Optimal. Leaf size=91 \[ \frac {768 \sqrt {e^{a+b x}}}{b^5}-\frac {384 x \sqrt {e^{a+b x}}}{b^4}+\frac {96 x^2 \sqrt {e^{a+b x}}}{b^3}-\frac {16 x^3 \sqrt {e^{a+b x}}}{b^2}+\frac {2 x^4 \sqrt {e^{a+b x}}}{b} \]
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Rubi [A] time = 0.14, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2176, 2194} \[ -\frac {16 x^3 \sqrt {e^{a+b x}}}{b^2}+\frac {96 x^2 \sqrt {e^{a+b x}}}{b^3}-\frac {384 x \sqrt {e^{a+b x}}}{b^4}+\frac {768 \sqrt {e^{a+b x}}}{b^5}+\frac {2 x^4 \sqrt {e^{a+b x}}}{b} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rubi steps
\begin {align*} \int \sqrt {e^{a+b x}} x^4 \, dx &=\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*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.49 \[ \frac {2 \left (b^4 x^4-8 b^3 x^3+48 b^2 x^2-192 b x+384\right ) \sqrt {e^{a+b x}}}{b^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 43, normalized size = 0.47 \[ \frac {2 \, {\left (b^{4} x^{4} - 8 \, b^{3} x^{3} + 48 \, b^{2} x^{2} - 192 \, b x + 384\right )} e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 43, normalized size = 0.47 \[ \frac {2 \, {\left (b^{4} x^{4} - 8 \, b^{3} x^{3} + 48 \, b^{2} x^{2} - 192 \, b x + 384\right )} e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 43, normalized size = 0.47 \[ \frac {2 \left (x^{4} b^{4}-8 b^{3} x^{3}+48 b^{2} x^{2}-192 b x +384\right ) \sqrt {{\mathrm e}^{b x +a}}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 60, normalized size = 0.66 \[ \frac {2 \, {\left (b^{4} x^{4} e^{\left (\frac {1}{2} \, a\right )} - 8 \, b^{3} x^{3} e^{\left (\frac {1}{2} \, a\right )} + 48 \, b^{2} x^{2} e^{\left (\frac {1}{2} \, a\right )} - 192 \, b x e^{\left (\frac {1}{2} \, a\right )} + 384 \, e^{\left (\frac {1}{2} \, a\right )}\right )} e^{\left (\frac {1}{2} \, b x\right )}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 45, normalized size = 0.49 \[ \sqrt {{\mathrm {e}}^{a+b\,x}}\,\left (\frac {768}{b^5}-\frac {384\,x}{b^4}+\frac {2\,x^4}{b}-\frac {16\,x^3}{b^2}+\frac {96\,x^2}{b^3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 51, normalized size = 0.56 \[ \begin {cases} \frac {\left (2 b^{4} x^{4} - 16 b^{3} x^{3} + 96 b^{2} x^{2} - 384 b x + 768\right ) \sqrt {e^{a + b x}}}{b^{5}} & \text {for}\: b^{5} \neq 0 \\\frac {x^{5}}{5} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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