a^{3} b^{2} d^{5}-60 \ln \left (d x +c \right ) x^{2} b^{5} c^{3} d^{2}-90 b^{5} c^{5}-120 \ln \left (d x +c \right ) x \,b^{5} c^{4} d}{6 d^{6} \left (d x +c \right )^{2}}\) | \(442\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (127) = 254\).
Time = 0.22 (sec) , antiderivative size = 416, normalized size of antiderivative = 3.13 \[ \int \frac {(a+b x)^5}{(c+d x)^3} \, dx=\frac {2 \, b^{5} d^{5} x^{5} - 27 \, b^{5} c^{5} + 105 \, a b^{4} c^{4} d - 150 \, a^{2} b^{3} c^{3} d^{2} + 90 \, a^{3} b^{2} c^{2} d^{3} - 15 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5} - 5 \, {\left (b^{5} c d^{4} - 3 \, a b^{4} d^{5}\right )} x^{4} + 20 \, {\left (b^{5} c^{2} d^{3} - 3 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \, {\left (21 \, b^{5} c^{3} d^{2} - 55 \, a b^{4} c^{2} d^{3} + 40 \, a^{2} b^{3} c d^{4}\right )} x^{2} + 6 \, {\left (b^{5} c^{4} d + 5 \, a b^{4} c^{3} d^{2} - 20 \, a^{2} b^{3} c^{2} d^{3} + 20 \, a^{3} b^{2} c d^{4} - 5 \, a^{4} b d^{5}\right )} x - 60 \, {\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} - a^{3} b^{2} c^{2} d^{3} + {\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (b^{5} c^{4} d - 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} - a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (d x + c\right )}{6 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (121) = 242\).
Time = 0.96 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b x)^5}{(c+d x)^3} \, dx=\frac {b^{5} x^{3}}{3 d^{3}} + \frac {10 b^{2} \left (a d - b c\right )^{3} \log {\left (c + d x \right )}}{d^{6}} + x^{2} \cdot \left (\frac {5 a b^{4}}{2 d^{3}} - \frac {3 b^{5} c}{2 d^{4}}\right ) + x \left (\frac {10 a^{2} b^{3}}{d^{3}} - \frac {15 a b^{4} c}{d^{4}} + \frac {6 b^{5} c^{2}}{d^{5}}\right ) + \frac {- a^{5} d^{5} - 5 a^{4} b c d^{4} + 30 a^{3} b^{2} c^{2} d^{3} - 50 a^{2} b^{3} c^{3} d^{2} + 35 a b^{4} c^{4} d - 9 b^{5} c^{5} + x \left (- 10 a^{4} b d^{5} + 40 a^{3} b^{2} c d^{4} - 60 a^{2} b^{3} c^{2} d^{3} + 40 a b^{4} c^{3} d^{2} - 10 b^{5} c^{4} d\right )}{2 c^{2} d^{6} + 4 c d^{7} x + 2 d^{8} x^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (127) = 254\).
Time = 0.20 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.04 \[ \int \frac {(a+b x)^5}{(c+d x)^3} \, dx=-\frac {9 \, b^{5} c^{5} - 35 \, a b^{4} c^{4} d + 50 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + a^{5} d^{5} + 10 \, {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x}{2 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} + \frac {2 \, b^{5} d^{2} x^{3} - 3 \, {\left (3 \, b^{5} c d - 5 \, a b^{4} d^{2}\right )} x^{2} + 6 \, {\left (6 \, b^{5} c^{2} - 15 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} x}{6 \, d^{5}} - \frac {10 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (d x + c\right )}{d^{6}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (127) = 254\).
Time = 0.29 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.98 \[ \int \frac {(a+b x)^5}{(c+d x)^3} \, dx=-\frac {10 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{6}} - \frac {9 \, b^{5} c^{5} - 35 \, a b^{4} c^{4} d + 50 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + a^{5} d^{5} + 10 \, {\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x}{2 \, {\left (d x + c\right )}^{2} d^{6}} + \frac {2 \, b^{5} d^{6} x^{3} - 9 \, b^{5} c d^{5} x^{2} + 15 \, a b^{4} d^{6} x^{2} + 36 \, b^{5} c^{2} d^{4} x - 90 \, a b^{4} c d^{5} x + 60 \, a^{2} b^{3} d^{6} x}{6 \, d^{9}} \]
[In]
[Out]
Time = 0.11 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.19 \[ \int \frac {(a+b x)^5}{(c+d x)^3} \, dx=x^2\,\left (\frac {5\,a\,b^4}{2\,d^3}-\frac {3\,b^5\,c}{2\,d^4}\right )-\frac {\frac {a^5\,d^5+5\,a^4\,b\,c\,d^4-30\,a^3\,b^2\,c^2\,d^3+50\,a^2\,b^3\,c^3\,d^2-35\,a\,b^4\,c^4\,d+9\,b^5\,c^5}{2\,d}+x\,\left (5\,a^4\,b\,d^4-20\,a^3\,b^2\,c\,d^3+30\,a^2\,b^3\,c^2\,d^2-20\,a\,b^4\,c^3\,d+5\,b^5\,c^4\right )}{c^2\,d^5+2\,c\,d^6\,x+d^7\,x^2}-x\,\left (\frac {3\,c\,\left (\frac {5\,a\,b^4}{d^3}-\frac {3\,b^5\,c}{d^4}\right )}{d}-\frac {10\,a^2\,b^3}{d^3}+\frac {3\,b^5\,c^2}{d^5}\right )-\frac {\ln \left (c+d\,x\right )\,\left (-10\,a^3\,b^2\,d^3+30\,a^2\,b^3\,c\,d^2-30\,a\,b^4\,c^2\,d+10\,b^5\,c^3\right )}{d^6}+\frac {b^5\,x^3}{3\,d^3} \]
[In]
[Out]