Integrand size = 15, antiderivative size = 91 \[ \int \sqrt {e^{a+b x}} x^4 \, dx=\frac {768 \sqrt {e^{a+b x}}}{b^5}-\frac {384 \sqrt {e^{a+b x}} x}{b^4}+\frac {96 \sqrt {e^{a+b x}} x^2}{b^3}-\frac {16 \sqrt {e^{a+b x}} x^3}{b^2}+\frac {2 \sqrt {e^{a+b x}} x^4}{b} \]
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Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2207, 2225} \[ \int \sqrt {e^{a+b x}} x^4 \, dx=\frac {768 \sqrt {e^{a+b x}}}{b^5}-\frac {384 x \sqrt {e^{a+b x}}}{b^4}+\frac {96 x^2 \sqrt {e^{a+b x}}}{b^3}-\frac {16 x^3 \sqrt {e^{a+b x}}}{b^2}+\frac {2 x^4 \sqrt {e^{a+b x}}}{b} \]
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Rule 2207
Rule 2225
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {e^{a+b x}} x^4}{b}-\frac {8 \int \sqrt {e^{a+b x}} x^3 \, dx}{b} \\ & = -\frac {16 \sqrt {e^{a+b x}} x^3}{b^2}+\frac {2 \sqrt {e^{a+b x}} x^4}{b}+\frac {48 \int \sqrt {e^{a+b x}} x^2 \, dx}{b^2} \\ & = \frac {96 \sqrt {e^{a+b x}} x^2}{b^3}-\frac {16 \sqrt {e^{a+b x}} x^3}{b^2}+\frac {2 \sqrt {e^{a+b x}} x^4}{b}-\frac {192 \int \sqrt {e^{a+b x}} x \, dx}{b^3} \\ & = -\frac {384 \sqrt {e^{a+b x}} x}{b^4}+\frac {96 \sqrt {e^{a+b x}} x^2}{b^3}-\frac {16 \sqrt {e^{a+b x}} x^3}{b^2}+\frac {2 \sqrt {e^{a+b x}} x^4}{b}+\frac {384 \int \sqrt {e^{a+b x}} \, dx}{b^4} \\ & = \frac {768 \sqrt {e^{a+b x}}}{b^5}-\frac {384 \sqrt {e^{a+b x}} x}{b^4}+\frac {96 \sqrt {e^{a+b x}} x^2}{b^3}-\frac {16 \sqrt {e^{a+b x}} x^3}{b^2}+\frac {2 \sqrt {e^{a+b x}} x^4}{b} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.49 \[ \int \sqrt {e^{a+b x}} x^4 \, dx=\frac {2 \sqrt {e^{a+b x}} \left (384-192 b x+48 b^2 x^2-8 b^3 x^3+b^4 x^4\right )}{b^5} \]
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Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.47
method | result | size |
gosper | \(\frac {2 \left (b^{4} x^{4}-8 b^{3} x^{3}+48 b^{2} x^{2}-192 b x +384\right ) \sqrt {{\mathrm e}^{b x +a}}}{b^{5}}\) | \(43\) |
risch | \(\frac {2 \left (b^{4} x^{4}-8 b^{3} x^{3}+48 b^{2} x^{2}-192 b x +384\right ) \sqrt {{\mathrm e}^{b x +a}}}{b^{5}}\) | \(43\) |
parallelrisch | \(\frac {2 x^{4} \sqrt {{\mathrm e}^{b x +a}}\, b^{4}-16 \sqrt {{\mathrm e}^{b x +a}}\, x^{3} b^{3}+96 \sqrt {{\mathrm e}^{b x +a}}\, x^{2} b^{2}-384 b \sqrt {{\mathrm e}^{b x +a}}\, x +768 \sqrt {{\mathrm e}^{b x +a}}}{b^{5}}\) | \(76\) |
meijerg | \(-\frac {32 \,{\mathrm e}^{-\frac {5 a}{2}-\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}} \sqrt {{\mathrm e}^{b x +a}}\, \left (24-\frac {\left (\frac {5 b^{4} x^{4} {\mathrm e}^{2 a}}{16}-\frac {5 b^{3} x^{3} {\mathrm e}^{\frac {3 a}{2}}}{2}+15 b^{2} x^{2} {\mathrm e}^{a}-60 b x \,{\mathrm e}^{\frac {a}{2}}+120\right ) {\mathrm e}^{\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}}}{5}\right )}{b^{5}}\) | \(84\) |
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Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.47 \[ \int \sqrt {e^{a+b x}} x^4 \, dx=\frac {2 \, {\left (b^{4} x^{4} - 8 \, b^{3} x^{3} + 48 \, b^{2} x^{2} - 192 \, b x + 384\right )} e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{b^{5}} \]
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Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.56 \[ \int \sqrt {e^{a+b x}} x^4 \, dx=\begin {cases} \frac {\left (2 b^{4} x^{4} - 16 b^{3} x^{3} + 96 b^{2} x^{2} - 384 b x + 768\right ) \sqrt {e^{a + b x}}}{b^{5}} & \text {for}\: b^{5} \neq 0 \\\frac {x^{5}}{5} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.66 \[ \int \sqrt {e^{a+b x}} x^4 \, dx=\frac {2 \, {\left (b^{4} x^{4} e^{\left (\frac {1}{2} \, a\right )} - 8 \, b^{3} x^{3} e^{\left (\frac {1}{2} \, a\right )} + 48 \, b^{2} x^{2} e^{\left (\frac {1}{2} \, a\right )} - 192 \, b x e^{\left (\frac {1}{2} \, a\right )} + 384 \, e^{\left (\frac {1}{2} \, a\right )}\right )} e^{\left (\frac {1}{2} \, b x\right )}}{b^{5}} \]
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Time = 0.32 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.47 \[ \int \sqrt {e^{a+b x}} x^4 \, dx=\frac {2 \, {\left (b^{4} x^{4} - 8 \, b^{3} x^{3} + 48 \, b^{2} x^{2} - 192 \, b x + 384\right )} e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{b^{5}} \]
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Time = 0.17 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.49 \[ \int \sqrt {e^{a+b x}} x^4 \, dx=\sqrt {{\mathrm {e}}^{a+b\,x}}\,\left (\frac {768}{b^5}-\frac {384\,x}{b^4}+\frac {2\,x^4}{b}-\frac {16\,x^3}{b^2}+\frac {96\,x^2}{b^3}\right ) \]
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