Integrand size = 29, antiderivative size = 172 \[ \int (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {(b c-a d)^3 q r x}{4 d^3}-\frac {(b c-a d)^2 q r (a+b x)^2}{8 b d^2}+\frac {(b c-a d) q r (a+b x)^3}{12 b d}-\frac {p r (a+b x)^4}{16 b}-\frac {q r (a+b x)^4}{16 b}-\frac {(b c-a d)^4 q r \log (c+d x)}{4 b d^4}+\frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2581, 32, 45} \[ \int (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {q r (b c-a d)^4 \log (c+d x)}{4 b d^4}+\frac {q r x (b c-a d)^3}{4 d^3}-\frac {q r (a+b x)^2 (b c-a d)^2}{8 b d^2}+\frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}+\frac {q r (a+b x)^3 (b c-a d)}{12 b d}-\frac {p r (a+b x)^4}{16 b}-\frac {q r (a+b x)^4}{16 b} \]
[In]
[Out]
Rule 32
Rule 45
Rule 2581
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {1}{4} (p r) \int (a+b x)^3 \, dx-\frac {(d q r) \int \frac {(a+b x)^4}{c+d x} \, dx}{4 b} \\ & = -\frac {p r (a+b x)^4}{16 b}+\frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {(d q r) \int \left (-\frac {b (b c-a d)^3}{d^4}+\frac {b (b c-a d)^2 (a+b x)}{d^3}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (a+b x)^3}{d}+\frac {(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{4 b} \\ & = \frac {(b c-a d)^3 q r x}{4 d^3}-\frac {(b c-a d)^2 q r (a+b x)^2}{8 b d^2}+\frac {(b c-a d) q r (a+b x)^3}{12 b d}-\frac {p r (a+b x)^4}{16 b}-\frac {q r (a+b x)^4}{16 b}-\frac {(b c-a d)^4 q r \log (c+d x)}{4 b d^4}+\frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.90 \[ \int (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {\frac {r \left (12 b d (b c-a d)^3 (p+4 q) x-18 b^2 (b c-a d)^2 (p+2 q) (c+d x)^2+4 b^3 (b c-a d) (3 p+4 q) (c+d x)^3-3 b^4 (p+q) (c+d x)^4-12 (b c-a d)^4 q \log (c+d x)\right )}{12 d^4}+(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(603\) vs. \(2(158)=316\).
Time = 301.79 (sec) , antiderivative size = 604, normalized size of antiderivative = 3.51
method | result | size |
parallelrisch | \(\frac {12 a^{4} d^{4} p r +30 a^{3} b c \,d^{3} p r -48 a^{2} b^{2} c^{2} d^{2} q r +42 a \,b^{3} c^{3} d q r +24 x^{2} a \,b^{3} c \,d^{3} q r +72 x \,a^{2} b^{2} c \,d^{3} q r -48 x a \,b^{3} c^{2} d^{2} q r +120 \ln \left (b x +a \right ) a^{3} b c \,d^{3} p r +168 \ln \left (d x +c \right ) a^{3} b c \,d^{3} q r -72 \ln \left (d x +c \right ) a^{2} b^{2} c^{2} d^{2} q r +48 \ln \left (d x +c \right ) a \,b^{3} c^{3} d q r -48 \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) a^{4} d^{4}+48 a^{4} d^{4} q r -12 b^{4} c^{4} q r +72 x^{2} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) a^{2} b^{2} d^{4}+48 x \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) a^{3} b \,d^{4}-120 \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) a^{3} b c \,d^{3}+60 \ln \left (b x +a \right ) a^{4} d^{4} p r +48 \ln \left (d x +c \right ) a^{4} d^{4} q r -12 \ln \left (d x +c \right ) b^{4} c^{4} q r -3 x^{4} b^{4} d^{4} p r -3 x^{4} b^{4} d^{4} q r +48 x^{3} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) a \,b^{3} d^{4}+12 x^{4} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) b^{4} d^{4}-12 x^{3} a \,b^{3} d^{4} p r -16 x^{3} a \,b^{3} d^{4} q r +4 x^{3} b^{4} c \,d^{3} q r -18 x^{2} a^{2} b^{2} d^{4} p r -36 x^{2} a^{2} b^{2} d^{4} q r -6 x^{2} b^{4} c^{2} d^{2} q r -12 x \,a^{3} b \,d^{4} p r -48 x \,a^{3} b \,d^{4} q r +12 x \,b^{4} c^{3} d q r +12 a^{3} b c \,d^{3} q r}{48 b \,d^{4}}\) | \(604\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (158) = 316\).
Time = 0.30 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.73 \[ \int (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {3 \, {\left (b^{4} d^{4} p + b^{4} d^{4} q\right )} r x^{4} + 4 \, {\left (3 \, a b^{3} d^{4} p - {\left (b^{4} c d^{3} - 4 \, a b^{3} d^{4}\right )} q\right )} r x^{3} + 6 \, {\left (3 \, a^{2} b^{2} d^{4} p + {\left (b^{4} c^{2} d^{2} - 4 \, a b^{3} c d^{3} + 6 \, a^{2} b^{2} d^{4}\right )} q\right )} r x^{2} + 12 \, {\left (a^{3} b d^{4} p - {\left (b^{4} c^{3} d - 4 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} - 4 \, a^{3} b d^{4}\right )} q\right )} r x - 12 \, {\left (b^{4} d^{4} p r x^{4} + 4 \, a b^{3} d^{4} p r x^{3} + 6 \, a^{2} b^{2} d^{4} p r x^{2} + 4 \, a^{3} b d^{4} p r x + a^{4} d^{4} p r\right )} \log \left (b x + a\right ) - 12 \, {\left (b^{4} d^{4} q r x^{4} + 4 \, a b^{3} d^{4} q r x^{3} + 6 \, a^{2} b^{2} d^{4} q r x^{2} + 4 \, a^{3} b d^{4} q r x - {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3}\right )} q r\right )} \log \left (d x + c\right ) - 12 \, {\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x\right )} \log \left (e\right ) - 12 \, {\left (b^{4} d^{4} r x^{4} + 4 \, a b^{3} d^{4} r x^{3} + 6 \, a^{2} b^{2} d^{4} r x^{2} + 4 \, a^{3} b d^{4} r x\right )} \log \left (f\right )}{48 \, b d^{4}} \]
[In]
[Out]
Timed out. \[ \int (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.66 \[ \int (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {1}{4} \, {\left (b^{3} x^{4} + 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} + 4 \, a^{3} x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) + \frac {{\left (\frac {12 \, a^{4} f p \log \left (b x + a\right )}{b} - \frac {3 \, b^{3} d^{3} f {\left (p + q\right )} x^{4} + 4 \, {\left (a b^{2} d^{3} f {\left (3 \, p + 4 \, q\right )} - b^{3} c d^{2} f q\right )} x^{3} + 6 \, {\left (3 \, a^{2} b d^{3} f {\left (p + 2 \, q\right )} + b^{3} c^{2} d f q - 4 \, a b^{2} c d^{2} f q\right )} x^{2} + 12 \, {\left (a^{3} d^{3} f {\left (p + 4 \, q\right )} - b^{3} c^{3} f q + 4 \, a b^{2} c^{2} d f q - 6 \, a^{2} b c d^{2} f q\right )} x}{d^{3}} - \frac {12 \, {\left (b^{3} c^{4} f q - 4 \, a b^{2} c^{3} d f q + 6 \, a^{2} b c^{2} d^{2} f q - 4 \, a^{3} c d^{3} f q\right )} \log \left (d x + c\right )}{d^{4}}\right )} r}{48 \, f} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (158) = 316\).
Time = 3.74 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.41 \[ \int (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {a^{4} p r \log \left (b x + a\right )}{4 \, b} - \frac {1}{16} \, {\left (b^{3} p r + b^{3} q r - 4 \, b^{3} r \log \left (f\right ) - 4 \, b^{3} \log \left (e\right )\right )} x^{4} - \frac {{\left (3 \, a b^{2} d p r - b^{3} c q r + 4 \, a b^{2} d q r - 12 \, a b^{2} d r \log \left (f\right ) - 12 \, a b^{2} d \log \left (e\right )\right )} x^{3}}{12 \, d} + \frac {1}{4} \, {\left (b^{3} p r x^{4} + 4 \, a b^{2} p r x^{3} + 6 \, a^{2} b p r x^{2} + 4 \, a^{3} p r x\right )} \log \left (b x + a\right ) + \frac {1}{4} \, {\left (b^{3} q r x^{4} + 4 \, a b^{2} q r x^{3} + 6 \, a^{2} b q r x^{2} + 4 \, a^{3} q r x\right )} \log \left (d x + c\right ) - \frac {{\left (3 \, a^{2} b d^{2} p r + b^{3} c^{2} q r - 4 \, a b^{2} c d q r + 6 \, a^{2} b d^{2} q r - 12 \, a^{2} b d^{2} r \log \left (f\right ) - 12 \, a^{2} b d^{2} \log \left (e\right )\right )} x^{2}}{8 \, d^{2}} - \frac {{\left (a^{3} d^{3} p r - b^{3} c^{3} q r + 4 \, a b^{2} c^{2} d q r - 6 \, a^{2} b c d^{2} q r + 4 \, a^{3} d^{3} q r - 4 \, a^{3} d^{3} r \log \left (f\right ) - 4 \, a^{3} d^{3} \log \left (e\right )\right )} x}{4 \, d^{3}} - \frac {{\left (b^{3} c^{4} q r - 4 \, a b^{2} c^{3} d q r + 6 \, a^{2} b c^{2} d^{2} q r - 4 \, a^{3} c d^{3} q r\right )} \log \left (-d x - c\right )}{4 \, d^{4}} \]
[In]
[Out]
Time = 1.57 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.91 \[ \int (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=x^2\,\left (\frac {\left (\frac {b^2\,r\,\left (4\,a\,d\,p+b\,c\,p+5\,a\,d\,q\right )}{4\,d}-\frac {b^2\,r\,\left (p+q\right )\,\left (4\,a\,d+4\,b\,c\right )}{16\,d}\right )\,\left (4\,a\,d+4\,b\,c\right )}{8\,b\,d}-\frac {a\,b\,r\,\left (3\,a\,d\,p+2\,b\,c\,p+5\,a\,d\,q\right )}{4\,d}+\frac {a\,b^2\,c\,r\,\left (p+q\right )}{8\,d}\right )-x^3\,\left (\frac {b^2\,r\,\left (4\,a\,d\,p+b\,c\,p+5\,a\,d\,q\right )}{12\,d}-\frac {b^2\,r\,\left (p+q\right )\,\left (4\,a\,d+4\,b\,c\right )}{48\,d}\right )+\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (a^3\,x+\frac {3\,a^2\,b\,x^2}{2}+a\,b^2\,x^3+\frac {b^3\,x^4}{4}\right )-x\,\left (\frac {\left (4\,a\,d+4\,b\,c\right )\,\left (\frac {\left (\frac {b^2\,r\,\left (4\,a\,d\,p+b\,c\,p+5\,a\,d\,q\right )}{4\,d}-\frac {b^2\,r\,\left (p+q\right )\,\left (4\,a\,d+4\,b\,c\right )}{16\,d}\right )\,\left (4\,a\,d+4\,b\,c\right )}{4\,b\,d}-\frac {a\,b\,r\,\left (3\,a\,d\,p+2\,b\,c\,p+5\,a\,d\,q\right )}{2\,d}+\frac {a\,b^2\,c\,r\,\left (p+q\right )}{4\,d}\right )}{4\,b\,d}+\frac {a^2\,r\,\left (2\,a\,d\,p+3\,b\,c\,p+5\,a\,d\,q\right )}{2\,d}-\frac {a\,c\,\left (\frac {b^2\,r\,\left (4\,a\,d\,p+b\,c\,p+5\,a\,d\,q\right )}{4\,d}-\frac {b^2\,r\,\left (p+q\right )\,\left (4\,a\,d+4\,b\,c\right )}{16\,d}\right )}{b\,d}\right )-\frac {\ln \left (c+d\,x\right )\,\left (-4\,q\,r\,a^3\,c\,d^3+6\,q\,r\,a^2\,b\,c^2\,d^2-4\,q\,r\,a\,b^2\,c^3\,d+q\,r\,b^3\,c^4\right )}{4\,d^4}-\frac {b^3\,r\,x^4\,\left (p+q\right )}{16}+\frac {a^4\,p\,r\,\ln \left (a+b\,x\right )}{4\,b} \]
[In]
[Out]