Integrand size = 26, antiderivative size = 127 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {(b d-a e)^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (1+2 p)}+\frac {e (b d-a e) (a+b x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (1+p)}+\frac {e^2 (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (3+2 p)} \]
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Time = 0.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {660, 45} \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {e (a+b x)^2 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (p+1)}+\frac {(a+b x) (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+1)}+\frac {e^2 (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+3)} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (a b+b^2 x\right )^{2 p} (d+e x)^2 \, dx \\ & = \left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (\frac {(b d-a e)^2 \left (a b+b^2 x\right )^{2 p}}{b^2}+\frac {2 e (b d-a e) \left (a b+b^2 x\right )^{1+2 p}}{b^3}+\frac {e^2 \left (a b+b^2 x\right )^{2+2 p}}{b^4}\right ) \, dx \\ & = \frac {(b d-a e)^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (1+2 p)}+\frac {e (b d-a e) (a+b x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (1+p)}+\frac {e^2 (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (3+2 p)} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.59 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {(a+b x) \left ((a+b x)^2\right )^p \left (\frac {(b d-a e)^2}{1+2 p}+\frac {e (b d-a e) (a+b x)}{1+p}+\frac {e^2 (a+b x)^2}{3+2 p}\right )}{b^3} \]
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Time = 2.68 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.38
method | result | size |
gosper | \(\frac {\left (b x +a \right ) \left (2 b^{2} e^{2} p^{2} x^{2}+4 b^{2} d e \,p^{2} x +3 b^{2} e^{2} p \,x^{2}-2 a b \,e^{2} p x +2 b^{2} d^{2} p^{2}+8 b^{2} d e p x +x^{2} b^{2} e^{2}-2 a b d e p -x a b \,e^{2}+5 b^{2} d^{2} p +3 b^{2} d e x +a^{2} e^{2}-3 a b d e +3 b^{2} d^{2}\right ) \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p}}{b^{3} \left (4 p^{3}+12 p^{2}+11 p +3\right )}\) | \(175\) |
risch | \(\frac {\left (2 b^{3} e^{2} p^{2} x^{3}+2 a \,b^{2} e^{2} p^{2} x^{2}+4 b^{3} d e \,p^{2} x^{2}+3 b^{3} e^{2} p \,x^{3}+4 a \,b^{2} d e \,p^{2} x +a \,b^{2} e^{2} p \,x^{2}+2 b^{3} d^{2} p^{2} x +8 b^{3} d e p \,x^{2}+e^{2} x^{3} b^{3}-2 a^{2} b \,e^{2} p x +2 a \,b^{2} d^{2} p^{2}+6 a \,b^{2} d e p x +5 b^{3} d^{2} p x +3 b^{3} d e \,x^{2}-2 a^{2} b d e p +5 a \,b^{2} d^{2} p +3 b^{3} d^{2} x +a^{3} e^{2}-3 a^{2} b d e +3 a \,b^{2} d^{2}\right ) \left (\left (b x +a \right )^{2}\right )^{p}}{\left (1+p \right ) \left (3+2 p \right ) \left (1+2 p \right ) b^{3}}\) | \(250\) |
norman | \(\frac {e^{2} x^{3} {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{3+2 p}+\frac {a \left (2 b^{2} d^{2} p^{2}-2 a b d e p +5 b^{2} d^{2} p +a^{2} e^{2}-3 a b d e +3 b^{2} d^{2}\right ) {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{b^{3} \left (4 p^{3}+12 p^{2}+11 p +3\right )}+\frac {\left (a e p +2 b d p +3 b d \right ) e \,x^{2} {\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{b \left (2 p^{2}+5 p +3\right )}-\frac {\left (-4 a b d e \,p^{2}-2 b^{2} d^{2} p^{2}+2 a^{2} e^{2} p -6 a b d e p -5 b^{2} d^{2} p -3 b^{2} d^{2}\right ) x \,{\mathrm e}^{p \ln \left (b^{2} x^{2}+2 a b x +a^{2}\right )}}{b^{2} \left (4 p^{3}+12 p^{2}+11 p +3\right )}\) | \(278\) |
parallelrisch | \(\frac {2 x^{3} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{3} e^{2} p^{2}+3 x^{3} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{3} e^{2} p +2 x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b^{2} e^{2} p^{2}+4 x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{3} d e \,p^{2}+x^{3} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{3} e^{2}+x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b^{2} e^{2} p +8 x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{3} d e p +4 x \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b^{2} d e \,p^{2}+2 x \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{3} d^{2} p^{2}+3 x^{2} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{3} d e -2 x \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{3} b \,e^{2} p +6 x \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b^{2} d e p +5 x \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{3} d^{2} p +2 \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b^{2} d^{2} p^{2}+3 x \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a \,b^{3} d^{2}-2 \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{3} b d e p +5 \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b^{2} d^{2} p +\left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{4} e^{2}-3 \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{3} b d e +3 \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{p} a^{2} b^{2} d^{2}}{\left (3+2 p \right ) \left (1+p \right ) \left (1+2 p \right ) a \,b^{3}}\) | \(627\) |
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Time = 0.41 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.96 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {{\left (2 \, a b^{2} d^{2} p^{2} + 3 \, a b^{2} d^{2} - 3 \, a^{2} b d e + a^{3} e^{2} + {\left (2 \, b^{3} e^{2} p^{2} + 3 \, b^{3} e^{2} p + b^{3} e^{2}\right )} x^{3} + {\left (3 \, b^{3} d e + 2 \, {\left (2 \, b^{3} d e + a b^{2} e^{2}\right )} p^{2} + {\left (8 \, b^{3} d e + a b^{2} e^{2}\right )} p\right )} x^{2} + {\left (5 \, a b^{2} d^{2} - 2 \, a^{2} b d e\right )} p + {\left (3 \, b^{3} d^{2} + 2 \, {\left (b^{3} d^{2} + 2 \, a b^{2} d e\right )} p^{2} + {\left (5 \, b^{3} d^{2} + 6 \, a b^{2} d e - 2 \, a^{2} b e^{2}\right )} p\right )} x\right )} {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{4 \, b^{3} p^{3} + 12 \, b^{3} p^{2} + 11 \, b^{3} p + 3 \, b^{3}} \]
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\[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\text {Too large to display} \]
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Time = 0.23 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.24 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {{\left (b x + a\right )} {\left (b x + a\right )}^{2 \, p} d^{2}}{b {\left (2 \, p + 1\right )}} + \frac {{\left (b^{2} {\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )} {\left (b x + a\right )}^{2 \, p} d e}{{\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} + \frac {{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{3} + {\left (2 \, p^{2} + p\right )} a b^{2} x^{2} - 2 \, a^{2} b p x + a^{3}\right )} {\left (b x + a\right )}^{2 \, p} e^{2}}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 608 vs. \(2 (127) = 254\).
Time = 0.29 (sec) , antiderivative size = 608, normalized size of antiderivative = 4.79 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx=\frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} e^{2} p^{2} x^{3} + 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} d e p^{2} x^{2} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} e^{2} p^{2} x^{2} + 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} e^{2} p x^{3} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} d^{2} p^{2} x + 4 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} d e p^{2} x + 8 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} d e p x^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} e^{2} p x^{2} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} e^{2} x^{3} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} d^{2} p^{2} + 5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} d^{2} p x + 6 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} d e p x - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b e^{2} p x + 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} d e x^{2} + 5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} d^{2} p - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b d e p + 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} d^{2} x + 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} d^{2} - 3 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b d e + {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{3} e^{2}}{4 \, b^{3} p^{3} + 12 \, b^{3} p^{2} + 11 \, b^{3} p + 3 \, b^{3}} \]
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Time = 9.90 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.97 \[ \int (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx={\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^p\,\left (\frac {e^2\,x^3\,\left (2\,p^2+3\,p+1\right )}{4\,p^3+12\,p^2+11\,p+3}+\frac {x\,\left (-2\,a^2\,b\,e^2\,p+4\,a\,b^2\,d\,e\,p^2+6\,a\,b^2\,d\,e\,p+2\,b^3\,d^2\,p^2+5\,b^3\,d^2\,p+3\,b^3\,d^2\right )}{b^3\,\left (4\,p^3+12\,p^2+11\,p+3\right )}+\frac {a\,\left (a^2\,e^2-2\,a\,b\,d\,e\,p-3\,a\,b\,d\,e+2\,b^2\,d^2\,p^2+5\,b^2\,d^2\,p+3\,b^2\,d^2\right )}{b^3\,\left (4\,p^3+12\,p^2+11\,p+3\right )}+\frac {e\,x^2\,\left (2\,p+1\right )\,\left (3\,b\,d+a\,e\,p+2\,b\,d\,p\right )}{b\,\left (4\,p^3+12\,p^2+11\,p+3\right )}\right ) \]
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