Optimal. Leaf size=320 \[ \frac {b^2 e^3 (b c-a d) \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}-\frac {b^2 e^2 (c+d x) (b c-a d) e^{\frac {e}{c+d x}}}{2 d^4}-\frac {b^2 e (c+d x)^2 (b c-a d) e^{\frac {e}{c+d x}}}{2 d^4}-\frac {b^2 (c+d x)^3 (b c-a d) e^{\frac {e}{c+d x}}}{d^4}-\frac {3 b e^2 (b c-a d)^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {e (b c-a d)^3 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^4}+\frac {3 b e (c+d x) (b c-a d)^2 e^{\frac {e}{c+d x}}}{2 d^4}+\frac {3 b (c+d x)^2 (b c-a d)^2 e^{\frac {e}{c+d x}}}{2 d^4}-\frac {(c+d x) (b c-a d)^3 e^{\frac {e}{c+d x}}}{d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4} \]
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Rubi [A] time = 0.32, antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 12, 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} \[ \frac {b^3 e^4 \text {Gamma}\left (-4,-\frac {e}{c+d x}\right )}{d^4}+\frac {b^2 e^3 (b c-a d) \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}-\frac {b^2 e^2 (c+d x) (b c-a d) e^{\frac {e}{c+d x}}}{2 d^4}-\frac {b^2 e (c+d x)^2 (b c-a d) e^{\frac {e}{c+d x}}}{2 d^4}-\frac {b^2 (c+d x)^3 (b c-a d) e^{\frac {e}{c+d x}}}{d^4}-\frac {3 b e^2 (b c-a d)^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {e (b c-a d)^3 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^4}+\frac {3 b e (c+d x) (b c-a d)^2 e^{\frac {e}{c+d x}}}{2 d^4}+\frac {3 b (c+d x)^2 (b c-a d)^2 e^{\frac {e}{c+d x}}}{2 d^4}-\frac {(c+d x) (b c-a d)^3 e^{\frac {e}{c+d x}}}{d^4} \]
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)^3 \, dx &=\int \left (\frac {(-b c+a d)^3 e^{\frac {e}{c+d x}}}{d^3}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {3 b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^3 e^{\frac {e}{c+d x}} (c+d x)^3}{d^3}\right ) \, dx\\ &=\frac {b^3 \int e^{\frac {e}{c+d x}} (c+d x)^3 \, dx}{d^3}-\frac {\left (3 b^2 (b c-a d)\right ) \int e^{\frac {e}{c+d x}} (c+d x)^2 \, dx}{d^3}+\frac {\left (3 b (b c-a d)^2\right ) \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{d^3}-\frac {(b c-a d)^3 \int e^{\frac {e}{c+d x}} \, dx}{d^3}\\ &=-\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4}-\frac {\left (b^2 (b c-a d) e\right ) \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{d^3}+\frac {\left (3 b (b c-a d)^2 e\right ) \int e^{\frac {e}{c+d x}} \, dx}{2 d^3}-\frac {\left ((b c-a d)^3 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{d^3}\\ &=-\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {(b c-a d)^3 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4}-\frac {\left (b^2 (b c-a d) e^2\right ) \int e^{\frac {e}{c+d x}} \, dx}{2 d^3}+\frac {\left (3 b (b c-a d)^2 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{2 d^3}\\ &=-\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}-\frac {b^2 (b c-a d) e^2 e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {(b c-a d)^3 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^4}-\frac {3 b (b c-a d)^2 e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4}-\frac {\left (b^2 (b c-a d) e^3\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{2 d^3}\\ &=-\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}-\frac {b^2 (b c-a d) e^2 e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {(b c-a d)^3 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^4}-\frac {3 b (b c-a d)^2 e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^2 (b c-a d) e^3 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 292, normalized size = 0.91 \[ \frac {d x e^{\frac {e}{c+d x}} \left (24 a^3 d^3+36 a^2 b d^2 (d x+e)+12 a b^2 d \left (-4 c e+2 d^2 x^2+d e x+e^2\right )+b^3 \left (18 c^2 e-2 c e (3 d x+5 e)+6 d^3 x^3+2 d^2 e x^2+d e^2 x+e^3\right )\right )-e \left (24 a^3 d^3+36 a^2 b d^2 (e-2 c)+12 a b^2 d \left (6 c^2-6 c e+e^2\right )+b^3 \left (-24 c^3+36 c^2 e-12 c e^2+e^3\right )\right ) \text {Ei}\left (\frac {e}{c+d x}\right )}{24 d^4}-\frac {c e^{\frac {e}{c+d x}} \left (-24 a^3 d^3+36 a^2 b d^2 (c-e)-12 a b^2 d \left (2 c^2-5 c e+e^2\right )+b^3 \left (6 c^3-26 c^2 e+11 c e^2-e^3\right )\right )}{24 d^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 377, normalized size = 1.18 \[ -\frac {{\left (b^{3} e^{4} - 12 \, {\left (b^{3} c - a b^{2} d\right )} e^{3} + 36 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} e^{2} - 24 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e\right )} {\rm Ei}\left (\frac {e}{d x + c}\right ) - {\left (6 \, b^{3} d^{4} x^{4} - 6 \, b^{3} c^{4} + 24 \, a b^{2} c^{3} d - 36 \, a^{2} b c^{2} d^{2} + 24 \, a^{3} c d^{3} + b^{3} c e^{3} + 2 \, {\left (12 \, a b^{2} d^{4} + b^{3} d^{3} e\right )} x^{3} - {\left (11 \, b^{3} c^{2} - 12 \, a b^{2} c d\right )} e^{2} + {\left (36 \, a^{2} b d^{4} + b^{3} d^{2} e^{2} - 6 \, {\left (b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} e\right )} x^{2} + 2 \, {\left (13 \, b^{3} c^{3} - 30 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2}\right )} e + {\left (24 \, a^{3} d^{4} + b^{3} d e^{3} - 2 \, {\left (5 \, b^{3} c d - 6 \, a b^{2} d^{2}\right )} e^{2} + 6 \, {\left (3 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + 6 \, a^{2} b d^{3}\right )} e\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{24 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 830, normalized size = 2.59 \[ \frac {{\left (\frac {24 \, b^{3} c^{3} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{6}}{{\left (d x + c\right )}^{4}} - \frac {72 \, a b^{2} c^{2} d {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{6}}{{\left (d x + c\right )}^{4}} + \frac {72 \, a^{2} b c d^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{6}}{{\left (d x + c\right )}^{4}} - \frac {24 \, a^{3} d^{3} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{6}}{{\left (d x + c\right )}^{4}} + 6 \, b^{3} e^{\left (\frac {e}{d x + c} + 5\right )} - \frac {24 \, b^{3} c e^{\left (\frac {e}{d x + c} + 5\right )}}{d x + c} + \frac {36 \, b^{3} c^{2} e^{\left (\frac {e}{d x + c} + 5\right )}}{{\left (d x + c\right )}^{2}} - \frac {24 \, b^{3} c^{3} e^{\left (\frac {e}{d x + c} + 5\right )}}{{\left (d x + c\right )}^{3}} + \frac {24 \, a b^{2} d e^{\left (\frac {e}{d x + c} + 5\right )}}{d x + c} - \frac {72 \, a b^{2} c d e^{\left (\frac {e}{d x + c} + 5\right )}}{{\left (d x + c\right )}^{2}} + \frac {72 \, a b^{2} c^{2} d e^{\left (\frac {e}{d x + c} + 5\right )}}{{\left (d x + c\right )}^{3}} + \frac {36 \, a^{2} b d^{2} e^{\left (\frac {e}{d x + c} + 5\right )}}{{\left (d x + c\right )}^{2}} - \frac {72 \, a^{2} b c d^{2} e^{\left (\frac {e}{d x + c} + 5\right )}}{{\left (d x + c\right )}^{3}} + \frac {24 \, a^{3} d^{3} e^{\left (\frac {e}{d x + c} + 5\right )}}{{\left (d x + c\right )}^{3}} - \frac {36 \, b^{3} c^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{7}}{{\left (d x + c\right )}^{4}} + \frac {72 \, a b^{2} c d {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{7}}{{\left (d x + c\right )}^{4}} - \frac {36 \, a^{2} b d^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{7}}{{\left (d x + c\right )}^{4}} + \frac {2 \, b^{3} e^{\left (\frac {e}{d x + c} + 6\right )}}{d x + c} - \frac {12 \, b^{3} c e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{2}} + \frac {36 \, b^{3} c^{2} e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{3}} + \frac {12 \, a b^{2} d e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{2}} - \frac {72 \, a b^{2} c d e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{3}} + \frac {36 \, a^{2} b d^{2} e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{3}} + \frac {12 \, b^{3} c {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{8}}{{\left (d x + c\right )}^{4}} - \frac {12 \, a b^{2} d {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{8}}{{\left (d x + c\right )}^{4}} + \frac {b^{3} e^{\left (\frac {e}{d x + c} + 7\right )}}{{\left (d x + c\right )}^{2}} - \frac {12 \, b^{3} c e^{\left (\frac {e}{d x + c} + 7\right )}}{{\left (d x + c\right )}^{3}} + \frac {12 \, a b^{2} d e^{\left (\frac {e}{d x + c} + 7\right )}}{{\left (d x + c\right )}^{3}} - \frac {b^{3} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{9}}{{\left (d x + c\right )}^{4}} + \frac {b^{3} e^{\left (\frac {e}{d x + c} + 8\right )}}{{\left (d x + c\right )}^{3}}\right )} {\left (d x + c\right )}^{4} e^{\left (-5\right )}}{24 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 682, normalized size = 2.13 \[ -\frac {\left (\left (-\Ei \left (1, -\frac {e}{d x +c}\right )-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}\right ) a^{3}-\frac {3 \left (-\Ei \left (1, -\frac {e}{d x +c}\right )-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}\right ) a^{2} b c}{d}+\frac {3 \left (-\frac {\Ei \left (1, -\frac {e}{d x +c}\right )}{2}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\left (d x +c \right )^{2} {\mathrm e}^{\frac {e}{d x +c}}}{2 e^{2}}\right ) a^{2} b e}{d}+\frac {3 \left (-\Ei \left (1, -\frac {e}{d x +c}\right )-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}\right ) a \,b^{2} c^{2}}{d^{2}}-\frac {6 \left (-\frac {\Ei \left (1, -\frac {e}{d x +c}\right )}{2}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\left (d x +c \right )^{2} {\mathrm e}^{\frac {e}{d x +c}}}{2 e^{2}}\right ) a \,b^{2} c e}{d^{2}}+\frac {3 \left (-\frac {\Ei \left (1, -\frac {e}{d x +c}\right )}{6}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{6 e}-\frac {\left (d x +c \right )^{2} {\mathrm e}^{\frac {e}{d x +c}}}{6 e^{2}}-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{3 e^{3}}\right ) a \,b^{2} e^{2}}{d^{2}}-\frac {\left (-\Ei \left (1, -\frac {e}{d x +c}\right )-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}\right ) b^{3} c^{3}}{d^{3}}+\frac {3 \left (-\frac {\Ei \left (1, -\frac {e}{d x +c}\right )}{2}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\left (d x +c \right )^{2} {\mathrm e}^{\frac {e}{d x +c}}}{2 e^{2}}\right ) b^{3} c^{2} e}{d^{3}}-\frac {3 \left (-\frac {\Ei \left (1, -\frac {e}{d x +c}\right )}{6}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{6 e}-\frac {\left (d x +c \right )^{2} {\mathrm e}^{\frac {e}{d x +c}}}{6 e^{2}}-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{3 e^{3}}\right ) b^{3} c \,e^{2}}{d^{3}}+\frac {\left (-\frac {\Ei \left (1, -\frac {e}{d x +c}\right )}{24}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{24 e}-\frac {\left (d x +c \right )^{2} {\mathrm e}^{\frac {e}{d x +c}}}{24 e^{2}}-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{12 e^{3}}-\frac {\left (d x +c \right )^{4} {\mathrm e}^{\frac {e}{d x +c}}}{4 e^{4}}\right ) b^{3} e^{3}}{d^{3}}\right ) e}{d} \]
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 \[ \frac {{\left (6 \, b^{3} d^{3} x^{4} + 2 \, {\left (12 \, a b^{2} d^{3} + b^{3} d^{2} e\right )} x^{3} + {\left (36 \, a^{2} b d^{3} + 12 \, a b^{2} d^{2} e - {\left (6 \, c d e - d e^{2}\right )} b^{3}\right )} x^{2} + {\left (24 \, a^{3} d^{3} + 36 \, a^{2} b d^{2} e - 12 \, {\left (4 \, c d e - d e^{2}\right )} a b^{2} + {\left (18 \, c^{2} e - 10 \, c e^{2} + e^{3}\right )} b^{3}\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{24 \, d^{3}} + \int -\frac {{\left (36 \, a^{2} b c^{2} d^{2} e - 12 \, {\left (4 \, c^{3} d e - c^{2} d e^{2}\right )} a b^{2} + {\left (18 \, c^{4} e - 10 \, c^{3} e^{2} + c^{2} e^{3}\right )} b^{3} - {\left (24 \, a^{3} d^{4} e - 36 \, {\left (2 \, c d^{3} e - d^{3} e^{2}\right )} a^{2} b + 12 \, {\left (6 \, c^{2} d^{2} e - 6 \, c d^{2} e^{2} + d^{2} e^{3}\right )} a b^{2} - {\left (24 \, c^{3} d e - 36 \, c^{2} d e^{2} + 12 \, c d e^{3} - d e^{4}\right )} b^{3}\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{24 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}\,{d x} \]
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 \[ \int {\mathrm {e}}^{\frac {e}{c+d\,x}}\,{\left (a+b\,x\right )}^3 \,d x \]
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 \[ \int \left (a + b x\right )^{3} e^{\frac {e}{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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