Integrand size = 15, antiderivative size = 110 \[ \int (a+b x)^n (c+d x)^3 \, dx=\frac {(b c-a d)^3 (a+b x)^{1+n}}{b^4 (1+n)}+\frac {3 d (b c-a d)^2 (a+b x)^{2+n}}{b^4 (2+n)}+\frac {3 d^2 (b c-a d) (a+b x)^{3+n}}{b^4 (3+n)}+\frac {d^3 (a+b x)^{4+n}}{b^4 (4+n)} \]
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Time = 0.03 (sec) , antiderivative size = 110, 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} \[ \int (a+b x)^n (c+d x)^3 \, dx=\frac {3 d^2 (b c-a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac {(b c-a d)^3 (a+b x)^{n+1}}{b^4 (n+1)}+\frac {3 d (b c-a d)^2 (a+b x)^{n+2}}{b^4 (n+2)}+\frac {d^3 (a+b x)^{n+4}}{b^4 (n+4)} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d)^3 (a+b x)^n}{b^3}+\frac {3 d (b c-a d)^2 (a+b x)^{1+n}}{b^3}+\frac {3 d^2 (b c-a d) (a+b x)^{2+n}}{b^3}+\frac {d^3 (a+b x)^{3+n}}{b^3}\right ) \, dx \\ & = \frac {(b c-a d)^3 (a+b x)^{1+n}}{b^4 (1+n)}+\frac {3 d (b c-a d)^2 (a+b x)^{2+n}}{b^4 (2+n)}+\frac {3 d^2 (b c-a d) (a+b x)^{3+n}}{b^4 (3+n)}+\frac {d^3 (a+b x)^{4+n}}{b^4 (4+n)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85 \[ \int (a+b x)^n (c+d x)^3 \, dx=\frac {(a+b x)^{1+n} \left (\frac {(b c-a d)^3}{1+n}+\frac {3 d (b c-a d)^2 (a+b x)}{2+n}+\frac {3 d^2 (b c-a d) (a+b x)^2}{3+n}+\frac {d^3 (a+b x)^3}{4+n}\right )}{b^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(388\) vs. \(2(110)=220\).
Time = 0.67 (sec) , antiderivative size = 389, normalized size of antiderivative = 3.54
method | result | size |
gosper | \(-\frac {\left (b x +a \right )^{1+n} \left (-b^{3} d^{3} n^{3} x^{3}-3 b^{3} c \,d^{2} n^{3} x^{2}-6 b^{3} d^{3} n^{2} x^{3}+3 a \,b^{2} d^{3} n^{2} x^{2}-3 b^{3} c^{2} d \,n^{3} x -21 b^{3} c \,d^{2} n^{2} x^{2}-11 b^{3} d^{3} n \,x^{3}+6 a \,b^{2} c \,d^{2} n^{2} x +9 a \,b^{2} d^{3} n \,x^{2}-b^{3} c^{3} n^{3}-24 b^{3} c^{2} d \,n^{2} x -42 b^{3} c \,d^{2} n \,x^{2}-6 d^{3} x^{3} b^{3}-6 a^{2} b \,d^{3} n x +3 a \,b^{2} c^{2} d \,n^{2}+30 a \,b^{2} c \,d^{2} n x +6 x^{2} a \,b^{2} d^{3}-9 b^{3} c^{3} n^{2}-57 b^{3} c^{2} d n x -24 x^{2} b^{3} c \,d^{2}-6 a^{2} b c \,d^{2} n -6 x \,a^{2} b \,d^{3}+21 a \,b^{2} c^{2} d n +24 x a \,b^{2} c \,d^{2}-26 b^{3} c^{3} n -36 x \,b^{3} c^{2} d +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right )}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\) | \(389\) |
norman | \(\frac {d^{3} x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{4+n}+\frac {\left (3 a \,b^{2} c^{2} d \,n^{3}+b^{3} c^{3} n^{3}-6 a^{2} b c \,d^{2} n^{2}+21 a \,b^{2} c^{2} d \,n^{2}+9 b^{3} c^{3} n^{2}+6 a^{3} d^{3} n -24 a^{2} b c \,d^{2} n +36 a \,b^{2} c^{2} d n +26 b^{3} c^{3} n +24 b^{3} c^{3}\right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}+\frac {\left (a d n +3 b c n +12 b c \right ) d^{2} x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+7 n +12\right )}-\frac {a \left (-b^{3} c^{3} n^{3}+3 a \,b^{2} c^{2} d \,n^{2}-9 b^{3} c^{3} n^{2}-6 a^{2} b c \,d^{2} n +21 a \,b^{2} c^{2} d n -26 b^{3} c^{3} n +6 a^{3} d^{3}-24 a^{2} b c \,d^{2}+36 a \,b^{2} c^{2} d -24 b^{3} c^{3}\right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}-\frac {3 \left (-a b c d \,n^{2}-b^{2} c^{2} n^{2}+a^{2} d^{2} n -4 a b c d n -7 b^{2} c^{2} n -12 b^{2} c^{2}\right ) d \,x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+9 n^{2}+26 n +24\right )}\) | \(433\) |
risch | \(-\frac {\left (-b^{4} d^{3} n^{3} x^{4}-a \,b^{3} d^{3} n^{3} x^{3}-3 b^{4} c \,d^{2} n^{3} x^{3}-6 b^{4} d^{3} n^{2} x^{4}-3 a \,b^{3} c \,d^{2} n^{3} x^{2}-3 a \,b^{3} d^{3} n^{2} x^{3}-3 b^{4} c^{2} d \,n^{3} x^{2}-21 b^{4} c \,d^{2} n^{2} x^{3}-11 b^{4} d^{3} n \,x^{4}+3 a^{2} b^{2} d^{3} n^{2} x^{2}-3 a \,b^{3} c^{2} d \,n^{3} x -15 a \,b^{3} c \,d^{2} n^{2} x^{2}-2 a \,b^{3} d^{3} n \,x^{3}-b^{4} c^{3} n^{3} x -24 b^{4} c^{2} d \,n^{2} x^{2}-42 b^{4} c \,d^{2} n \,x^{3}-6 d^{3} x^{4} b^{4}+6 a^{2} b^{2} c \,d^{2} n^{2} x +3 a^{2} b^{2} d^{3} n \,x^{2}-a \,b^{3} c^{3} n^{3}-21 a \,b^{3} c^{2} d \,n^{2} x -12 a \,b^{3} c \,d^{2} n \,x^{2}-9 b^{4} c^{3} n^{2} x -57 b^{4} c^{2} d n \,x^{2}-24 x^{3} b^{4} c \,d^{2}-6 a^{3} b \,d^{3} n x +3 a^{2} b^{2} c^{2} d \,n^{2}+24 a^{2} b^{2} c \,d^{2} n x -9 a \,b^{3} c^{3} n^{2}-36 a \,b^{3} c^{2} d n x -26 b^{4} c^{3} n x -36 x^{2} b^{4} c^{2} d -6 a^{3} b c \,d^{2} n +21 a^{2} b^{2} c^{2} d n -26 a \,b^{3} c^{3} n -24 b^{4} c^{3} x +6 a^{4} d^{3}-24 a^{3} b c \,d^{2}+36 a^{2} b^{2} c^{2} d -24 a \,b^{3} c^{3}\right ) \left (b x +a \right )^{n}}{\left (3+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) b^{4}}\) | \(552\) |
parallelrisch | \(\frac {x^{4} \left (b x +a \right )^{n} a \,b^{4} d^{3} n^{3}+6 x^{4} \left (b x +a \right )^{n} a \,b^{4} d^{3} n^{2}+x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d^{3} n^{3}+11 x^{4} \left (b x +a \right )^{n} a \,b^{4} d^{3} n +3 x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d^{3} n^{2}+2 x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d^{3} n -3 x^{2} \left (b x +a \right )^{n} a^{3} b^{2} d^{3} n^{2}+x \left (b x +a \right )^{n} a \,b^{4} c^{3} n^{3}-24 x \left (b x +a \right )^{n} a^{3} b^{2} c \,d^{2} n +24 \left (b x +a \right )^{n} a^{2} b^{3} c^{3}+36 x \left (b x +a \right )^{n} a^{2} b^{3} c^{2} d n +3 x^{3} \left (b x +a \right )^{n} a \,b^{4} c \,d^{2} n^{3}+21 x^{3} \left (b x +a \right )^{n} a \,b^{4} c \,d^{2} n^{2}+3 x^{2} \left (b x +a \right )^{n} a^{2} b^{3} c \,d^{2} n^{3}+3 x^{2} \left (b x +a \right )^{n} a \,b^{4} c^{2} d \,n^{3}+42 x^{3} \left (b x +a \right )^{n} a \,b^{4} c \,d^{2} n +15 x^{2} \left (b x +a \right )^{n} a^{2} b^{3} c \,d^{2} n^{2}+24 x^{2} \left (b x +a \right )^{n} a \,b^{4} c^{2} d \,n^{2}+3 x \left (b x +a \right )^{n} a^{2} b^{3} c^{2} d \,n^{3}+12 x^{2} \left (b x +a \right )^{n} a^{2} b^{3} c \,d^{2} n +\left (b x +a \right )^{n} a^{2} b^{3} c^{3} n^{3}+9 \left (b x +a \right )^{n} a^{2} b^{3} c^{3} n^{2}+26 \left (b x +a \right )^{n} a^{2} b^{3} c^{3} n +24 \left (b x +a \right )^{n} a^{4} b c \,d^{2}-36 \left (b x +a \right )^{n} a^{3} b^{2} c^{2} d +24 x^{3} \left (b x +a \right )^{n} a \,b^{4} c \,d^{2}+36 x^{2} \left (b x +a \right )^{n} a \,b^{4} c^{2} d +6 x^{4} \left (b x +a \right )^{n} a \,b^{4} d^{3}+24 x \left (b x +a \right )^{n} a \,b^{4} c^{3}-6 \left (b x +a \right )^{n} a^{5} d^{3}-3 x^{2} \left (b x +a \right )^{n} a^{3} b^{2} d^{3} n +9 x \left (b x +a \right )^{n} a \,b^{4} c^{3} n^{2}+6 x \left (b x +a \right )^{n} a^{4} b \,d^{3} n +26 x \left (b x +a \right )^{n} a \,b^{4} c^{3} n -3 \left (b x +a \right )^{n} a^{3} b^{2} c^{2} d \,n^{2}+6 \left (b x +a \right )^{n} a^{4} b c \,d^{2} n -21 \left (b x +a \right )^{n} a^{3} b^{2} c^{2} d n +57 x^{2} \left (b x +a \right )^{n} a \,b^{4} c^{2} d n -6 x \left (b x +a \right )^{n} a^{3} b^{2} c \,d^{2} n^{2}+21 x \left (b x +a \right )^{n} a^{2} b^{3} c^{2} d \,n^{2}}{\left (n^{3}+9 n^{2}+26 n +24\right ) a \left (1+n \right ) b^{4}}\) | \(865\) |
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Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (110) = 220\).
Time = 0.23 (sec) , antiderivative size = 497, normalized size of antiderivative = 4.52 \[ \int (a+b x)^n (c+d x)^3 \, dx=\frac {{\left (a b^{3} c^{3} n^{3} + 24 \, a b^{3} c^{3} - 36 \, a^{2} b^{2} c^{2} d + 24 \, a^{3} b c d^{2} - 6 \, a^{4} d^{3} + {\left (b^{4} d^{3} n^{3} + 6 \, b^{4} d^{3} n^{2} + 11 \, b^{4} d^{3} n + 6 \, b^{4} d^{3}\right )} x^{4} + {\left (24 \, b^{4} c d^{2} + {\left (3 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} n^{3} + 3 \, {\left (7 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} n^{2} + 2 \, {\left (21 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} n\right )} x^{3} + 3 \, {\left (3 \, a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )} n^{2} + 3 \, {\left (12 \, b^{4} c^{2} d + {\left (b^{4} c^{2} d + a b^{3} c d^{2}\right )} n^{3} + {\left (8 \, b^{4} c^{2} d + 5 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} n^{2} + {\left (19 \, b^{4} c^{2} d + 4 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} n\right )} x^{2} + {\left (26 \, a b^{3} c^{3} - 21 \, a^{2} b^{2} c^{2} d + 6 \, a^{3} b c d^{2}\right )} n + {\left (24 \, b^{4} c^{3} + {\left (b^{4} c^{3} + 3 \, a b^{3} c^{2} d\right )} n^{3} + 3 \, {\left (3 \, b^{4} c^{3} + 7 \, a b^{3} c^{2} d - 2 \, a^{2} b^{2} c d^{2}\right )} n^{2} + 2 \, {\left (13 \, b^{4} c^{3} + 18 \, a b^{3} c^{2} d - 12 \, a^{2} b^{2} c d^{2} + 3 \, a^{3} b d^{3}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 4058 vs. \(2 (95) = 190\).
Time = 1.12 (sec) , antiderivative size = 4058, normalized size of antiderivative = 36.89 \[ \int (a+b x)^n (c+d x)^3 \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (110) = 220\).
Time = 0.21 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.24 \[ \int (a+b x)^n (c+d x)^3 \, dx=\frac {3 \, {\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c^{2} d}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {{\left (b x + a\right )}^{n + 1} c^{3}}{b {\left (n + 1\right )}} + \frac {3 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c d^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} d^{3}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 833 vs. \(2 (110) = 220\).
Time = 0.28 (sec) , antiderivative size = 833, normalized size of antiderivative = 7.57 \[ \int (a+b x)^n (c+d x)^3 \, dx=\frac {{\left (b x + a\right )}^{n} b^{4} d^{3} n^{3} x^{4} + 3 \, {\left (b x + a\right )}^{n} b^{4} c d^{2} n^{3} x^{3} + {\left (b x + a\right )}^{n} a b^{3} d^{3} n^{3} x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d^{3} n^{2} x^{4} + 3 \, {\left (b x + a\right )}^{n} b^{4} c^{2} d n^{3} x^{2} + 3 \, {\left (b x + a\right )}^{n} a b^{3} c d^{2} n^{3} x^{2} + 21 \, {\left (b x + a\right )}^{n} b^{4} c d^{2} n^{2} x^{3} + 3 \, {\left (b x + a\right )}^{n} a b^{3} d^{3} n^{2} x^{3} + 11 \, {\left (b x + a\right )}^{n} b^{4} d^{3} n x^{4} + {\left (b x + a\right )}^{n} b^{4} c^{3} n^{3} x + 3 \, {\left (b x + a\right )}^{n} a b^{3} c^{2} d n^{3} x + 24 \, {\left (b x + a\right )}^{n} b^{4} c^{2} d n^{2} x^{2} + 15 \, {\left (b x + a\right )}^{n} a b^{3} c d^{2} n^{2} x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d^{3} n^{2} x^{2} + 42 \, {\left (b x + a\right )}^{n} b^{4} c d^{2} n x^{3} + 2 \, {\left (b x + a\right )}^{n} a b^{3} d^{3} n x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d^{3} x^{4} + {\left (b x + a\right )}^{n} a b^{3} c^{3} n^{3} + 9 \, {\left (b x + a\right )}^{n} b^{4} c^{3} n^{2} x + 21 \, {\left (b x + a\right )}^{n} a b^{3} c^{2} d n^{2} x - 6 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c d^{2} n^{2} x + 57 \, {\left (b x + a\right )}^{n} b^{4} c^{2} d n x^{2} + 12 \, {\left (b x + a\right )}^{n} a b^{3} c d^{2} n x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d^{3} n x^{2} + 24 \, {\left (b x + a\right )}^{n} b^{4} c d^{2} x^{3} + 9 \, {\left (b x + a\right )}^{n} a b^{3} c^{3} n^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c^{2} d n^{2} + 26 \, {\left (b x + a\right )}^{n} b^{4} c^{3} n x + 36 \, {\left (b x + a\right )}^{n} a b^{3} c^{2} d n x - 24 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c d^{2} n x + 6 \, {\left (b x + a\right )}^{n} a^{3} b d^{3} n x + 36 \, {\left (b x + a\right )}^{n} b^{4} c^{2} d x^{2} + 26 \, {\left (b x + a\right )}^{n} a b^{3} c^{3} n - 21 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c^{2} d n + 6 \, {\left (b x + a\right )}^{n} a^{3} b c d^{2} n + 24 \, {\left (b x + a\right )}^{n} b^{4} c^{3} x + 24 \, {\left (b x + a\right )}^{n} a b^{3} c^{3} - 36 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c^{2} d + 24 \, {\left (b x + a\right )}^{n} a^{3} b c d^{2} - 6 \, {\left (b x + a\right )}^{n} a^{4} d^{3}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
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Time = 1.69 (sec) , antiderivative size = 478, normalized size of antiderivative = 4.35 \[ \int (a+b x)^n (c+d x)^3 \, dx=\frac {x\,{\left (a+b\,x\right )}^n\,\left (6\,a^3\,b\,d^3\,n-6\,a^2\,b^2\,c\,d^2\,n^2-24\,a^2\,b^2\,c\,d^2\,n+3\,a\,b^3\,c^2\,d\,n^3+21\,a\,b^3\,c^2\,d\,n^2+36\,a\,b^3\,c^2\,d\,n+b^4\,c^3\,n^3+9\,b^4\,c^3\,n^2+26\,b^4\,c^3\,n+24\,b^4\,c^3\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {d^3\,x^4\,{\left (a+b\,x\right )}^n\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}+\frac {a\,{\left (a+b\,x\right )}^n\,\left (-6\,a^3\,d^3+6\,a^2\,b\,c\,d^2\,n+24\,a^2\,b\,c\,d^2-3\,a\,b^2\,c^2\,d\,n^2-21\,a\,b^2\,c^2\,d\,n-36\,a\,b^2\,c^2\,d+b^3\,c^3\,n^3+9\,b^3\,c^3\,n^2+26\,b^3\,c^3\,n+24\,b^3\,c^3\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {3\,d\,x^2\,\left (n+1\right )\,{\left (a+b\,x\right )}^n\,\left (-a^2\,d^2\,n+a\,b\,c\,d\,n^2+4\,a\,b\,c\,d\,n+b^2\,c^2\,n^2+7\,b^2\,c^2\,n+12\,b^2\,c^2\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {d^2\,x^3\,{\left (a+b\,x\right )}^n\,\left (12\,b\,c+a\,d\,n+3\,b\,c\,n\right )\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )} \]
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