Integrand size = 21, antiderivative size = 80 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=-\frac {1}{3 a^2 d (c+d x)^3}-\frac {b}{3 a^2 d \left (a+b (c+d x)^3\right )}-\frac {2 b \log (c+d x)}{a^3 d}+\frac {2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {379, 272, 46} \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=-\frac {2 b \log (c+d x)}{a^3 d}+\frac {2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d}-\frac {b}{3 a^2 d \left (a+b (c+d x)^3\right )}-\frac {1}{3 a^2 d (c+d x)^3} \]
[In]
[Out]
Rule 46
Rule 272
Rule 379
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^4 \left (a+b x^3\right )^2} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x^2 (a+b x)^2} \, dx,x,(c+d x)^3\right )}{3 d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{a^2 x^2}-\frac {2 b}{a^3 x}+\frac {b^2}{a^2 (a+b x)^2}+\frac {2 b^2}{a^3 (a+b x)}\right ) \, dx,x,(c+d x)^3\right )}{3 d} \\ & = -\frac {1}{3 a^2 d (c+d x)^3}-\frac {b}{3 a^2 d \left (a+b (c+d x)^3\right )}-\frac {2 b \log (c+d x)}{a^3 d}+\frac {2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=-\frac {a \left (\frac {1}{(c+d x)^3}+\frac {b}{a+b (c+d x)^3}\right )+6 b \log (c+d x)-2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d} \]
[In]
[Out]
Time = 4.10 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.58
method | result | size |
default | \(-\frac {1}{3 a^{2} d \left (d x +c \right )^{3}}-\frac {2 b \ln \left (d x +c \right )}{a^{3} d}+\frac {b^{2} \left (-\frac {a}{3 d b \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}+\frac {2 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 b d}\right )}{a^{3}}\) | \(126\) |
risch | \(\frac {-\frac {2 b \,d^{2} x^{3}}{3 a^{2}}-\frac {2 b c d \,x^{2}}{a^{2}}-\frac {2 b x \,c^{2}}{a^{2}}-\frac {2 c^{3} b +a}{3 d \,a^{2}}}{\left (d x +c \right )^{3} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}-\frac {2 b \ln \left (d x +c \right )}{a^{3} d}+\frac {2 b \ln \left (-b \,d^{3} x^{3}-3 b c \,d^{2} x^{2}-3 b \,c^{2} d x -c^{3} b -a \right )}{3 a^{3} d}\) | \(157\) |
norman | \(\frac {-\frac {2 b c d \,x^{2}}{a^{2}}-\frac {2 b x \,c^{2}}{a^{2}}+\frac {-2 b^{2} c^{3} d^{5}-a b \,d^{5}}{3 a^{2} d^{6} b}-\frac {2 b \,d^{2} x^{3}}{3 a^{2}}}{\left (d x +c \right )^{3} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}-\frac {2 b \ln \left (d x +c \right )}{a^{3} d}+\frac {2 b \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}{3 a^{3} d}\) | \(167\) |
parallelrisch | \(-\frac {a^{2} b \,d^{5}+18 \ln \left (d x +c \right ) x^{2} a \,b^{2} c \,d^{7}-6 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x^{2} a \,b^{2} c \,d^{7}+18 \ln \left (d x +c \right ) x a \,b^{2} c^{2} d^{6}-6 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x a \,b^{2} c^{2} d^{6}+2 a \,b^{2} c^{3} d^{5}+6 x^{2} a \,b^{2} c \,d^{7}+6 x a \,b^{2} c^{2} d^{6}+36 \ln \left (d x +c \right ) x^{5} b^{3} c \,d^{10}-12 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x^{5} b^{3} c \,d^{10}+90 \ln \left (d x +c \right ) x^{4} b^{3} c^{2} d^{9}-30 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x^{4} b^{3} c^{2} d^{9}+120 \ln \left (d x +c \right ) x^{3} b^{3} c^{3} d^{8}-40 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x^{3} b^{3} c^{3} d^{8}+90 \ln \left (d x +c \right ) x^{2} b^{3} c^{4} d^{7}-30 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x^{2} b^{3} c^{4} d^{7}+36 \ln \left (d x +c \right ) x \,b^{3} c^{5} d^{6}-12 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x \,b^{3} c^{5} d^{6}+6 \ln \left (d x +c \right ) x^{3} a \,b^{2} d^{8}-2 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x^{3} a \,b^{2} d^{8}+6 \ln \left (d x +c \right ) a \,b^{2} c^{3} d^{5}-2 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) a \,b^{2} c^{3} d^{5}+2 x^{3} a \,b^{2} d^{8}+6 \ln \left (d x +c \right ) x^{6} b^{3} d^{11}-2 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) x^{6} b^{3} d^{11}+6 \ln \left (d x +c \right ) b^{3} c^{6} d^{5}-2 \ln \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right ) b^{3} c^{6} d^{5}}{3 a^{3} b \,d^{6} \left (d x +c \right )^{3} \left (b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +c^{3} b +a \right )}\) | \(829\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (74) = 148\).
Time = 0.38 (sec) , antiderivative size = 431, normalized size of antiderivative = 5.39 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=-\frac {2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + a^{2} - 2 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + {\left (20 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} x^{2} + 3 \, {\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d x\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 6 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + {\left (20 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + a b c^{3} + 3 \, {\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} x^{2} + 3 \, {\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d x\right )} \log \left (d x + c\right )}{3 \, {\left (a^{3} b d^{7} x^{6} + 6 \, a^{3} b c d^{6} x^{5} + 15 \, a^{3} b c^{2} d^{5} x^{4} + {\left (20 \, a^{3} b c^{3} + a^{4}\right )} d^{4} x^{3} + 3 \, {\left (5 \, a^{3} b c^{4} + a^{4} c\right )} d^{3} x^{2} + 3 \, {\left (2 \, a^{3} b c^{5} + a^{4} c^{2}\right )} d^{2} x + {\left (a^{3} b c^{6} + a^{4} c^{3}\right )} d\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (70) = 140\).
Time = 1.65 (sec) , antiderivative size = 250, normalized size of antiderivative = 3.12 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=\frac {- a - 2 b c^{3} - 6 b c^{2} d x - 6 b c d^{2} x^{2} - 2 b d^{3} x^{3}}{3 a^{3} c^{3} d + 3 a^{2} b c^{6} d + 45 a^{2} b c^{2} d^{5} x^{4} + 18 a^{2} b c d^{6} x^{5} + 3 a^{2} b d^{7} x^{6} + x^{3} \cdot \left (3 a^{3} d^{4} + 60 a^{2} b c^{3} d^{4}\right ) + x^{2} \cdot \left (9 a^{3} c d^{3} + 45 a^{2} b c^{4} d^{3}\right ) + x \left (9 a^{3} c^{2} d^{2} + 18 a^{2} b c^{5} d^{2}\right )} - \frac {2 b \log {\left (\frac {c}{d} + x \right )}}{a^{3} d} + \frac {2 b \log {\left (\frac {3 c^{2} x}{d^{2}} + \frac {3 c x^{2}}{d} + x^{3} + \frac {a + b c^{3}}{b d^{3}} \right )}}{3 a^{3} d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (74) = 148\).
Time = 0.20 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.78 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=-\frac {2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 6 \, b c^{2} d x + 2 \, b c^{3} + a}{3 \, {\left (a^{2} b d^{7} x^{6} + 6 \, a^{2} b c d^{6} x^{5} + 15 \, a^{2} b c^{2} d^{5} x^{4} + {\left (20 \, a^{2} b c^{3} + a^{3}\right )} d^{4} x^{3} + 3 \, {\left (5 \, a^{2} b c^{4} + a^{3} c\right )} d^{3} x^{2} + 3 \, {\left (2 \, a^{2} b c^{5} + a^{3} c^{2}\right )} d^{2} x + {\left (a^{2} b c^{6} + a^{3} c^{3}\right )} d\right )}} + \frac {2 \, b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{3} d} - \frac {2 \, b \log \left (d x + c\right )}{a^{3} d} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=\frac {2 \, b \log \left ({\left | -b - \frac {a}{{\left (d x + c\right )}^{3}} \right |}\right )}{3 \, a^{3} d} + \frac {b^{2}}{3 \, a^{3} {\left (b + \frac {a}{{\left (d x + c\right )}^{3}}\right )} d} - \frac {1}{3 \, {\left (d x + c\right )}^{3} a^{2} d} \]
[In]
[Out]
Time = 6.16 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.64 \[ \int \frac {1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx=\frac {2\,b\,\ln \left (b\,c^3+3\,b\,c^2\,d\,x+3\,b\,c\,d^2\,x^2+b\,d^3\,x^3+a\right )}{3\,a^3\,d}-\frac {\frac {2\,b\,c^3+a}{3\,a^2\,d}+\frac {2\,b\,d^2\,x^3}{3\,a^2}+\frac {2\,b\,c^2\,x}{a^2}+\frac {2\,b\,c\,d\,x^2}{a^2}}{x\,\left (6\,b\,d\,c^5+3\,a\,d\,c^2\right )+x^3\,\left (20\,b\,c^3\,d^3+a\,d^3\right )+a\,c^3+b\,c^6+x^2\,\left (15\,b\,c^4\,d^2+3\,a\,c\,d^2\right )+b\,d^6\,x^6+15\,b\,c^2\,d^4\,x^4+6\,b\,c\,d^5\,x^5}-\frac {2\,b\,\ln \left (c+d\,x\right )}{a^3\,d} \]
[In]
[Out]