Integrand size = 21, antiderivative size = 30 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^p \, dx=\frac {\left (a+b (c+d x)^4\right )^{1+p}}{4 b d (1+p)} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {379, 267} \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^p \, dx=\frac {\left (a+b (c+d x)^4\right )^{p+1}}{4 b d (p+1)} \]
[In]
[Out]
Rule 267
Rule 379
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^3 \left (a+b x^4\right )^p \, dx,x,c+d x\right )}{d} \\ & = \frac {\left (a+b (c+d x)^4\right )^{1+p}}{4 b d (1+p)} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^p \, dx=\frac {\left (a+b (c+d x)^4\right )^{1+p}}{4 b d (1+p)} \]
[In]
[Out]
Time = 4.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {\left (a +b \left (d x +c \right )^{4}\right )^{1+p}}{4 b d \left (1+p \right )}\) | \(29\) |
default | \(\frac {\left (a +b \left (d x +c \right )^{4}\right )^{1+p}}{4 b d \left (1+p \right )}\) | \(29\) |
gosper | \(\frac {\left (b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a \right )^{1+p}}{4 b d \left (1+p \right )}\) | \(63\) |
risch | \(\frac {\left (b \,d^{4} x^{4}+4 b c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{2}+4 b \,c^{3} d x +b \,c^{4}+a \right ) \left (a +b \left (d x +c \right )^{4}\right )^{p}}{4 b d \left (1+p \right )}\) | \(72\) |
parallelrisch | \(\frac {d^{7} \left (a +b \left (d x +c \right )^{4}\right )^{p} x^{4} b +4 c \,d^{6} \left (a +b \left (d x +c \right )^{4}\right )^{p} x^{3} b +6 c^{2} d^{5} \left (a +b \left (d x +c \right )^{4}\right )^{p} x^{2} b +4 c^{3} \left (a +b \left (d x +c \right )^{4}\right )^{p} x b \,d^{4}+\left (a +b \left (d x +c \right )^{4}\right )^{p} b \,c^{4} d^{3}+\left (a +b \left (d x +c \right )^{4}\right )^{p} a \,d^{3}}{4 b \,d^{4} \left (1+p \right )}\) | \(146\) |
norman | \(\frac {c^{3} x \,{\mathrm e}^{p \ln \left (a +b \left (d x +c \right )^{4}\right )}}{1+p}+\frac {c \,d^{2} x^{3} {\mathrm e}^{p \ln \left (a +b \left (d x +c \right )^{4}\right )}}{1+p}+\frac {d^{3} x^{4} {\mathrm e}^{p \ln \left (a +b \left (d x +c \right )^{4}\right )}}{4+4 p}+\frac {3 c^{2} d \,x^{2} {\mathrm e}^{p \ln \left (a +b \left (d x +c \right )^{4}\right )}}{2 \left (1+p \right )}+\frac {\left (b \,c^{4}+a \right ) {\mathrm e}^{p \ln \left (a +b \left (d x +c \right )^{4}\right )}}{4 b d \left (1+p \right )}\) | \(147\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (28) = 56\).
Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.47 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^p \, dx=\frac {{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )} {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )}^{p}}{4 \, {\left (b d p + b d\right )}} \]
[In]
[Out]
Timed out. \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^p \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^p \, dx=\frac {{\left ({\left (d x + c\right )}^{4} b + a\right )}^{p + 1}}{4 \, b d {\left (p + 1\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (28) = 56\).
Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^p \, dx=\frac {{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )}^{p + 1}}{4 \, b d {\left (p + 1\right )}} \]
[In]
[Out]
Time = 6.01 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.93 \[ \int (c+d x)^3 \left (a+b (c+d x)^4\right )^p \, dx={\left (a+b\,{\left (c+d\,x\right )}^4\right )}^p\,\left (\frac {d^3\,x^4}{4\,\left (p+1\right )}+\frac {c^3\,x}{p+1}+\frac {b\,c^4+a}{4\,b\,d\,\left (p+1\right )}+\frac {3\,c^2\,d\,x^2}{2\,\left (p+1\right )}+\frac {c\,d^2\,x^3}{p+1}\right ) \]
[In]
[Out]