Optimal. Leaf size=346 \[ -\frac {4 b^3 e^4 (b c-a d) \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^5}-\frac {b^2 e^3 (b c-a d)^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^5}+\frac {b^2 e^2 (c+d x) (b c-a d)^2 e^{\frac {e}{c+d x}}}{d^5}+\frac {b^2 e (c+d x)^2 (b c-a d)^2 e^{\frac {e}{c+d x}}}{d^5}+\frac {2 b^2 (c+d x)^3 (b c-a d)^2 e^{\frac {e}{c+d x}}}{d^5}+\frac {2 b e^2 (b c-a d)^3 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^5}-\frac {e (b c-a d)^4 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^5}-\frac {2 b e (c+d x) (b c-a d)^3 e^{\frac {e}{c+d x}}}{d^5}-\frac {2 b (c+d x)^2 (b c-a d)^3 e^{\frac {e}{c+d x}}}{d^5}+\frac {(c+d x) (b c-a d)^4 e^{\frac {e}{c+d x}}}{d^5}-\frac {b^4 e^5 \Gamma \left (-5,-\frac {e}{c+d x}\right )}{d^5} \]
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Rubi [A] time = 0.36, antiderivative size = 346, normalized size of antiderivative = 1.00, 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 2206
Rule 2210
Rule 2214
Rule 2218
Rule 2226
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*}
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Mathematica [A] time = 0.50, size = 468, normalized size = 1.35 \[ \frac {d x e^{\frac {e}{c+d x}} \left (120 a^4 d^4+240 a^3 b d^3 (d x+e)+120 a^2 b^2 d^2 \left (-4 c e+2 d^2 x^2+d e x+e^2\right )+20 a b^3 d \left (18 c^2 e-2 c e (3 d x+5 e)+6 d^3 x^3+2 d^2 e x^2+d e^2 x+e^3\right )+b^4 \left (-96 c^3 e+2 c^2 e (18 d x+43 e)-2 c e \left (8 d^2 x^2+7 d e x+9 e^2\right )+24 d^4 x^4+6 d^3 e x^3+2 d^2 e^2 x^2+d e^3 x+e^4\right )\right )-e \left (120 a^4 d^4-240 a^3 b d^3 (2 c-e)+120 a^2 b^2 d^2 \left (6 c^2-6 c e+e^2\right )-20 a b^3 d \left (24 c^3-36 c^2 e+12 c e^2-e^3\right )+b^4 \left (120 c^4-240 c^3 e+120 c^2 e^2-20 c e^3+e^4\right )\right ) \text {Ei}\left (\frac {e}{c+d x}\right )}{120 d^5}+\frac {c e^{\frac {e}{c+d x}} \left (120 a^4 d^4-240 a^3 b d^3 (c-e)+120 a^2 b^2 d^2 \left (2 c^2-5 c e+e^2\right )-20 a b^3 d \left (6 c^3-26 c^2 e+11 c e^2-e^3\right )+b^4 \left (24 c^4-154 c^3 e+102 c^2 e^2-19 c e^3+e^4\right )\right )}{120 d^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 638, normalized size = 1.84 \[ -\frac {{\left (b^{4} e^{5} - 20 \, {\left (b^{4} c - a b^{3} d\right )} e^{4} + 120 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} e^{3} - 240 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} e^{2} + 120 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} e\right )} {\rm Ei}\left (\frac {e}{d x + c}\right ) - {\left (24 \, b^{4} d^{5} x^{5} + 24 \, b^{4} c^{5} - 120 \, a b^{3} c^{4} d + 240 \, a^{2} b^{2} c^{3} d^{2} - 240 \, a^{3} b c^{2} d^{3} + 120 \, a^{4} c d^{4} + b^{4} c e^{4} + 6 \, {\left (20 \, a b^{3} d^{5} + b^{4} d^{4} e\right )} x^{4} - {\left (19 \, b^{4} c^{2} - 20 \, a b^{3} c d\right )} e^{3} + 2 \, {\left (120 \, a^{2} b^{2} d^{5} + b^{4} d^{3} e^{2} - 4 \, {\left (2 \, b^{4} c d^{3} - 5 \, a b^{3} d^{4}\right )} e\right )} x^{3} + 2 \, {\left (51 \, b^{4} c^{3} - 110 \, a b^{3} c^{2} d + 60 \, a^{2} b^{2} c d^{2}\right )} e^{2} + {\left (240 \, a^{3} b d^{5} + b^{4} d^{2} e^{3} - 2 \, {\left (7 \, b^{4} c d^{2} - 10 \, a b^{3} d^{3}\right )} e^{2} + 12 \, {\left (3 \, b^{4} c^{2} d^{2} - 10 \, a b^{3} c d^{3} + 10 \, a^{2} b^{2} d^{4}\right )} e\right )} x^{2} - 2 \, {\left (77 \, b^{4} c^{4} - 260 \, a b^{3} c^{3} d + 300 \, a^{2} b^{2} c^{2} d^{2} - 120 \, a^{3} b c d^{3}\right )} e + {\left (120 \, a^{4} d^{5} + b^{4} d e^{4} - 2 \, {\left (9 \, b^{4} c d - 10 \, a b^{3} d^{2}\right )} e^{3} + 2 \, {\left (43 \, b^{4} c^{2} d - 100 \, a b^{3} c d^{2} + 60 \, a^{2} b^{2} d^{3}\right )} e^{2} - 24 \, {\left (4 \, b^{4} c^{3} d - 15 \, a b^{3} c^{2} d^{2} + 20 \, a^{2} b^{2} c d^{3} - 10 \, a^{3} b d^{4}\right )} e\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{120 \, d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.75, size = 1427, normalized size = 4.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 1146, normalized size = 3.31 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (24 \, b^{4} d^{4} x^{5} + 6 \, {\left (20 \, a b^{3} d^{4} + b^{4} d^{3} e\right )} x^{4} + 2 \, {\left (120 \, a^{2} b^{2} d^{4} + 20 \, a b^{3} d^{3} e - {\left (8 \, c d^{2} e - d^{2} e^{2}\right )} b^{4}\right )} x^{3} + {\left (240 \, a^{3} b d^{4} + 120 \, a^{2} b^{2} d^{3} e - 20 \, {\left (6 \, c d^{2} e - d^{2} e^{2}\right )} a b^{3} + {\left (36 \, c^{2} d e - 14 \, c d e^{2} + d e^{3}\right )} b^{4}\right )} x^{2} + {\left (120 \, a^{4} d^{4} + 240 \, a^{3} b d^{3} e - 120 \, {\left (4 \, c d^{2} e - d^{2} e^{2}\right )} a^{2} b^{2} + 20 \, {\left (18 \, c^{2} d e - 10 \, c d e^{2} + d e^{3}\right )} a b^{3} - {\left (96 \, c^{3} e - 86 \, c^{2} e^{2} + 18 \, c e^{3} - e^{4}\right )} b^{4}\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{120 \, d^{4}} + \int -\frac {{\left (240 \, a^{3} b c^{2} d^{3} e - 120 \, {\left (4 \, c^{3} d^{2} e - c^{2} d^{2} e^{2}\right )} a^{2} b^{2} + 20 \, {\left (18 \, c^{4} d e - 10 \, c^{3} d e^{2} + c^{2} d e^{3}\right )} a b^{3} - {\left (96 \, c^{5} e - 86 \, c^{4} e^{2} + 18 \, c^{3} e^{3} - c^{2} e^{4}\right )} b^{4} - {\left (120 \, a^{4} d^{5} e - 240 \, {\left (2 \, c d^{4} e - d^{4} e^{2}\right )} a^{3} b + 120 \, {\left (6 \, c^{2} d^{3} e - 6 \, c d^{3} e^{2} + d^{3} e^{3}\right )} a^{2} b^{2} - 20 \, {\left (24 \, c^{3} d^{2} e - 36 \, c^{2} d^{2} e^{2} + 12 \, c d^{2} e^{3} - d^{2} e^{4}\right )} a b^{3} + {\left (120 \, c^{4} d e - 240 \, c^{3} d e^{2} + 120 \, c^{2} d e^{3} - 20 \, c d e^{4} + d e^{5}\right )} b^{4}\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{120 \, {\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {e}}^{\frac {e}{c+d\,x}}\,{\left (a+b\,x\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b x\right )^{4} e^{\frac {e}{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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