Optimal. Leaf size=320 \[ -\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}-\frac {b^2 (b c-a d) e^2 e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {(b c-a d)^3 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^4}-\frac {3 b (b c-a d)^2 e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^2 (b c-a d) e^3 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4} \]
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Rubi [A]
time = 0.21, 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 = {2258, 2237,
2241, 2245, 2250} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2237
Rule 2241
Rule 2245
Rule 2250
Rule 2258
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 = 0.24, size = 292, normalized size = 0.91 \begin {gather*} -\frac {c \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 ) e^{\frac {e}{c+d x}}}{24 d^4}+\frac {d e^{\frac {e}{c+d x}} x \left (24 a^3 d^3+36 a^2 b d^2 (e+d x)+12 a b^2 d \left (-4 c e+e^2+d e x+2 d^2 x^2\right )+b^3 \left (18 c^2 e+e^3+d e^2 x+2 d^2 e x^2+6 d^3 x^3-2 c e (5 e+3 d x)\right )\right )-e \left (24 a^3 d^3+36 a^2 b d^2 (-2 c+e)+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} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(681\) vs.
\(2(306)=612\).
time = 0.07, size = 682, normalized size = 2.13 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.08, size = 370, normalized size = 1.16 \begin {gather*} -\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} + 24 \, a b^{2} d^{4} x^{3} + 36 \, a^{2} b d^{4} x^{2} + 24 \, a^{3} d^{4} x - 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} + {\left (b^{3} d x + b^{3} c\right )} e^{3} + {\left (b^{3} d^{2} x^{2} - 11 \, b^{3} c^{2} + 12 \, a b^{2} c d - 2 \, {\left (5 \, b^{3} c d - 6 \, a b^{2} d^{2}\right )} x\right )} e^{2} + 2 \, {\left (b^{3} d^{3} x^{3} + 13 \, b^{3} c^{3} - 30 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 3 \, {\left (b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (3 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + 6 \, a^{2} b d^{3}\right )} x\right )} e\right )} e^{\left (\frac {e}{d x + c}\right )}}{24 \, d^{4}} \end {gather*}
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 \begin {gather*} \int \left (a + b x\right )^{3} e^{\frac {e}{c + d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 830 vs.
\(2 (316) = 632\).
time = 1.27, size = 830, normalized size = 2.59 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \int {\mathrm {e}}^{\frac {e}{c+d\,x}}\,{\left (a+b\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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