Integrand size = 16, antiderivative size = 102 \[ \int x (a+b x)^n \left (c+d x^2\right ) \, dx=-\frac {a \left (b^2 c+a^2 d\right ) (a+b x)^{1+n}}{b^4 (1+n)}+\frac {\left (b^2 c+3 a^2 d\right ) (a+b x)^{2+n}}{b^4 (2+n)}-\frac {3 a d (a+b x)^{3+n}}{b^4 (3+n)}+\frac {d (a+b x)^{4+n}}{b^4 (4+n)} \]
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Time = 0.04 (sec) , antiderivative size = 102, 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 = {786} \[ \int x (a+b x)^n \left (c+d x^2\right ) \, dx=-\frac {a \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^4 (n+1)}+\frac {\left (3 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^4 (n+2)}-\frac {3 a d (a+b x)^{n+3}}{b^4 (n+3)}+\frac {d (a+b x)^{n+4}}{b^4 (n+4)} \]
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Rule 786
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a \left (-b^2 c-a^2 d\right ) (a+b x)^n}{b^3}+\frac {\left (b^2 c+3 a^2 d\right ) (a+b x)^{1+n}}{b^3}-\frac {3 a d (a+b x)^{2+n}}{b^3}+\frac {d (a+b x)^{3+n}}{b^3}\right ) \, dx \\ & = -\frac {a \left (b^2 c+a^2 d\right ) (a+b x)^{1+n}}{b^4 (1+n)}+\frac {\left (b^2 c+3 a^2 d\right ) (a+b x)^{2+n}}{b^4 (2+n)}-\frac {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 = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.07 \[ \int x (a+b x)^n \left (c+d x^2\right ) \, dx=\frac {(a+b x)^{1+n} \left (-6 a^3 d+6 a^2 b d (1+n) x+b^3 \left (3+4 n+n^2\right ) x \left (c (4+n)+d (2+n) x^2\right )-a b^2 \left (c \left (12+7 n+n^2\right )+3 d \left (2+3 n+n^2\right ) x^2\right )\right )}{b^4 (1+n) (2+n) (3+n) (4+n)} \]
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Time = 0.40 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.91
method | result | size |
gosper | \(-\frac {\left (b x +a \right )^{1+n} \left (-b^{3} d \,n^{3} x^{3}-6 b^{3} d \,n^{2} x^{3}+3 a \,b^{2} d \,n^{2} x^{2}-b^{3} c \,n^{3} x -11 b^{3} d n \,x^{3}+9 a \,b^{2} d n \,x^{2}-8 b^{3} c \,n^{2} x -6 x^{3} b^{3} d -6 a^{2} b d n x +a \,b^{2} c \,n^{2}+6 x^{2} b^{2} d a -19 b^{3} c n x -6 a^{2} b d x +7 a \,b^{2} c n -12 b^{3} c x +6 d \,a^{3}+12 a \,b^{2} c \right )}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}\) | \(195\) |
norman | \(\frac {d \,x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{4+n}+\frac {n a \left (b^{2} c \,n^{2}+7 b^{2} c n +6 a^{2} d +12 b^{2} c \right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}+\frac {n a d \,x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+7 n +12\right )}-\frac {a^{2} \left (b^{2} c \,n^{2}+7 b^{2} c n +6 a^{2} d +12 b^{2} c \right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right )}-\frac {\left (-b^{2} c \,n^{2}+3 a^{2} d n -7 b^{2} c n -12 b^{2} c \right ) x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+9 n^{2}+26 n +24\right )}\) | \(241\) |
risch | \(-\frac {\left (-b^{4} d \,n^{3} x^{4}-a \,b^{3} d \,n^{3} x^{3}-6 b^{4} d \,n^{2} x^{4}-3 a \,b^{3} d \,n^{2} x^{3}-b^{4} c \,n^{3} x^{2}-11 b^{4} d n \,x^{4}+3 a^{2} b^{2} d \,n^{2} x^{2}-a \,b^{3} c \,n^{3} x -2 a \,b^{3} d n \,x^{3}-8 b^{4} c \,n^{2} x^{2}-6 x^{4} b^{4} d +3 a^{2} b^{2} d n \,x^{2}-7 a \,b^{3} c \,n^{2} x -19 b^{4} c n \,x^{2}-6 a^{3} b d n x +a^{2} b^{2} c \,n^{2}-12 a \,b^{3} c n x -12 b^{4} c \,x^{2}+7 a^{2} b^{2} c n +6 a^{4} d +12 a^{2} b^{2} c \right ) \left (b x +a \right )^{n}}{\left (3+n \right ) \left (4+n \right ) \left (2+n \right ) \left (1+n \right ) b^{4}}\) | \(261\) |
parallelrisch | \(\frac {-\left (b x +a \right )^{n} a^{3} b^{2} c \,n^{2}-7 \left (b x +a \right )^{n} a^{3} b^{2} c n -12 \left (b x +a \right )^{n} a^{3} b^{2} c +12 x \left (b x +a \right )^{n} a^{2} b^{3} c n +11 x^{4} \left (b x +a \right )^{n} a \,b^{4} d n +3 x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d \,n^{2}+x^{2} \left (b x +a \right )^{n} a \,b^{4} c \,n^{3}+2 x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d n -3 x^{2} \left (b x +a \right )^{n} a^{3} b^{2} d \,n^{2}+8 x^{2} \left (b x +a \right )^{n} a \,b^{4} c \,n^{2}+x \left (b x +a \right )^{n} a^{2} b^{3} c \,n^{3}-3 x^{2} \left (b x +a \right )^{n} a^{3} b^{2} d n +19 x^{2} \left (b x +a \right )^{n} a \,b^{4} c n +7 x \left (b x +a \right )^{n} a^{2} b^{3} c \,n^{2}+6 x \left (b x +a \right )^{n} a^{4} b d n +x^{4} \left (b x +a \right )^{n} a \,b^{4} d \,n^{3}+6 x^{4} \left (b x +a \right )^{n} a \,b^{4} d \,n^{2}+x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d \,n^{3}-6 \left (b x +a \right )^{n} a^{5} d +6 x^{4} \left (b x +a \right )^{n} a \,b^{4} d +12 x^{2} \left (b x +a \right )^{n} a \,b^{4} c}{a \left (n^{2}+7 n +12\right ) \left (2+n \right ) \left (1+n \right ) b^{4}}\) | \(420\) |
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Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (102) = 204\).
Time = 0.29 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.45 \[ \int x (a+b x)^n \left (c+d x^2\right ) \, dx=-\frac {{\left (a^{2} b^{2} c n^{2} + 7 \, a^{2} b^{2} c n + 12 \, a^{2} b^{2} c + 6 \, a^{4} d - {\left (b^{4} d n^{3} + 6 \, b^{4} d n^{2} + 11 \, b^{4} d n + 6 \, b^{4} d\right )} x^{4} - {\left (a b^{3} d n^{3} + 3 \, a b^{3} d n^{2} + 2 \, a b^{3} d n\right )} x^{3} - {\left (b^{4} c n^{3} + 12 \, b^{4} c + {\left (8 \, b^{4} c - 3 \, a^{2} b^{2} d\right )} n^{2} + {\left (19 \, b^{4} c - 3 \, a^{2} b^{2} d\right )} n\right )} x^{2} - {\left (a b^{3} c n^{3} + 7 \, a b^{3} c n^{2} + 6 \, {\left (2 \, a b^{3} c + a^{3} b d\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2181 vs. \(2 (90) = 180\).
Time = 0.78 (sec) , antiderivative size = 2181, normalized size of antiderivative = 21.38 \[ \int x (a+b x)^n \left (c+d x^2\right ) \, dx=\text {Too large to display} \]
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Time = 0.20 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.43 \[ \int x (a+b x)^n \left (c+d x^2\right ) \, dx=\frac {{\left (b^{2} {\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )} {\left (b x + a\right )}^{n} c}{{\left (n^{2} + 3 \, n + 2\right )} b^{2}} + \frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b n x - 6 \, a^{4}\right )} {\left (b x + a\right )}^{n} d}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (102) = 204\).
Time = 0.28 (sec) , antiderivative size = 410, normalized size of antiderivative = 4.02 \[ \int x (a+b x)^n \left (c+d x^2\right ) \, dx=\frac {{\left (b x + a\right )}^{n} b^{4} d n^{3} x^{4} + {\left (b x + a\right )}^{n} a b^{3} d n^{3} x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d n^{2} x^{4} + {\left (b x + a\right )}^{n} b^{4} c n^{3} x^{2} + 3 \, {\left (b x + a\right )}^{n} a b^{3} d n^{2} x^{3} + 11 \, {\left (b x + a\right )}^{n} b^{4} d n x^{4} + {\left (b x + a\right )}^{n} a b^{3} c n^{3} x + 8 \, {\left (b x + a\right )}^{n} b^{4} c n^{2} x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d n^{2} x^{2} + 2 \, {\left (b x + a\right )}^{n} a b^{3} d n x^{3} + 6 \, {\left (b x + a\right )}^{n} b^{4} d x^{4} + 7 \, {\left (b x + a\right )}^{n} a b^{3} c n^{2} x + 19 \, {\left (b x + a\right )}^{n} b^{4} c n x^{2} - 3 \, {\left (b x + a\right )}^{n} a^{2} b^{2} d n x^{2} - {\left (b x + a\right )}^{n} a^{2} b^{2} c n^{2} + 12 \, {\left (b x + a\right )}^{n} a b^{3} c n x + 6 \, {\left (b x + a\right )}^{n} a^{3} b d n x + 12 \, {\left (b x + a\right )}^{n} b^{4} c x^{2} - 7 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c n - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{2} c - 6 \, {\left (b x + a\right )}^{n} a^{4} d}{b^{4} n^{4} + 10 \, b^{4} n^{3} + 35 \, b^{4} n^{2} + 50 \, b^{4} n + 24 \, b^{4}} \]
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Time = 11.56 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.50 \[ \int x (a+b x)^n \left (c+d x^2\right ) \, dx={\left (a+b\,x\right )}^n\,\left (\frac {d\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{n^4+10\,n^3+35\,n^2+50\,n+24}-\frac {a^2\,\left (6\,d\,a^2+c\,b^2\,n^2+7\,c\,b^2\,n+12\,c\,b^2\right )}{b^4\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {x^2\,\left (n+1\right )\,\left (-3\,d\,a^2\,n+c\,b^2\,n^2+7\,c\,b^2\,n+12\,c\,b^2\right )}{b^2\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,n\,x\,\left (6\,d\,a^2+c\,b^2\,n^2+7\,c\,b^2\,n+12\,c\,b^2\right )}{b^3\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}+\frac {a\,d\,n\,x^3\,\left (n^2+3\,n+2\right )}{b\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}\right ) \]
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