Integrand size = 15, antiderivative size = 103 \[ \int \frac {(a+b x)^4}{(c+d x)^3} \, dx=-\frac {b^3 (3 b c-4 a d) x}{d^4}+\frac {b^4 x^2}{2 d^3}-\frac {(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac {4 b (b c-a d)^3}{d^5 (c+d x)}+\frac {6 b^2 (b c-a d)^2 \log (c+d x)}{d^5} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 103, 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} \[ \int \frac {(a+b x)^4}{(c+d x)^3} \, dx=-\frac {b^3 x (3 b c-4 a d)}{d^4}+\frac {6 b^2 (b c-a d)^2 \log (c+d x)}{d^5}+\frac {4 b (b c-a d)^3}{d^5 (c+d x)}-\frac {(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac {b^4 x^2}{2 d^3} \]
[In]
[Out]
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {b^3 (3 b c-4 a d)}{d^4}+\frac {b^4 x}{d^3}+\frac {(-b c+a d)^4}{d^4 (c+d x)^3}-\frac {4 b (b c-a d)^3}{d^4 (c+d x)^2}+\frac {6 b^2 (b c-a d)^2}{d^4 (c+d x)}\right ) \, dx \\ & = -\frac {b^3 (3 b c-4 a d) x}{d^4}+\frac {b^4 x^2}{2 d^3}-\frac {(b c-a d)^4}{2 d^5 (c+d x)^2}+\frac {4 b (b c-a d)^3}{d^5 (c+d x)}+\frac {6 b^2 (b c-a d)^2 \log (c+d x)}{d^5} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.62 \[ \int \frac {(a+b x)^4}{(c+d x)^3} \, dx=\frac {-a^4 d^4-4 a^3 b d^3 (c+2 d x)+6 a^2 b^2 c d^2 (3 c+4 d x)+4 a b^3 d \left (-5 c^3-4 c^2 d x+4 c d^2 x^2+2 d^3 x^3\right )+b^4 \left (7 c^4+2 c^3 d x-11 c^2 d^2 x^2-4 c d^3 x^3+d^4 x^4\right )+12 b^2 (b c-a d)^2 (c+d x)^2 \log (c+d x)}{2 d^5 (c+d x)^2} \]
[In]
[Out]
Time = 0.22 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.67
method | result | size |
default | \(\frac {b^{3} \left (\frac {1}{2} b d \,x^{2}+4 a d x -3 b c x \right )}{d^{4}}-\frac {a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}}{2 d^{5} \left (d x +c \right )^{2}}+\frac {6 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{5}}-\frac {4 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{5} \left (d x +c \right )}\) | \(172\) |
norman | \(\frac {-\frac {a^{4} d^{4}+4 a^{3} b c \,d^{3}-18 a^{2} b^{2} c^{2} d^{2}+36 a \,b^{3} c^{3} d -18 b^{4} c^{4}}{2 d^{5}}+\frac {b^{4} x^{4}}{2 d}-\frac {2 \left (2 a^{3} b \,d^{3}-6 a^{2} b^{2} c \,d^{2}+12 a \,b^{3} c^{2} d -6 b^{4} c^{3}\right ) x}{d^{4}}+\frac {2 b^{3} \left (2 a d -b c \right ) x^{3}}{d^{2}}}{\left (d x +c \right )^{2}}+\frac {6 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (d x +c \right )}{d^{5}}\) | \(178\) |
risch | \(\frac {b^{4} x^{2}}{2 d^{3}}+\frac {4 a \,b^{3} x}{d^{3}}-\frac {3 b^{4} c x}{d^{4}}+\frac {\left (-4 a^{3} b \,d^{3}+12 a^{2} b^{2} c \,d^{2}-12 a \,b^{3} c^{2} d +4 b^{4} c^{3}\right ) x -\frac {a^{4} d^{4}+4 a^{3} b c \,d^{3}-18 a^{2} b^{2} c^{2} d^{2}+20 a \,b^{3} c^{3} d -7 b^{4} c^{4}}{2 d}}{d^{4} \left (d x +c \right )^{2}}+\frac {6 b^{2} \ln \left (d x +c \right ) a^{2}}{d^{3}}-\frac {12 b^{3} \ln \left (d x +c \right ) a c}{d^{4}}+\frac {6 b^{4} \ln \left (d x +c \right ) c^{2}}{d^{5}}\) | \(192\) |
parallelrisch | \(\frac {d^{4} x^{4} b^{4}+12 \ln \left (d x +c \right ) x^{2} a^{2} b^{2} d^{4}-24 \ln \left (d x +c \right ) x^{2} a \,b^{3} c \,d^{3}+12 \ln \left (d x +c \right ) x^{2} b^{4} c^{2} d^{2}+8 a \,b^{3} d^{4} x^{3}-4 b^{4} c \,d^{3} x^{3}+24 \ln \left (d x +c \right ) x \,a^{2} b^{2} c \,d^{3}-48 \ln \left (d x +c \right ) x a \,b^{3} c^{2} d^{2}+24 \ln \left (d x +c \right ) x \,b^{4} c^{3} d +12 \ln \left (d x +c \right ) a^{2} b^{2} c^{2} d^{2}-24 \ln \left (d x +c \right ) a \,b^{3} c^{3} d +12 \ln \left (d x +c \right ) b^{4} c^{4}-8 a^{3} b \,d^{4} x +24 a^{2} b^{2} c \,d^{3} x -48 a \,b^{3} c^{2} d^{2} x +24 b^{4} c^{3} d x -a^{4} d^{4}-4 a^{3} b c \,d^{3}+18 a^{2} b^{2} c^{2} d^{2}-36 a \,b^{3} c^{3} d +18 b^{4} c^{4}}{2 d^{5} \left (d x +c \right )^{2}}\) | \(307\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (99) = 198\).
Time = 0.23 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.83 \[ \int \frac {(a+b x)^4}{(c+d x)^3} \, dx=\frac {b^{4} d^{4} x^{4} + 7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} - 4 \, {\left (b^{4} c d^{3} - 2 \, a b^{3} d^{4}\right )} x^{3} - {\left (11 \, b^{4} c^{2} d^{2} - 16 \, a b^{3} c d^{3}\right )} x^{2} + 2 \, {\left (b^{4} c^{3} d - 8 \, a b^{3} c^{2} d^{2} + 12 \, a^{2} b^{2} c d^{3} - 4 \, a^{3} b d^{4}\right )} x + 12 \, {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2} + {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (b^{4} c^{3} d - 2 \, a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \, {\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}} \]
[In]
[Out]
Time = 0.69 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b x)^4}{(c+d x)^3} \, dx=\frac {b^{4} x^{2}}{2 d^{3}} + \frac {6 b^{2} \left (a d - b c\right )^{2} \log {\left (c + d x \right )}}{d^{5}} + x \left (\frac {4 a b^{3}}{d^{3}} - \frac {3 b^{4} c}{d^{4}}\right ) + \frac {- a^{4} d^{4} - 4 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} - 20 a b^{3} c^{3} d + 7 b^{4} c^{4} + x \left (- 8 a^{3} b d^{4} + 24 a^{2} b^{2} c d^{3} - 24 a b^{3} c^{2} d^{2} + 8 b^{4} c^{3} d\right )}{2 c^{2} d^{5} + 4 c d^{6} x + 2 d^{7} x^{2}} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.85 \[ \int \frac {(a+b x)^4}{(c+d x)^3} \, dx=\frac {7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} + 8 \, {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x}{2 \, {\left (d^{7} x^{2} + 2 \, c d^{6} x + c^{2} d^{5}\right )}} + \frac {b^{4} d x^{2} - 2 \, {\left (3 \, b^{4} c - 4 \, a b^{3} d\right )} x}{2 \, d^{4}} + \frac {6 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left (d x + c\right )}{d^{5}} \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.78 \[ \int \frac {(a+b x)^4}{(c+d x)^3} \, dx=\frac {6 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{5}} + \frac {b^{4} d^{3} x^{2} - 6 \, b^{4} c d^{2} x + 8 \, a b^{3} d^{3} x}{2 \, d^{6}} + \frac {7 \, b^{4} c^{4} - 20 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - a^{4} d^{4} + 8 \, {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x}{2 \, {\left (d x + c\right )}^{2} d^{5}} \]
[In]
[Out]
Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.90 \[ \int \frac {(a+b x)^4}{(c+d x)^3} \, dx=x\,\left (\frac {4\,a\,b^3}{d^3}-\frac {3\,b^4\,c}{d^4}\right )-\frac {\frac {a^4\,d^4+4\,a^3\,b\,c\,d^3-18\,a^2\,b^2\,c^2\,d^2+20\,a\,b^3\,c^3\,d-7\,b^4\,c^4}{2\,d}-x\,\left (-4\,a^3\,b\,d^3+12\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+4\,b^4\,c^3\right )}{c^2\,d^4+2\,c\,d^5\,x+d^6\,x^2}+\frac {b^4\,x^2}{2\,d^3}+\frac {\ln \left (c+d\,x\right )\,\left (6\,a^2\,b^2\,d^2-12\,a\,b^3\,c\,d+6\,b^4\,c^2\right )}{d^5} \]
[In]
[Out]