Integrand size = 16, antiderivative size = 104 \[ \int x^2 (a+b x)^n (c+d x) \, dx=\frac {a^2 (b c-a d) (a+b x)^{1+n}}{b^4 (1+n)}-\frac {a (2 b c-3 a d) (a+b x)^{2+n}}{b^4 (2+n)}+\frac {(b c-3 a d) (a+b x)^{3+n}}{b^4 (3+n)}+\frac {d (a+b x)^{4+n}}{b^4 (4+n)} \]
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Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {78} \[ \int x^2 (a+b x)^n (c+d x) \, dx=\frac {a^2 (b c-a d) (a+b x)^{n+1}}{b^4 (n+1)}-\frac {a (2 b c-3 a d) (a+b x)^{n+2}}{b^4 (n+2)}+\frac {(b c-3 a d) (a+b x)^{n+3}}{b^4 (n+3)}+\frac {d (a+b x)^{n+4}}{b^4 (n+4)} \]
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Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^2 (-b c+a d) (a+b x)^n}{b^3}+\frac {a (-2 b c+3 a d) (a+b x)^{1+n}}{b^3}+\frac {(b c-3 a d) (a+b x)^{2+n}}{b^3}+\frac {d (a+b x)^{3+n}}{b^3}\right ) \, dx \\ & = \frac {a^2 (b c-a d) (a+b x)^{1+n}}{b^4 (1+n)}-\frac {a (2 b c-3 a d) (a+b x)^{2+n}}{b^4 (2+n)}+\frac {(b c-3 a d) (a+b x)^{3+n}}{b^4 (3+n)}+\frac {d (a+b x)^{4+n}}{b^4 (4+n)} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.84 \[ \int x^2 (a+b x)^n (c+d x) \, dx=\frac {(a+b x)^{1+n} \left (\frac {a^2 (b c-a d)}{1+n}+\frac {a (-2 b c+3 a d) (a+b x)}{2+n}+\frac {(b c-3 a d) (a+b x)^2}{3+n}+\frac {d (a+b x)^3}{4+n}\right )}{b^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(211\) vs. \(2(104)=208\).
Time = 0.65 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.04
method | result | size |
norman | \(\frac {d \,x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{4+n}+\frac {\left (a d n +b c n +4 b c \right ) x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+7 n +12\right )}-\frac {2 a^{3} \left (-b c n +3 a d -4 b c \right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}+\frac {2 n \,a^{2} \left (-b c n +3 a d -4 b c \right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}-\frac {\left (-b c n +3 a d -4 b c \right ) a n \,x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+9 n^{2}+26 n +24\right )}\) | \(212\) |
gosper | \(-\frac {\left (b x +a \right )^{1+n} \left (-b^{3} d \,n^{3} x^{3}-b^{3} c \,n^{3} x^{2}-6 b^{3} d \,n^{2} x^{3}+3 a \,b^{2} d \,n^{2} x^{2}-7 b^{3} c \,n^{2} x^{2}-11 b^{3} d n \,x^{3}+2 a \,b^{2} c \,n^{2} x +9 a \,b^{2} d n \,x^{2}-14 b^{3} c n \,x^{2}-6 b^{3} d \,x^{3}-6 a^{2} b d n x +10 a \,b^{2} c n x +6 a \,b^{2} d \,x^{2}-8 b^{3} c \,x^{2}-2 a^{2} b c n -6 a^{2} b d x +8 a \,b^{2} c x +6 a^{3} d -8 a^{2} b c \right )}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\) | \(222\) |
risch | \(-\frac {\left (-b^{4} d \,n^{3} x^{4}-a \,b^{3} d \,n^{3} x^{3}-b^{4} c \,n^{3} x^{3}-6 b^{4} d \,n^{2} x^{4}-a \,b^{3} c \,n^{3} x^{2}-3 a \,b^{3} d \,n^{2} x^{3}-7 b^{4} c \,n^{2} x^{3}-11 b^{4} d n \,x^{4}+3 a^{2} b^{2} d \,n^{2} x^{2}-5 a \,b^{3} c \,n^{2} x^{2}-2 a \,b^{3} d n \,x^{3}-14 b^{4} c n \,x^{3}-6 d \,x^{4} b^{4}+2 a^{2} b^{2} c \,n^{2} x +3 a^{2} b^{2} d n \,x^{2}-4 a \,b^{3} c n \,x^{2}-8 x^{3} b^{4} c -6 a^{3} b d n x +8 a^{2} b^{2} c n x -2 a^{3} b c n +6 a^{4} d -8 a^{3} b c \right ) \left (b x +a \right )^{n}}{\left (3+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) b^{4}}\) | \(276\) |
parallelrisch | \(\frac {8 x^{3} \left (b x +a \right )^{n} a \,b^{4} c +2 \left (b x +a \right )^{n} a^{4} b c n +6 x^{4} \left (b x +a \right )^{n} a \,b^{4} d +14 x^{3} \left (b x +a \right )^{n} a \,b^{4} c n +3 x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d \,n^{2}+7 x^{3} \left (b x +a \right )^{n} a \,b^{4} c \,n^{2}+5 x^{2} \left (b x +a \right )^{n} a^{2} b^{3} c \,n^{2}-3 x^{2} \left (b x +a \right )^{n} a^{3} b^{2} d n -3 x^{2} \left (b x +a \right )^{n} a^{3} b^{2} d \,n^{2}-2 x \left (b x +a \right )^{n} a^{3} b^{2} c \,n^{2}+x^{4} \left (b x +a \right )^{n} a \,b^{4} d \,n^{3}+6 x^{4} \left (b x +a \right )^{n} a \,b^{4} d \,n^{2}+x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d \,n^{3}+x^{3} \left (b x +a \right )^{n} a \,b^{4} c \,n^{3}+11 x^{4} \left (b x +a \right )^{n} a \,b^{4} d n +6 x \left (b x +a \right )^{n} a^{4} b d n -8 x \left (b x +a \right )^{n} a^{3} b^{2} c n +4 x^{2} \left (b x +a \right )^{n} a^{2} b^{3} c n +8 \left (b x +a \right )^{n} a^{4} b c +x^{2} \left (b x +a \right )^{n} a^{2} b^{3} c \,n^{3}+2 x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d n -6 \left (b x +a \right )^{n} a^{5} d}{\left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) b^{4} a}\) | \(441\) |
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Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (104) = 208\).
Time = 0.24 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.41 \[ \int x^2 (a+b x)^n (c+d x) \, dx=\frac {{\left (2 \, a^{3} b c n + 8 \, a^{3} b c - 6 \, a^{4} d + {\left (b^{4} d n^{3} + 6 \, b^{4} d n^{2} + 11 \, b^{4} d n + 6 \, b^{4} d\right )} x^{4} + {\left (8 \, b^{4} c + {\left (b^{4} c + a b^{3} d\right )} n^{3} + {\left (7 \, b^{4} c + 3 \, a b^{3} d\right )} n^{2} + 2 \, {\left (7 \, b^{4} c + a b^{3} d\right )} n\right )} x^{3} + {\left (a b^{3} c n^{3} + {\left (5 \, a b^{3} c - 3 \, a^{2} b^{2} d\right )} n^{2} + {\left (4 \, a b^{3} c - 3 \, a^{2} b^{2} d\right )} n\right )} x^{2} - 2 \, {\left (a^{2} b^{2} c n^{2} + {\left (4 \, a^{2} b^{2} c - 3 \, a^{3} b d\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2462 vs. \(2 (92) = 184\).
Time = 0.84 (sec) , antiderivative size = 2462, normalized size of antiderivative = 23.67 \[ \int x^2 (a+b x)^n (c+d x) \, dx=\text {Too large to display} \]
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none
Time = 0.20 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.65 \[ \int x^2 (a+b x)^n (c+d x) \, dx=\frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (104) = 208\).
Time = 0.27 (sec) , antiderivative size = 431, normalized size of antiderivative = 4.14 \[ \int x^2 (a+b x)^n (c+d x) \, dx=\frac {{\left (b x + a\right )}^{n} b^{4} d n^{3} x^{4} + {\left (b x + a\right )}^{n} b^{4} c n^{3} x^{3} + {\left (b x + a\right )}^{n} a b^{3} d n^{3} x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d n^{2} x^{4} + {\left (b x + a\right )}^{n} a b^{3} c n^{3} x^{2} + 7 \, {\left (b x + a\right )}^{n} b^{4} c n^{2} x^{3} + 3 \, {\left (b x + a\right )}^{n} a b^{3} d n^{2} x^{3} + 11 \, {\left (b x + a\right )}^{n} b^{4} d n x^{4} + 5 \, {\left (b x + a\right )}^{n} a b^{3} c n^{2} x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d n^{2} x^{2} + 14 \, {\left (b x + a\right )}^{n} b^{4} c n x^{3} + 2 \, {\left (b x + a\right )}^{n} a b^{3} d n x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d x^{4} - 2 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c n^{2} x + 4 \, {\left (b x + a\right )}^{n} a b^{3} c n x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d n x^{2} + 8 \, {\left (b x + a\right )}^{n} b^{4} c x^{3} - 8 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c n x + 6 \, {\left (b x + a\right )}^{n} a^{3} b d n x + 2 \, {\left (b x + a\right )}^{n} a^{3} b c n + 8 \, {\left (b x + a\right )}^{n} a^{3} b c - 6 \, {\left (b x + a\right )}^{n} a^{4} d}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
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Time = 1.22 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.15 \[ \int x^2 (a+b x)^n (c+d x) \, dx={\left (a+b\,x\right )}^n\,\left (\frac {d\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}+\frac {2\,a^3\,\left (4\,b\,c-3\,a\,d+b\,c\,n\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {x^3\,\left (4\,b\,c+a\,d\,n+b\,c\,n\right )\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}-\frac {2\,a^2\,n\,x\,\left (4\,b\,c-3\,a\,d+b\,c\,n\right )}{b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,n\,x^2\,\left (n+1\right )\,\left (4\,b\,c-3\,a\,d+b\,c\,n\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right ) \]
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