Integrand size = 18, antiderivative size = 124 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^2} \, dx=\frac {x}{b^2 d^2}-\frac {a^4}{b^3 (b c-a d)^2 (a+b x)}-\frac {c^4}{d^3 (b c-a d)^2 (c+d x)}-\frac {2 a^3 (2 b c-a d) \log (a+b x)}{b^3 (b c-a d)^3}-\frac {2 c^3 (b c-2 a d) \log (c+d x)}{d^3 (b c-a d)^3} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {90} \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {a^4}{b^3 (a+b x) (b c-a d)^2}-\frac {2 a^3 (2 b c-a d) \log (a+b x)}{b^3 (b c-a d)^3}-\frac {c^4}{d^3 (c+d x) (b c-a d)^2}-\frac {2 c^3 (b c-2 a d) \log (c+d x)}{d^3 (b c-a d)^3}+\frac {x}{b^2 d^2} \]
[In]
[Out]
Rule 90
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{b^2 d^2}+\frac {a^4}{b^2 (b c-a d)^2 (a+b x)^2}+\frac {2 a^3 (-2 b c+a d)}{b^2 (b c-a d)^3 (a+b x)}+\frac {c^4}{d^2 (-b c+a d)^2 (c+d x)^2}+\frac {2 c^3 (b c-2 a d)}{d^2 (-b c+a d)^3 (c+d x)}\right ) \, dx \\ & = \frac {x}{b^2 d^2}-\frac {a^4}{b^3 (b c-a d)^2 (a+b x)}-\frac {c^4}{d^3 (b c-a d)^2 (c+d x)}-\frac {2 a^3 (2 b c-a d) \log (a+b x)}{b^3 (b c-a d)^3}-\frac {2 c^3 (b c-2 a d) \log (c+d x)}{d^3 (b c-a d)^3} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.99 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^2} \, dx=\frac {x}{b^2 d^2}-\frac {a^4}{b^3 (b c-a d)^2 (a+b x)}-\frac {c^4}{d^3 (b c-a d)^2 (c+d x)}+\frac {2 a^3 (-2 b c+a d) \log (a+b x)}{b^3 (b c-a d)^3}+\frac {2 c^3 (b c-2 a d) \log (c+d x)}{d^3 (-b c+a d)^3} \]
[In]
[Out]
Time = 0.50 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.01
method | result | size |
default | \(\frac {x}{b^{2} d^{2}}-\frac {c^{4}}{d^{3} \left (a d -b c \right )^{2} \left (d x +c \right )}-\frac {2 c^{3} \left (2 a d -b c \right ) \ln \left (d x +c \right )}{d^{3} \left (a d -b c \right )^{3}}-\frac {a^{4}}{b^{3} \left (a d -b c \right )^{2} \left (b x +a \right )}-\frac {2 a^{3} \left (a d -2 b c \right ) \ln \left (b x +a \right )}{b^{3} \left (a d -b c \right )^{3}}\) | \(125\) |
norman | \(\frac {\frac {x^{3}}{b d}-\frac {\left (2 a^{4} d^{4}-a^{3} b c \,d^{3}-a \,b^{3} c^{3} d +2 b^{4} c^{4}\right ) x}{d^{3} b^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (2 a^{3} d^{3}-a^{2} b c \,d^{2}-a \,b^{2} c^{2} d +2 b^{3} c^{3}\right ) a c}{d^{3} b^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{\left (b x +a \right ) \left (d x +c \right )}-\frac {2 a^{3} \left (a d -2 b c \right ) \ln \left (b x +a \right )}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{3}}-\frac {2 c^{3} \left (2 a d -b c \right ) \ln \left (d x +c \right )}{d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(288\) |
risch | \(\frac {x}{b^{2} d^{2}}+\frac {-\frac {\left (a^{4} d^{4}+b^{4} c^{4}\right ) x}{b d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (a d +b c \right ) a c \left (a^{2} d^{2}-a b c d +b^{2} c^{2}\right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b d}}{b^{2} d^{2} \left (b x +a \right ) \left (d x +c \right )}-\frac {4 c^{3} \ln \left (-d x -c \right ) a}{d^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {2 c^{4} \ln \left (-d x -c \right ) b}{d^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {2 a^{4} \ln \left (b x +a \right ) d}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{3}}+\frac {4 a^{3} \ln \left (b x +a \right ) c}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2}}\) | \(358\) |
parallelrisch | \(-\frac {-3 a^{4} b \,c^{2} d^{3}+3 a^{2} b^{3} c^{4} d +2 a^{5} d^{5} x +4 \ln \left (d x +c \right ) x \,a^{2} b^{3} c^{3} d^{2}+2 \ln \left (d x +c \right ) x a \,b^{4} c^{4} d -a^{3} b^{2} d^{5} x^{3}+b^{5} c^{3} d^{2} x^{3}-4 \ln \left (b x +a \right ) x^{2} a^{3} b^{2} c \,d^{4}+4 \ln \left (d x +c \right ) x^{2} a \,b^{4} c^{3} d^{2}-2 \ln \left (b x +a \right ) x \,a^{4} b c \,d^{4}-4 \ln \left (b x +a \right ) x \,a^{3} b^{2} c^{2} d^{3}+2 \ln \left (b x +a \right ) x \,a^{5} d^{5}-2 \ln \left (d x +c \right ) x \,b^{5} c^{5}+2 \ln \left (b x +a \right ) a^{5} c \,d^{4}-2 \ln \left (d x +c \right ) a \,b^{4} c^{5}-3 a \,b^{4} c^{2} d^{3} x^{3}-3 a^{4} b c \,d^{4} x +a^{3} b^{2} c^{2} d^{3} x -a^{2} b^{3} c^{3} d^{2} x +3 a \,b^{4} c^{4} d x +3 a^{2} b^{3} c \,d^{4} x^{3}-2 b^{5} c^{5} x +2 \ln \left (b x +a \right ) x^{2} a^{4} b \,d^{5}-2 \ln \left (d x +c \right ) x^{2} b^{5} c^{4} d -4 \ln \left (b x +a \right ) a^{4} b \,c^{2} d^{3}+4 \ln \left (d x +c \right ) a^{2} b^{3} c^{4} d +2 a^{5} c \,d^{4}-2 a \,b^{4} c^{5}}{\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d x +c \right ) \left (b x +a \right ) b^{3} d^{3}}\) | \(480\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 537 vs. \(2 (124) = 248\).
Time = 0.24 (sec) , antiderivative size = 537, normalized size of antiderivative = 4.33 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {a b^{4} c^{5} - a^{2} b^{3} c^{4} d + a^{4} b c^{2} d^{3} - a^{5} c d^{4} - {\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{3} - {\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} + 2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{2} + {\left (b^{5} c^{5} - 2 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} - 3 \, a^{3} b^{2} c^{2} d^{3} + 2 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} x + 2 \, {\left (2 \, a^{4} b c^{2} d^{3} - a^{5} c d^{4} + {\left (2 \, a^{3} b^{2} c d^{4} - a^{4} b d^{5}\right )} x^{2} + {\left (2 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} - a^{5} d^{5}\right )} x\right )} \log \left (b x + a\right ) + 2 \, {\left (a b^{4} c^{5} - 2 \, a^{2} b^{3} c^{4} d + {\left (b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2}\right )} x^{2} + {\left (b^{5} c^{5} - a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{6} c^{4} d^{3} - 3 \, a^{2} b^{5} c^{3} d^{4} + 3 \, a^{3} b^{4} c^{2} d^{5} - a^{4} b^{3} c d^{6} + {\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{2} + {\left (b^{7} c^{4} d^{3} - 2 \, a b^{6} c^{3} d^{4} + 2 \, a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (112) = 224\).
Time = 22.62 (sec) , antiderivative size = 695, normalized size of antiderivative = 5.60 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^2} \, dx=- \frac {2 a^{3} \left (a d - 2 b c\right ) \log {\left (x + \frac {\frac {2 a^{7} d^{6} \left (a d - 2 b c\right )}{b \left (a d - b c\right )^{3}} - \frac {8 a^{6} c d^{5} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{3}} + \frac {12 a^{5} b c^{2} d^{4} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{3}} - \frac {8 a^{4} b^{2} c^{3} d^{3} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{3}} + 2 a^{4} c d^{3} + \frac {2 a^{3} b^{3} c^{4} d^{2} \left (a d - 2 b c\right )}{\left (a d - b c\right )^{3}} - 4 a^{3} b c^{2} d^{2} - 4 a^{2} b^{2} c^{3} d + 2 a b^{3} c^{4}}{2 a^{4} d^{4} - 4 a^{3} b c d^{3} - 4 a b^{3} c^{3} d + 2 b^{4} c^{4}} \right )}}{b^{3} \left (a d - b c\right )^{3}} - \frac {2 c^{3} \cdot \left (2 a d - b c\right ) \log {\left (x + \frac {\frac {2 a^{4} b^{2} c^{3} d^{3} \cdot \left (2 a d - b c\right )}{\left (a d - b c\right )^{3}} + 2 a^{4} c d^{3} - \frac {8 a^{3} b^{3} c^{4} d^{2} \cdot \left (2 a d - b c\right )}{\left (a d - b c\right )^{3}} - 4 a^{3} b c^{2} d^{2} + \frac {12 a^{2} b^{4} c^{5} d \left (2 a d - b c\right )}{\left (a d - b c\right )^{3}} - 4 a^{2} b^{2} c^{3} d - \frac {8 a b^{5} c^{6} \cdot \left (2 a d - b c\right )}{\left (a d - b c\right )^{3}} + 2 a b^{3} c^{4} + \frac {2 b^{6} c^{7} \cdot \left (2 a d - b c\right )}{d \left (a d - b c\right )^{3}}}{2 a^{4} d^{4} - 4 a^{3} b c d^{3} - 4 a b^{3} c^{3} d + 2 b^{4} c^{4}} \right )}}{d^{3} \left (a d - b c\right )^{3}} + \frac {- a^{4} c d^{3} - a b^{3} c^{4} + x \left (- a^{4} d^{4} - b^{4} c^{4}\right )}{a^{3} b^{3} c d^{5} - 2 a^{2} b^{4} c^{2} d^{4} + a b^{5} c^{3} d^{3} + x^{2} \left (a^{2} b^{4} d^{6} - 2 a b^{5} c d^{5} + b^{6} c^{2} d^{4}\right ) + x \left (a^{3} b^{3} d^{6} - a^{2} b^{4} c d^{5} - a b^{5} c^{2} d^{4} + b^{6} c^{3} d^{3}\right )} + \frac {x}{b^{2} d^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (124) = 248\).
Time = 0.21 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.38 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {2 \, {\left (2 \, a^{3} b c - a^{4} d\right )} \log \left (b x + a\right )}{b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}} - \frac {2 \, {\left (b c^{4} - 2 \, a c^{3} d\right )} \log \left (d x + c\right )}{b^{3} c^{3} d^{3} - 3 \, a b^{2} c^{2} d^{4} + 3 \, a^{2} b c d^{5} - a^{3} d^{6}} - \frac {a b^{3} c^{4} + a^{4} c d^{3} + {\left (b^{4} c^{4} + a^{4} d^{4}\right )} x}{a b^{5} c^{3} d^{3} - 2 \, a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + {\left (b^{6} c^{2} d^{4} - 2 \, a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{2} + {\left (b^{6} c^{3} d^{3} - a b^{5} c^{2} d^{4} - a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x} + \frac {x}{b^{2} d^{2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (124) = 248\).
Time = 0.29 (sec) , antiderivative size = 312, normalized size of antiderivative = 2.52 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {a^{4} b^{3}}{{\left (b^{8} c^{2} - 2 \, a b^{7} c d + a^{2} b^{6} d^{2}\right )} {\left (b x + a\right )}} - \frac {2 \, {\left (b^{2} c^{4} - 2 \, a b c^{3} d\right )} \log \left ({\left | \frac {b c}{b x + a} - \frac {a d}{b x + a} + d \right |}\right )}{b^{4} c^{3} d^{3} - 3 \, a b^{3} c^{2} d^{4} + 3 \, a^{2} b^{2} c d^{5} - a^{3} b d^{6}} + \frac {2 \, {\left (b c + a d\right )} \log \left (\frac {{\left | b x + a \right |}}{{\left (b x + a\right )}^{2} {\left | b \right |}}\right )}{b^{3} d^{3}} + \frac {{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3} + \frac {2 \, b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}}{{\left (b c - a d\right )} {\left (b x + a\right )} b}\right )} {\left (b x + a\right )}}{{\left (b c - a d\right )}^{2} b^{3} {\left (\frac {b c}{b x + a} - \frac {a d}{b x + a} + d\right )} d^{2}} \]
[In]
[Out]
Time = 0.75 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.21 \[ \int \frac {x^4}{(a+b x)^2 (c+d x)^2} \, dx=\frac {x}{b^2\,d^2}-\frac {\frac {x\,\left (a^4\,d^4+b^4\,c^4\right )}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {a\,c\,\left (a^3\,d^3+b^3\,c^3\right )}{b\,d\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}}{x\,\left (c\,b^3\,d^2+a\,b^2\,d^3\right )+b^3\,d^3\,x^2+a\,b^2\,c\,d^2}+\frac {\ln \left (a+b\,x\right )\,\left (2\,a^4\,d-4\,a^3\,b\,c\right )}{-a^3\,b^3\,d^3+3\,a^2\,b^4\,c\,d^2-3\,a\,b^5\,c^2\,d+b^6\,c^3}+\frac {\ln \left (c+d\,x\right )\,\left (2\,b\,c^4-4\,a\,c^3\,d\right )}{a^3\,d^6-3\,a^2\,b\,c\,d^5+3\,a\,b^2\,c^2\,d^4-b^3\,c^3\,d^3} \]
[In]
[Out]