Integrand size = 18, antiderivative size = 135 \[ \int x^2 (a+b x)^n \left (c+d x^2\right ) \, dx=\frac {a^2 \left (b^2 c+a^2 d\right ) (a+b x)^{1+n}}{b^5 (1+n)}-\frac {2 a \left (b^2 c+2 a^2 d\right ) (a+b x)^{2+n}}{b^5 (2+n)}+\frac {\left (b^2 c+6 a^2 d\right ) (a+b x)^{3+n}}{b^5 (3+n)}-\frac {4 a d (a+b x)^{4+n}}{b^5 (4+n)}+\frac {d (a+b x)^{5+n}}{b^5 (5+n)} \]
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Time = 0.06 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {962} \[ \int x^2 (a+b x)^n \left (c+d x^2\right ) \, dx=\frac {a^2 \left (a^2 d+b^2 c\right ) (a+b x)^{n+1}}{b^5 (n+1)}-\frac {2 a \left (2 a^2 d+b^2 c\right ) (a+b x)^{n+2}}{b^5 (n+2)}+\frac {\left (6 a^2 d+b^2 c\right ) (a+b x)^{n+3}}{b^5 (n+3)}-\frac {4 a d (a+b x)^{n+4}}{b^5 (n+4)}+\frac {d (a+b x)^{n+5}}{b^5 (n+5)} \]
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Rule 962
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (a^2 b^2 c+a^4 d\right ) (a+b x)^n}{b^4}-\frac {2 \left (a b^2 c+2 a^3 d\right ) (a+b x)^{1+n}}{b^4}+\frac {\left (b^2 c+6 a^2 d\right ) (a+b x)^{2+n}}{b^4}-\frac {4 a d (a+b x)^{3+n}}{b^4}+\frac {d (a+b x)^{4+n}}{b^4}\right ) \, dx \\ & = \frac {a^2 \left (b^2 c+a^2 d\right ) (a+b x)^{1+n}}{b^5 (1+n)}-\frac {2 a \left (b^2 c+2 a^2 d\right ) (a+b x)^{2+n}}{b^5 (2+n)}+\frac {\left (b^2 c+6 a^2 d\right ) (a+b x)^{3+n}}{b^5 (3+n)}-\frac {4 a d (a+b x)^{4+n}}{b^5 (4+n)}+\frac {d (a+b x)^{5+n}}{b^5 (5+n)} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.84 \[ \int x^2 (a+b x)^n \left (c+d x^2\right ) \, dx=\frac {(a+b x)^{1+n} \left (\frac {a^2 b^2 c+a^4 d}{1+n}-\frac {2 a \left (b^2 c+2 a^2 d\right ) (a+b x)}{2+n}+\frac {\left (b^2 c+6 a^2 d\right ) (a+b x)^2}{3+n}-\frac {4 a d (a+b x)^3}{4+n}+\frac {d (a+b x)^4}{5+n}\right )}{b^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(319\) vs. \(2(135)=270\).
Time = 0.37 (sec) , antiderivative size = 320, normalized size of antiderivative = 2.37
method | result | size |
norman | \(\frac {d \,x^{5} {\mathrm e}^{n \ln \left (b x +a \right )}}{5+n}+\frac {n a d \,x^{4} {\mathrm e}^{n \ln \left (b x +a \right )}}{b \left (n^{2}+9 n +20\right )}+\frac {\left (b^{2} c \,n^{2}+9 b^{2} c n +12 a^{2} d +20 b^{2} c \right ) a n \,x^{2} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{3} \left (n^{4}+14 n^{3}+71 n^{2}+154 n +120\right )}+\frac {2 a^{3} \left (b^{2} c \,n^{2}+9 b^{2} c n +12 a^{2} d +20 b^{2} c \right ) {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}-\frac {\left (-b^{2} c \,n^{2}+4 a^{2} d n -9 b^{2} c n -20 b^{2} c \right ) x^{3} {\mathrm e}^{n \ln \left (b x +a \right )}}{b^{2} \left (n^{3}+12 n^{2}+47 n +60\right )}-\frac {2 n \,a^{2} \left (b^{2} c \,n^{2}+9 b^{2} c n +12 a^{2} d +20 b^{2} c \right ) x \,{\mathrm e}^{n \ln \left (b x +a \right )}}{b^{4} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(320\) |
gosper | \(\frac {\left (b x +a \right )^{1+n} \left (b^{4} d \,n^{4} x^{4}+10 b^{4} d \,n^{3} x^{4}-4 a \,b^{3} d \,n^{3} x^{3}+b^{4} c \,n^{4} x^{2}+35 b^{4} d \,n^{2} x^{4}-24 a \,b^{3} d \,n^{2} x^{3}+12 b^{4} c \,n^{3} x^{2}+50 b^{4} d n \,x^{4}+12 a^{2} b^{2} d \,n^{2} x^{2}-2 a \,b^{3} c \,n^{3} x -44 a \,b^{3} d n \,x^{3}+49 b^{4} c \,n^{2} x^{2}+24 x^{4} b^{4} d +36 a^{2} b^{2} d n \,x^{2}-20 a \,b^{3} c \,n^{2} x -24 x^{3} a \,b^{3} d +78 b^{4} c n \,x^{2}-24 a^{3} b d n x +2 a^{2} b^{2} c \,n^{2}+24 x^{2} b^{2} d \,a^{2}-58 a \,b^{3} c n x +40 b^{4} c \,x^{2}-24 a^{3} b d x +18 a^{2} b^{2} c n -40 a \,b^{3} c x +24 a^{4} d +40 a^{2} b^{2} c \right )}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(328\) |
risch | \(\frac {\left (b^{5} d \,n^{4} x^{5}+a \,b^{4} d \,n^{4} x^{4}+10 b^{5} d \,n^{3} x^{5}+6 a \,b^{4} d \,n^{3} x^{4}+b^{5} c \,n^{4} x^{3}+35 b^{5} d \,n^{2} x^{5}-4 a^{2} b^{3} d \,n^{3} x^{3}+a \,b^{4} c \,n^{4} x^{2}+11 a \,b^{4} d \,n^{2} x^{4}+12 b^{5} c \,n^{3} x^{3}+50 b^{5} d n \,x^{5}-12 a^{2} b^{3} d \,n^{2} x^{3}+10 a \,b^{4} c \,n^{3} x^{2}+6 a d n \,x^{4} b^{4}+49 b^{5} c \,n^{2} x^{3}+24 d \,x^{5} b^{5}+12 a^{3} b^{2} d \,n^{2} x^{2}-2 a^{2} b^{3} c \,n^{3} x -8 a^{2} b^{3} d n \,x^{3}+29 a \,b^{4} c \,n^{2} x^{2}+78 b^{5} c n \,x^{3}+12 a^{3} b^{2} d n \,x^{2}-18 a^{2} b^{3} c \,n^{2} x +20 a \,b^{4} c n \,x^{2}+40 b^{5} c \,x^{3}-24 a^{4} b d n x +2 a^{3} b^{2} c \,n^{2}-40 a^{2} b^{3} c n x +18 a^{3} b^{2} c n +24 d \,a^{5}+40 a^{3} b^{2} c \right ) \left (b x +a \right )^{n}}{\left (4+n \right ) \left (5+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) b^{5}}\) | \(398\) |
parallelrisch | \(\frac {x^{3} \left (b x +a \right )^{n} b^{5} c \,n^{4}+50 x^{5} \left (b x +a \right )^{n} b^{5} d n +12 x^{3} \left (b x +a \right )^{n} b^{5} c \,n^{3}+49 x^{3} \left (b x +a \right )^{n} b^{5} c \,n^{2}+78 x^{3} \left (b x +a \right )^{n} b^{5} c n +2 \left (b x +a \right )^{n} a^{3} b^{2} c \,n^{2}+18 \left (b x +a \right )^{n} a^{3} b^{2} c n +24 x^{5} \left (b x +a \right )^{n} b^{5} d +40 x^{3} \left (b x +a \right )^{n} b^{5} c +40 \left (b x +a \right )^{n} a^{3} b^{2} c -40 x \left (b x +a \right )^{n} a^{2} b^{3} c n +x^{2} \left (b x +a \right )^{n} a \,b^{4} c \,n^{4}+6 x^{4} \left (b x +a \right )^{n} a \,b^{4} d n -12 x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d \,n^{2}+10 x^{2} \left (b x +a \right )^{n} a \,b^{4} c \,n^{3}-8 x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d n +12 x^{2} \left (b x +a \right )^{n} a^{3} b^{2} d \,n^{2}+29 x^{2} \left (b x +a \right )^{n} a \,b^{4} c \,n^{2}-2 x \left (b x +a \right )^{n} a^{2} b^{3} c \,n^{3}+12 x^{2} \left (b x +a \right )^{n} a^{3} b^{2} d n +20 x^{2} \left (b x +a \right )^{n} a \,b^{4} c n -18 x \left (b x +a \right )^{n} a^{2} b^{3} c \,n^{2}-24 x \left (b x +a \right )^{n} a^{4} b d n +x^{4} \left (b x +a \right )^{n} a \,b^{4} d \,n^{4}+6 x^{4} \left (b x +a \right )^{n} a \,b^{4} d \,n^{3}+11 x^{4} \left (b x +a \right )^{n} a \,b^{4} d \,n^{2}-4 x^{3} \left (b x +a \right )^{n} a^{2} b^{3} d \,n^{3}+24 \left (b x +a \right )^{n} a^{5} d +x^{5} \left (b x +a \right )^{n} b^{5} d \,n^{4}+10 x^{5} \left (b x +a \right )^{n} b^{5} d \,n^{3}+35 x^{5} \left (b x +a \right )^{n} b^{5} d \,n^{2}}{b^{5} \left (n^{5}+15 n^{4}+85 n^{3}+225 n^{2}+274 n +120\right )}\) | \(608\) |
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Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (135) = 270\).
Time = 0.30 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.73 \[ \int x^2 (a+b x)^n \left (c+d x^2\right ) \, dx=\frac {{\left (2 \, a^{3} b^{2} c n^{2} + 18 \, a^{3} b^{2} c n + 40 \, a^{3} b^{2} c + 24 \, a^{5} d + {\left (b^{5} d n^{4} + 10 \, b^{5} d n^{3} + 35 \, b^{5} d n^{2} + 50 \, b^{5} d n + 24 \, b^{5} d\right )} x^{5} + {\left (a b^{4} d n^{4} + 6 \, a b^{4} d n^{3} + 11 \, a b^{4} d n^{2} + 6 \, a b^{4} d n\right )} x^{4} + {\left (b^{5} c n^{4} + 40 \, b^{5} c + 4 \, {\left (3 \, b^{5} c - a^{2} b^{3} d\right )} n^{3} + {\left (49 \, b^{5} c - 12 \, a^{2} b^{3} d\right )} n^{2} + 2 \, {\left (39 \, b^{5} c - 4 \, a^{2} b^{3} d\right )} n\right )} x^{3} + {\left (a b^{4} c n^{4} + 10 \, a b^{4} c n^{3} + {\left (29 \, a b^{4} c + 12 \, a^{3} b^{2} d\right )} n^{2} + 4 \, {\left (5 \, a b^{4} c + 3 \, a^{3} b^{2} d\right )} n\right )} x^{2} - 2 \, {\left (a^{2} b^{3} c n^{3} + 9 \, a^{2} b^{3} c n^{2} + 4 \, {\left (5 \, a^{2} b^{3} c + 3 \, a^{4} b d\right )} n\right )} x\right )} {\left (b x + a\right )}^{n}}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 4134 vs. \(2 (122) = 244\).
Time = 1.26 (sec) , antiderivative size = 4134, normalized size of antiderivative = 30.62 \[ \int x^2 (a+b x)^n \left (c+d x^2\right ) \, dx=\text {Too large to display} \]
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Time = 0.20 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.56 \[ \int x^2 (a+b x)^n \left (c+d x^2\right ) \, dx=\frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} x^{3} + {\left (n^{2} + n\right )} a b^{2} x^{2} - 2 \, a^{2} b n x + 2 \, a^{3}\right )} {\left (b x + a\right )}^{n} c}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{3}} + \frac {{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} x^{2} - 24 \, a^{4} b n x + 24 \, a^{5}\right )} {\left (b x + a\right )}^{n} d}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (135) = 270\).
Time = 0.29 (sec) , antiderivative size = 624, normalized size of antiderivative = 4.62 \[ \int x^2 (a+b x)^n \left (c+d x^2\right ) \, dx=\frac {{\left (b x + a\right )}^{n} b^{5} d n^{4} x^{5} + {\left (b x + a\right )}^{n} a b^{4} d n^{4} x^{4} + 10 \, {\left (b x + a\right )}^{n} b^{5} d n^{3} x^{5} + {\left (b x + a\right )}^{n} b^{5} c n^{4} x^{3} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d n^{3} x^{4} + 35 \, {\left (b x + a\right )}^{n} b^{5} d n^{2} x^{5} + {\left (b x + a\right )}^{n} a b^{4} c n^{4} x^{2} + 12 \, {\left (b x + a\right )}^{n} b^{5} c n^{3} x^{3} - 4 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n^{3} x^{3} + 11 \, {\left (b x + a\right )}^{n} a b^{4} d n^{2} x^{4} + 50 \, {\left (b x + a\right )}^{n} b^{5} d n x^{5} + 10 \, {\left (b x + a\right )}^{n} a b^{4} c n^{3} x^{2} + 49 \, {\left (b x + a\right )}^{n} b^{5} c n^{2} x^{3} - 12 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n^{2} x^{3} + 6 \, {\left (b x + a\right )}^{n} a b^{4} d n x^{4} + 24 \, {\left (b x + a\right )}^{n} b^{5} d x^{5} - 2 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c n^{3} x + 29 \, {\left (b x + a\right )}^{n} a b^{4} c n^{2} x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d n^{2} x^{2} + 78 \, {\left (b x + a\right )}^{n} b^{5} c n x^{3} - 8 \, {\left (b x + a\right )}^{n} a^{2} b^{3} d n x^{3} - 18 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c n^{2} x + 20 \, {\left (b x + a\right )}^{n} a b^{4} c n x^{2} + 12 \, {\left (b x + a\right )}^{n} a^{3} b^{2} d n x^{2} + 40 \, {\left (b x + a\right )}^{n} b^{5} c x^{3} + 2 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c n^{2} - 40 \, {\left (b x + a\right )}^{n} a^{2} b^{3} c n x - 24 \, {\left (b x + a\right )}^{n} a^{4} b d n x + 18 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c n + 40 \, {\left (b x + a\right )}^{n} a^{3} b^{2} c + 24 \, {\left (b x + a\right )}^{n} a^{5} d}{b^{5} n^{5} + 15 \, b^{5} n^{4} + 85 \, b^{5} n^{3} + 225 \, b^{5} n^{2} + 274 \, b^{5} n + 120 \, b^{5}} \]
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Time = 11.63 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.69 \[ \int x^2 (a+b x)^n \left (c+d x^2\right ) \, dx={\left (a+b\,x\right )}^n\,\left (\frac {2\,a^3\,\left (12\,d\,a^2+c\,b^2\,n^2+9\,c\,b^2\,n+20\,c\,b^2\right )}{b^5\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {d\,x^5\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120}+\frac {x^3\,\left (n^2+3\,n+2\right )\,\left (-4\,d\,a^2\,n+c\,b^2\,n^2+9\,c\,b^2\,n+20\,c\,b^2\right )}{b^2\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}-\frac {2\,a^2\,n\,x\,\left (12\,d\,a^2+c\,b^2\,n^2+9\,c\,b^2\,n+20\,c\,b^2\right )}{b^4\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {a\,n\,x^2\,\left (n+1\right )\,\left (12\,d\,a^2+c\,b^2\,n^2+9\,c\,b^2\,n+20\,c\,b^2\right )}{b^3\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}+\frac {a\,d\,n\,x^4\,\left (n^3+6\,n^2+11\,n+6\right )}{b\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}\right ) \]
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