Integrand size = 15, antiderivative size = 111 \[ \int (a+b x)^3 (c+d x)^n \, dx=-\frac {(b c-a d)^3 (c+d x)^{1+n}}{d^4 (1+n)}+\frac {3 b (b c-a d)^2 (c+d x)^{2+n}}{d^4 (2+n)}-\frac {3 b^2 (b c-a d) (c+d x)^{3+n}}{d^4 (3+n)}+\frac {b^3 (c+d x)^{4+n}}{d^4 (4+n)} \]
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Time = 0.04 (sec) , antiderivative size = 111, 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} \[ \int (a+b x)^3 (c+d x)^n \, dx=-\frac {3 b^2 (b c-a d) (c+d x)^{n+3}}{d^4 (n+3)}-\frac {(b c-a d)^3 (c+d x)^{n+1}}{d^4 (n+1)}+\frac {3 b (b c-a d)^2 (c+d x)^{n+2}}{d^4 (n+2)}+\frac {b^3 (c+d x)^{n+4}}{d^4 (n+4)} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^3 (c+d x)^n}{d^3}+\frac {3 b (b c-a d)^2 (c+d x)^{1+n}}{d^3}-\frac {3 b^2 (b c-a d) (c+d x)^{2+n}}{d^3}+\frac {b^3 (c+d x)^{3+n}}{d^3}\right ) \, dx \\ & = -\frac {(b c-a d)^3 (c+d x)^{1+n}}{d^4 (1+n)}+\frac {3 b (b c-a d)^2 (c+d x)^{2+n}}{d^4 (2+n)}-\frac {3 b^2 (b c-a d) (c+d x)^{3+n}}{d^4 (3+n)}+\frac {b^3 (c+d x)^{4+n}}{d^4 (4+n)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.86 \[ \int (a+b x)^3 (c+d x)^n \, dx=\frac {(c+d x)^{1+n} \left (-\frac {(b c-a d)^3}{1+n}+\frac {3 b (b c-a d)^2 (c+d x)}{2+n}-\frac {3 b^2 (b c-a d) (c+d x)^2}{3+n}+\frac {b^3 (c+d x)^3}{4+n}\right )}{d^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(385\) vs. \(2(111)=222\).
Time = 0.41 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.48
method | result | size |
gosper | \(\frac {\left (d x +c \right )^{1+n} \left (b^{3} d^{3} n^{3} x^{3}+3 a \,b^{2} d^{3} n^{3} x^{2}+6 b^{3} d^{3} n^{2} x^{3}+3 a^{2} b \,d^{3} n^{3} x +21 a \,b^{2} d^{3} n^{2} x^{2}-3 b^{3} c \,d^{2} n^{2} x^{2}+11 b^{3} d^{3} n \,x^{3}+a^{3} d^{3} n^{3}+24 a^{2} b \,d^{3} n^{2} x -6 a \,b^{2} c \,d^{2} n^{2} x +42 a \,b^{2} d^{3} n \,x^{2}-9 b^{3} c \,d^{2} n \,x^{2}+6 d^{3} x^{3} b^{3}+9 a^{3} d^{3} n^{2}-3 a^{2} b c \,d^{2} n^{2}+57 a^{2} b \,d^{3} n x -30 a \,b^{2} c \,d^{2} n x +24 x^{2} a \,b^{2} d^{3}+6 b^{3} c^{2} d n x -6 x^{2} b^{3} c \,d^{2}+26 a^{3} d^{3} n -21 a^{2} b c \,d^{2} n +36 x \,a^{2} b \,d^{3}+6 a \,b^{2} c^{2} d n -24 x a \,b^{2} c \,d^{2}+6 x \,b^{3} c^{2} d +24 a^{3} d^{3}-36 a^{2} b c \,d^{2}+24 a \,b^{2} c^{2} d -6 b^{3} c^{3}\right )}{d^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\) | \(386\) |
norman | \(\frac {b^{3} x^{4} {\mathrm e}^{n \ln \left (d x +c \right )}}{4+n}+\frac {c \left (a^{3} d^{3} n^{3}+9 a^{3} d^{3} n^{2}-3 a^{2} b c \,d^{2} n^{2}+26 a^{3} d^{3} n -21 a^{2} b c \,d^{2} n +6 a \,b^{2} c^{2} d n +24 a^{3} d^{3}-36 a^{2} b c \,d^{2}+24 a \,b^{2} c^{2} d -6 b^{3} c^{3}\right ) {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}+\frac {\left (a^{3} d^{3} n^{3}+3 a^{2} b c \,d^{2} n^{3}+9 a^{3} d^{3} n^{2}+21 a^{2} b c \,d^{2} n^{2}-6 a \,b^{2} c^{2} d \,n^{2}+26 a^{3} d^{3} n +36 a^{2} b c \,d^{2} n -24 a \,b^{2} c^{2} d n +6 b^{3} c^{3} n +24 a^{3} d^{3}\right ) x \,{\mathrm e}^{n \ln \left (d x +c \right )}}{d^{3} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}+\frac {\left (3 a d n +b c n +12 a d \right ) b^{2} x^{3} {\mathrm e}^{n \ln \left (d x +c \right )}}{d \left (n^{2}+7 n +12\right )}+\frac {3 \left (a^{2} d^{2} n^{2}+a b c d \,n^{2}+7 a^{2} d^{2} n +4 a b c d n -b^{2} c^{2} n +12 a^{2} d^{2}\right ) b \,x^{2} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{2} \left (n^{3}+9 n^{2}+26 n +24\right )}\) | \(430\) |
risch | \(\frac {\left (b^{3} d^{4} n^{3} x^{4}+3 a \,b^{2} d^{4} n^{3} x^{3}+b^{3} c \,d^{3} n^{3} x^{3}+6 b^{3} d^{4} n^{2} x^{4}+3 a^{2} b \,d^{4} n^{3} x^{2}+3 a \,b^{2} c \,d^{3} n^{3} x^{2}+21 a \,b^{2} d^{4} n^{2} x^{3}+3 b^{3} c \,d^{3} n^{2} x^{3}+11 b^{3} d^{4} n \,x^{4}+a^{3} d^{4} n^{3} x +3 a^{2} b c \,d^{3} n^{3} x +24 a^{2} b \,d^{4} n^{2} x^{2}+15 a \,b^{2} c \,d^{3} n^{2} x^{2}+42 a \,b^{2} d^{4} n \,x^{3}-3 b^{3} c^{2} d^{2} n^{2} x^{2}+2 b^{3} c \,d^{3} n \,x^{3}+6 b^{3} x^{4} d^{4}+a^{3} c \,d^{3} n^{3}+9 a^{3} d^{4} n^{2} x +21 a^{2} b c \,d^{3} n^{2} x +57 a^{2} b \,d^{4} n \,x^{2}-6 a \,b^{2} c^{2} d^{2} n^{2} x +12 a \,b^{2} c \,d^{3} n \,x^{2}+24 a \,b^{2} d^{4} x^{3}-3 b^{3} c^{2} d^{2} n \,x^{2}+9 a^{3} c \,d^{3} n^{2}+26 a^{3} d^{4} n x -3 a^{2} b \,c^{2} d^{2} n^{2}+36 a^{2} b c \,d^{3} n x +36 a^{2} b \,d^{4} x^{2}-24 a \,b^{2} c^{2} d^{2} n x +6 b^{3} c^{3} d n x +26 a^{3} c \,d^{3} n +24 a^{3} d^{4} x -21 a^{2} b \,c^{2} d^{2} n +6 a \,b^{2} c^{3} d n +24 a^{3} c \,d^{3}-36 a^{2} b \,c^{2} d^{2}+24 a \,b^{2} c^{3} d -6 b^{3} c^{4}\right ) \left (d x +c \right )^{n}}{\left (3+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) d^{4}}\) | \(547\) |
parallelrisch | \(\frac {6 x^{4} \left (d x +c \right )^{n} b^{3} c \,d^{4}+\left (d x +c \right )^{n} a^{3} c^{2} d^{3} n^{3}+9 \left (d x +c \right )^{n} a^{3} c^{2} d^{3} n^{2}+24 x \left (d x +c \right )^{n} a^{3} c \,d^{4}+26 \left (d x +c \right )^{n} a^{3} c^{2} d^{3} n +6 x^{4} \left (d x +c \right )^{n} b^{3} c \,d^{4} n^{2}+x^{3} \left (d x +c \right )^{n} b^{3} c^{2} d^{3} n^{3}+11 x^{4} \left (d x +c \right )^{n} b^{3} c \,d^{4} n +3 x^{3} \left (d x +c \right )^{n} b^{3} c^{2} d^{3} n^{2}+2 x^{3} \left (d x +c \right )^{n} b^{3} c^{2} d^{3} n -3 x^{2} \left (d x +c \right )^{n} b^{3} c^{3} d^{2} n^{2}+26 x \left (d x +c \right )^{n} a^{3} c \,d^{4} n +6 x \left (d x +c \right )^{n} b^{3} c^{4} d n -3 \left (d x +c \right )^{n} a^{2} b \,c^{3} d^{2} n^{2}-21 \left (d x +c \right )^{n} a^{2} b \,c^{3} d^{2} n +6 \left (d x +c \right )^{n} a \,b^{2} c^{4} d n -36 \left (d x +c \right )^{n} a^{2} b \,c^{3} d^{2}+24 \left (d x +c \right )^{n} a \,b^{2} c^{4} d -3 x^{2} \left (d x +c \right )^{n} b^{3} c^{3} d^{2} n +9 x \left (d x +c \right )^{n} a^{3} c \,d^{4} n^{2}+36 x^{2} \left (d x +c \right )^{n} a^{2} b c \,d^{4}+x^{4} \left (d x +c \right )^{n} b^{3} c \,d^{4} n^{3}+x \left (d x +c \right )^{n} a^{3} c \,d^{4} n^{3}+24 x^{3} \left (d x +c \right )^{n} a \,b^{2} c \,d^{4}+3 x^{3} \left (d x +c \right )^{n} a \,b^{2} c \,d^{4} n^{3}+21 x^{3} \left (d x +c \right )^{n} a \,b^{2} c \,d^{4} n^{2}+3 x^{2} \left (d x +c \right )^{n} a^{2} b c \,d^{4} n^{3}+3 x^{2} \left (d x +c \right )^{n} a \,b^{2} c^{2} d^{3} n^{3}+42 x^{3} \left (d x +c \right )^{n} a \,b^{2} c \,d^{4} n +24 x^{2} \left (d x +c \right )^{n} a^{2} b c \,d^{4} n^{2}+15 x^{2} \left (d x +c \right )^{n} a \,b^{2} c^{2} d^{3} n^{2}+3 x \left (d x +c \right )^{n} a^{2} b \,c^{2} d^{3} n^{3}+57 x^{2} \left (d x +c \right )^{n} a^{2} b c \,d^{4} n +12 x^{2} \left (d x +c \right )^{n} a \,b^{2} c^{2} d^{3} n +21 x \left (d x +c \right )^{n} a^{2} b \,c^{2} d^{3} n^{2}-6 x \left (d x +c \right )^{n} a \,b^{2} c^{3} d^{2} n^{2}+36 x \left (d x +c \right )^{n} a^{2} b \,c^{2} d^{3} n -24 x \left (d x +c \right )^{n} a \,b^{2} c^{3} d^{2} n +24 \left (d x +c \right )^{n} a^{3} c^{2} d^{3}-6 \left (d x +c \right )^{n} b^{3} c^{5}}{\left (n^{3}+9 n^{2}+26 n +24\right ) c \left (1+n \right ) d^{4}}\) | \(865\) |
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Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (111) = 222\).
Time = 0.24 (sec) , antiderivative size = 496, normalized size of antiderivative = 4.47 \[ \int (a+b x)^3 (c+d x)^n \, dx=\frac {{\left (a^{3} c d^{3} n^{3} - 6 \, b^{3} c^{4} + 24 \, a b^{2} c^{3} d - 36 \, a^{2} b c^{2} d^{2} + 24 \, a^{3} c d^{3} + {\left (b^{3} d^{4} n^{3} + 6 \, b^{3} d^{4} n^{2} + 11 \, b^{3} d^{4} n + 6 \, b^{3} d^{4}\right )} x^{4} + {\left (24 \, a b^{2} d^{4} + {\left (b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} n^{3} + 3 \, {\left (b^{3} c d^{3} + 7 \, a b^{2} d^{4}\right )} n^{2} + 2 \, {\left (b^{3} c d^{3} + 21 \, a b^{2} d^{4}\right )} n\right )} x^{3} - 3 \, {\left (a^{2} b c^{2} d^{2} - 3 \, a^{3} c d^{3}\right )} n^{2} + 3 \, {\left (12 \, a^{2} b d^{4} + {\left (a b^{2} c d^{3} + a^{2} b d^{4}\right )} n^{3} - {\left (b^{3} c^{2} d^{2} - 5 \, a b^{2} c d^{3} - 8 \, a^{2} b d^{4}\right )} n^{2} - {\left (b^{3} c^{2} d^{2} - 4 \, a b^{2} c d^{3} - 19 \, a^{2} b d^{4}\right )} n\right )} x^{2} + {\left (6 \, a b^{2} c^{3} d - 21 \, a^{2} b c^{2} d^{2} + 26 \, a^{3} c d^{3}\right )} n + {\left (24 \, a^{3} d^{4} + {\left (3 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} n^{3} - 3 \, {\left (2 \, a b^{2} c^{2} d^{2} - 7 \, a^{2} b c d^{3} - 3 \, a^{3} d^{4}\right )} n^{2} + 2 \, {\left (3 \, b^{3} c^{3} d - 12 \, a b^{2} c^{2} d^{2} + 18 \, a^{2} b c d^{3} + 13 \, a^{3} d^{4}\right )} n\right )} x\right )} {\left (d x + c\right )}^{n}}{d^{4} n^{4} + 10 \, d^{4} n^{3} + 35 \, d^{4} n^{2} + 50 \, d^{4} n + 24 \, d^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 4058 vs. \(2 (95) = 190\).
Time = 1.02 (sec) , antiderivative size = 4058, normalized size of antiderivative = 36.56 \[ \int (a+b x)^3 (c+d x)^n \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (111) = 222\).
Time = 0.24 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.22 \[ \int (a+b x)^3 (c+d x)^n \, dx=\frac {3 \, {\left (d^{2} {\left (n + 1\right )} x^{2} + c d n x - c^{2}\right )} {\left (d x + c\right )}^{n} a^{2} b}{{\left (n^{2} + 3 \, n + 2\right )} d^{2}} + \frac {{\left (d x + c\right )}^{n + 1} a^{3}}{d {\left (n + 1\right )}} + \frac {3 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} d^{3} x^{3} + {\left (n^{2} + n\right )} c d^{2} x^{2} - 2 \, c^{2} d n x + 2 \, c^{3}\right )} {\left (d x + c\right )}^{n} a b^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{3}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c d^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} c^{2} d^{2} x^{2} + 6 \, c^{3} d n x - 6 \, c^{4}\right )} {\left (d x + c\right )}^{n} b^{3}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 833 vs. \(2 (111) = 222\).
Time = 0.28 (sec) , antiderivative size = 833, normalized size of antiderivative = 7.50 \[ \int (a+b x)^3 (c+d x)^n \, dx=\frac {{\left (d x + c\right )}^{n} b^{3} d^{4} n^{3} x^{4} + {\left (d x + c\right )}^{n} b^{3} c d^{3} n^{3} x^{3} + 3 \, {\left (d x + c\right )}^{n} a b^{2} d^{4} n^{3} x^{3} + 6 \, {\left (d x + c\right )}^{n} b^{3} d^{4} n^{2} x^{4} + 3 \, {\left (d x + c\right )}^{n} a b^{2} c d^{3} n^{3} x^{2} + 3 \, {\left (d x + c\right )}^{n} a^{2} b d^{4} n^{3} x^{2} + 3 \, {\left (d x + c\right )}^{n} b^{3} c d^{3} n^{2} x^{3} + 21 \, {\left (d x + c\right )}^{n} a b^{2} d^{4} n^{2} x^{3} + 11 \, {\left (d x + c\right )}^{n} b^{3} d^{4} n x^{4} + 3 \, {\left (d x + c\right )}^{n} a^{2} b c d^{3} n^{3} x + {\left (d x + c\right )}^{n} a^{3} d^{4} n^{3} x - 3 \, {\left (d x + c\right )}^{n} b^{3} c^{2} d^{2} n^{2} x^{2} + 15 \, {\left (d x + c\right )}^{n} a b^{2} c d^{3} n^{2} x^{2} + 24 \, {\left (d x + c\right )}^{n} a^{2} b d^{4} n^{2} x^{2} + 2 \, {\left (d x + c\right )}^{n} b^{3} c d^{3} n x^{3} + 42 \, {\left (d x + c\right )}^{n} a b^{2} d^{4} n x^{3} + 6 \, {\left (d x + c\right )}^{n} b^{3} d^{4} x^{4} + {\left (d x + c\right )}^{n} a^{3} c d^{3} n^{3} - 6 \, {\left (d x + c\right )}^{n} a b^{2} c^{2} d^{2} n^{2} x + 21 \, {\left (d x + c\right )}^{n} a^{2} b c d^{3} n^{2} x + 9 \, {\left (d x + c\right )}^{n} a^{3} d^{4} n^{2} x - 3 \, {\left (d x + c\right )}^{n} b^{3} c^{2} d^{2} n x^{2} + 12 \, {\left (d x + c\right )}^{n} a b^{2} c d^{3} n x^{2} + 57 \, {\left (d x + c\right )}^{n} a^{2} b d^{4} n x^{2} + 24 \, {\left (d x + c\right )}^{n} a b^{2} d^{4} x^{3} - 3 \, {\left (d x + c\right )}^{n} a^{2} b c^{2} d^{2} n^{2} + 9 \, {\left (d x + c\right )}^{n} a^{3} c d^{3} n^{2} + 6 \, {\left (d x + c\right )}^{n} b^{3} c^{3} d n x - 24 \, {\left (d x + c\right )}^{n} a b^{2} c^{2} d^{2} n x + 36 \, {\left (d x + c\right )}^{n} a^{2} b c d^{3} n x + 26 \, {\left (d x + c\right )}^{n} a^{3} d^{4} n x + 36 \, {\left (d x + c\right )}^{n} a^{2} b d^{4} x^{2} + 6 \, {\left (d x + c\right )}^{n} a b^{2} c^{3} d n - 21 \, {\left (d x + c\right )}^{n} a^{2} b c^{2} d^{2} n + 26 \, {\left (d x + c\right )}^{n} a^{3} c d^{3} n + 24 \, {\left (d x + c\right )}^{n} a^{3} d^{4} x - 6 \, {\left (d x + c\right )}^{n} b^{3} c^{4} + 24 \, {\left (d x + c\right )}^{n} a b^{2} c^{3} d - 36 \, {\left (d x + c\right )}^{n} a^{2} b c^{2} d^{2} + 24 \, {\left (d x + c\right )}^{n} a^{3} c d^{3}}{d^{4} n^{4} + 10 \, d^{4} n^{3} + 35 \, d^{4} n^{2} + 50 \, d^{4} n + 24 \, d^{4}} \]
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Time = 1.00 (sec) , antiderivative size = 478, normalized size of antiderivative = 4.31 \[ \int (a+b x)^3 (c+d x)^n \, dx=\frac {x\,{\left (c+d\,x\right )}^n\,\left (a^3\,d^4\,n^3+9\,a^3\,d^4\,n^2+26\,a^3\,d^4\,n+24\,a^3\,d^4+3\,a^2\,b\,c\,d^3\,n^3+21\,a^2\,b\,c\,d^3\,n^2+36\,a^2\,b\,c\,d^3\,n-6\,a\,b^2\,c^2\,d^2\,n^2-24\,a\,b^2\,c^2\,d^2\,n+6\,b^3\,c^3\,d\,n\right )}{d^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {b^3\,x^4\,{\left (c+d\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}+\frac {c\,{\left (c+d\,x\right )}^n\,\left (a^3\,d^3\,n^3+9\,a^3\,d^3\,n^2+26\,a^3\,d^3\,n+24\,a^3\,d^3-3\,a^2\,b\,c\,d^2\,n^2-21\,a^2\,b\,c\,d^2\,n-36\,a^2\,b\,c\,d^2+6\,a\,b^2\,c^2\,d\,n+24\,a\,b^2\,c^2\,d-6\,b^3\,c^3\right )}{d^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {3\,b\,x^2\,\left (n+1\right )\,{\left (c+d\,x\right )}^n\,\left (a^2\,d^2\,n^2+7\,a^2\,d^2\,n+12\,a^2\,d^2+a\,b\,c\,d\,n^2+4\,a\,b\,c\,d\,n-b^2\,c^2\,n\right )}{d^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {b^2\,x^3\,{\left (c+d\,x\right )}^n\,\left (12\,a\,d+3\,a\,d\,n+b\,c\,n\right )\,\left (n^2+3\,n+2\right )}{d\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )} \]
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