Optimal. Leaf size=346 \[ -\frac{4 b^3 e^4 (b c-a d) \text{Gamma}\left (-4,-\frac{e}{c+d x}\right )}{d^5}-\frac{b^4 e^5 \text{Gamma}\left (-5,-\frac{e}{c+d x}\right )}{d^5}-\frac{b^2 e^3 (b c-a d)^2 \text{Ei}\left (\frac{e}{c+d x}\right )}{d^5}+\frac{b^2 e^2 (c+d x) (b c-a d)^2 e^{\frac{e}{c+d x}}}{d^5}+\frac{b^2 e (c+d x)^2 (b c-a d)^2 e^{\frac{e}{c+d x}}}{d^5}+\frac{2 b^2 (c+d x)^3 (b c-a d)^2 e^{\frac{e}{c+d x}}}{d^5}+\frac{2 b e^2 (b c-a d)^3 \text{Ei}\left (\frac{e}{c+d x}\right )}{d^5}-\frac{e (b c-a d)^4 \text{Ei}\left (\frac{e}{c+d x}\right )}{d^5}-\frac{2 b e (c+d x) (b c-a d)^3 e^{\frac{e}{c+d x}}}{d^5}-\frac{2 b (c+d x)^2 (b c-a d)^3 e^{\frac{e}{c+d x}}}{d^5}+\frac{(c+d x) (b c-a d)^4 e^{\frac{e}{c+d x}}}{d^5} \]
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Rubi [A] time = 0.361779, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {2226, 2206, 2210, 2214, 2218} \[ -\frac{4 b^3 e^4 (b c-a d) \text{Gamma}\left (-4,-\frac{e}{c+d x}\right )}{d^5}-\frac{b^4 e^5 \text{Gamma}\left (-5,-\frac{e}{c+d x}\right )}{d^5}-\frac{b^2 e^3 (b c-a d)^2 \text{Ei}\left (\frac{e}{c+d x}\right )}{d^5}+\frac{b^2 e^2 (c+d x) (b c-a d)^2 e^{\frac{e}{c+d x}}}{d^5}+\frac{b^2 e (c+d x)^2 (b c-a d)^2 e^{\frac{e}{c+d x}}}{d^5}+\frac{2 b^2 (c+d x)^3 (b c-a d)^2 e^{\frac{e}{c+d x}}}{d^5}+\frac{2 b e^2 (b c-a d)^3 \text{Ei}\left (\frac{e}{c+d x}\right )}{d^5}-\frac{e (b c-a d)^4 \text{Ei}\left (\frac{e}{c+d x}\right )}{d^5}-\frac{2 b e (c+d x) (b c-a d)^3 e^{\frac{e}{c+d x}}}{d^5}-\frac{2 b (c+d x)^2 (b c-a d)^3 e^{\frac{e}{c+d x}}}{d^5}+\frac{(c+d x) (b c-a d)^4 e^{\frac{e}{c+d x}}}{d^5} \]
Antiderivative was successfully verified.
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Rule 2226
Rule 2206
Rule 2210
Rule 2214
Rule 2218
Rubi steps
\begin{align*} \int e^{\frac{e}{c+d x}} (a+b x)^4 \, dx &=\int \left (\frac{(-b c+a d)^4 e^{\frac{e}{c+d x}}}{d^4}-\frac{4 b (b c-a d)^3 e^{\frac{e}{c+d x}} (c+d x)}{d^4}+\frac{6 b^2 (b c-a d)^2 e^{\frac{e}{c+d x}} (c+d x)^2}{d^4}-\frac{4 b^3 (b c-a d) e^{\frac{e}{c+d x}} (c+d x)^3}{d^4}+\frac{b^4 e^{\frac{e}{c+d x}} (c+d x)^4}{d^4}\right ) \, dx\\ &=\frac{b^4 \int e^{\frac{e}{c+d x}} (c+d x)^4 \, dx}{d^4}-\frac{\left (4 b^3 (b c-a d)\right ) \int e^{\frac{e}{c+d x}} (c+d x)^3 \, dx}{d^4}+\frac{\left (6 b^2 (b c-a d)^2\right ) \int e^{\frac{e}{c+d x}} (c+d x)^2 \, dx}{d^4}-\frac{\left (4 b (b c-a d)^3\right ) \int e^{\frac{e}{c+d x}} (c+d x) \, dx}{d^4}+\frac{(b c-a d)^4 \int e^{\frac{e}{c+d x}} \, dx}{d^4}\\ &=\frac{(b c-a d)^4 e^{\frac{e}{c+d x}} (c+d x)}{d^5}-\frac{2 b (b c-a d)^3 e^{\frac{e}{c+d x}} (c+d x)^2}{d^5}+\frac{2 b^2 (b c-a d)^2 e^{\frac{e}{c+d x}} (c+d x)^3}{d^5}-\frac{b^4 e^5 \Gamma \left (-5,-\frac{e}{c+d x}\right )}{d^5}-\frac{4 b^3 (b c-a d) e^4 \Gamma \left (-4,-\frac{e}{c+d x}\right )}{d^5}+\frac{\left (2 b^2 (b c-a d)^2 e\right ) \int e^{\frac{e}{c+d x}} (c+d x) \, dx}{d^4}-\frac{\left (2 b (b c-a d)^3 e\right ) \int e^{\frac{e}{c+d x}} \, dx}{d^4}+\frac{\left ((b c-a d)^4 e\right ) \int \frac{e^{\frac{e}{c+d x}}}{c+d x} \, dx}{d^4}\\ &=\frac{(b c-a d)^4 e^{\frac{e}{c+d x}} (c+d x)}{d^5}-\frac{2 b (b c-a d)^3 e e^{\frac{e}{c+d x}} (c+d x)}{d^5}-\frac{2 b (b c-a d)^3 e^{\frac{e}{c+d x}} (c+d x)^2}{d^5}+\frac{b^2 (b c-a d)^2 e e^{\frac{e}{c+d x}} (c+d x)^2}{d^5}+\frac{2 b^2 (b c-a d)^2 e^{\frac{e}{c+d x}} (c+d x)^3}{d^5}-\frac{(b c-a d)^4 e \text{Ei}\left (\frac{e}{c+d x}\right )}{d^5}-\frac{b^4 e^5 \Gamma \left (-5,-\frac{e}{c+d x}\right )}{d^5}-\frac{4 b^3 (b c-a d) e^4 \Gamma \left (-4,-\frac{e}{c+d x}\right )}{d^5}+\frac{\left (b^2 (b c-a d)^2 e^2\right ) \int e^{\frac{e}{c+d x}} \, dx}{d^4}-\frac{\left (2 b (b c-a d)^3 e^2\right ) \int \frac{e^{\frac{e}{c+d x}}}{c+d x} \, dx}{d^4}\\ &=\frac{(b c-a d)^4 e^{\frac{e}{c+d x}} (c+d x)}{d^5}-\frac{2 b (b c-a d)^3 e e^{\frac{e}{c+d x}} (c+d x)}{d^5}+\frac{b^2 (b c-a d)^2 e^2 e^{\frac{e}{c+d x}} (c+d x)}{d^5}-\frac{2 b (b c-a d)^3 e^{\frac{e}{c+d x}} (c+d x)^2}{d^5}+\frac{b^2 (b c-a d)^2 e e^{\frac{e}{c+d x}} (c+d x)^2}{d^5}+\frac{2 b^2 (b c-a d)^2 e^{\frac{e}{c+d x}} (c+d x)^3}{d^5}-\frac{(b c-a d)^4 e \text{Ei}\left (\frac{e}{c+d x}\right )}{d^5}+\frac{2 b (b c-a d)^3 e^2 \text{Ei}\left (\frac{e}{c+d x}\right )}{d^5}-\frac{b^4 e^5 \Gamma \left (-5,-\frac{e}{c+d x}\right )}{d^5}-\frac{4 b^3 (b c-a d) e^4 \Gamma \left (-4,-\frac{e}{c+d x}\right )}{d^5}+\frac{\left (b^2 (b c-a d)^2 e^3\right ) \int \frac{e^{\frac{e}{c+d x}}}{c+d x} \, dx}{d^4}\\ &=\frac{(b c-a d)^4 e^{\frac{e}{c+d x}} (c+d x)}{d^5}-\frac{2 b (b c-a d)^3 e e^{\frac{e}{c+d x}} (c+d x)}{d^5}+\frac{b^2 (b c-a d)^2 e^2 e^{\frac{e}{c+d x}} (c+d x)}{d^5}-\frac{2 b (b c-a d)^3 e^{\frac{e}{c+d x}} (c+d x)^2}{d^5}+\frac{b^2 (b c-a d)^2 e e^{\frac{e}{c+d x}} (c+d x)^2}{d^5}+\frac{2 b^2 (b c-a d)^2 e^{\frac{e}{c+d x}} (c+d x)^3}{d^5}-\frac{(b c-a d)^4 e \text{Ei}\left (\frac{e}{c+d x}\right )}{d^5}+\frac{2 b (b c-a d)^3 e^2 \text{Ei}\left (\frac{e}{c+d x}\right )}{d^5}-\frac{b^2 (b c-a d)^2 e^3 \text{Ei}\left (\frac{e}{c+d x}\right )}{d^5}-\frac{b^4 e^5 \Gamma \left (-5,-\frac{e}{c+d x}\right )}{d^5}-\frac{4 b^3 (b c-a d) e^4 \Gamma \left (-4,-\frac{e}{c+d x}\right )}{d^5}\\ \end{align*}
Mathematica [A] time = 0.487314, size = 468, normalized size = 1.35 \[ \frac{d x e^{\frac{e}{c+d x}} \left (120 a^2 b^2 d^2 \left (-4 c e+2 d^2 x^2+d e x+e^2\right )+240 a^3 b d^3 (d x+e)+120 a^4 d^4+20 a b^3 d \left (18 c^2 e-2 c e (3 d x+5 e)+2 d^2 e x^2+6 d^3 x^3+d e^2 x+e^3\right )+b^4 \left (2 c^2 e (18 d x+43 e)-96 c^3 e-2 c e \left (8 d^2 x^2+7 d e x+9 e^2\right )+2 d^2 e^2 x^2+6 d^3 e x^3+24 d^4 x^4+d e^3 x+e^4\right )\right )-e \left (120 a^2 b^2 d^2 \left (6 c^2-6 c e+e^2\right )-240 a^3 b d^3 (2 c-e)+120 a^4 d^4-20 a b^3 d \left (-36 c^2 e+24 c^3+12 c e^2-e^3\right )+b^4 \left (120 c^2 e^2-240 c^3 e+120 c^4-20 c e^3+e^4\right )\right ) \text{Ei}\left (\frac{e}{c+d x}\right )}{120 d^5}+\frac{c e^{\frac{e}{c+d x}} \left (120 a^2 b^2 d^2 \left (2 c^2-5 c e+e^2\right )-240 a^3 b d^3 (c-e)+120 a^4 d^4-20 a b^3 d \left (-26 c^2 e+6 c^3+11 c e^2-e^3\right )+b^4 \left (102 c^2 e^2-154 c^3 e+24 c^4-19 c e^3+e^4\right )\right )}{120 d^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 1146, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (24 \, b^{4} d^{4} x^{5} + 6 \,{\left (20 \, a b^{3} d^{4} + b^{4} d^{3} e\right )} x^{4} + 2 \,{\left (120 \, a^{2} b^{2} d^{4} + 20 \, a b^{3} d^{3} e -{\left (8 \, c d^{2} e - d^{2} e^{2}\right )} b^{4}\right )} x^{3} +{\left (240 \, a^{3} b d^{4} + 120 \, a^{2} b^{2} d^{3} e - 20 \,{\left (6 \, c d^{2} e - d^{2} e^{2}\right )} a b^{3} +{\left (36 \, c^{2} d e - 14 \, c d e^{2} + d e^{3}\right )} b^{4}\right )} x^{2} +{\left (120 \, a^{4} d^{4} + 240 \, a^{3} b d^{3} e - 120 \,{\left (4 \, c d^{2} e - d^{2} e^{2}\right )} a^{2} b^{2} + 20 \,{\left (18 \, c^{2} d e - 10 \, c d e^{2} + d e^{3}\right )} a b^{3} -{\left (96 \, c^{3} e - 86 \, c^{2} e^{2} + 18 \, c e^{3} - e^{4}\right )} b^{4}\right )} x\right )} e^{\left (\frac{e}{d x + c}\right )}}{120 \, d^{4}} + \int -\frac{{\left (240 \, a^{3} b c^{2} d^{3} e - 120 \,{\left (4 \, c^{3} d^{2} e - c^{2} d^{2} e^{2}\right )} a^{2} b^{2} + 20 \,{\left (18 \, c^{4} d e - 10 \, c^{3} d e^{2} + c^{2} d e^{3}\right )} a b^{3} -{\left (96 \, c^{5} e - 86 \, c^{4} e^{2} + 18 \, c^{3} e^{3} - c^{2} e^{4}\right )} b^{4} -{\left (120 \, a^{4} d^{5} e - 240 \,{\left (2 \, c d^{4} e - d^{4} e^{2}\right )} a^{3} b + 120 \,{\left (6 \, c^{2} d^{3} e - 6 \, c d^{3} e^{2} + d^{3} e^{3}\right )} a^{2} b^{2} - 20 \,{\left (24 \, c^{3} d^{2} e - 36 \, c^{2} d^{2} e^{2} + 12 \, c d^{2} e^{3} - d^{2} e^{4}\right )} a b^{3} +{\left (120 \, c^{4} d e - 240 \, c^{3} d e^{2} + 120 \, c^{2} d e^{3} - 20 \, c d e^{4} + d e^{5}\right )} b^{4}\right )} x\right )} e^{\left (\frac{e}{d x + c}\right )}}{120 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b x\right )^{4} e^{\frac{e}{c + d x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{4} e^{\left (\frac{e}{d x + c}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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