Optimal. Leaf size=255 \[ \frac {b e^2 (b c-a d) \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}-\frac {e (b c-a d)^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}-\frac {b e (c+d x) (b c-a d) e^{\frac {e}{c+d x}}}{d^3}-\frac {b (c+d x)^2 (b c-a d) e^{\frac {e}{c+d x}}}{d^3}+\frac {(c+d x) (b c-a d)^2 e^{\frac {e}{c+d x}}}{d^3}-\frac {b^2 e^3 \text {Ei}\left (\frac {e}{c+d x}\right )}{6 d^3}+\frac {b^2 e^2 (c+d x) e^{\frac {e}{c+d x}}}{6 d^3}+\frac {b^2 e (c+d x)^2 e^{\frac {e}{c+d x}}}{6 d^3}+\frac {b^2 (c+d x)^3 e^{\frac {e}{c+d x}}}{3 d^3} \]
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Rubi [A] time = 0.26, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2226, 2206, 2210, 2214} \[ \frac {b e^2 (b c-a d) \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}-\frac {e (b c-a d)^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}-\frac {b e (c+d x) (b c-a d) e^{\frac {e}{c+d x}}}{d^3}-\frac {b (c+d x)^2 (b c-a d) e^{\frac {e}{c+d x}}}{d^3}+\frac {(c+d x) (b c-a d)^2 e^{\frac {e}{c+d x}}}{d^3}-\frac {b^2 e^3 \text {Ei}\left (\frac {e}{c+d x}\right )}{6 d^3}+\frac {b^2 e^2 (c+d x) e^{\frac {e}{c+d x}}}{6 d^3}+\frac {b^2 e (c+d x)^2 e^{\frac {e}{c+d x}}}{6 d^3}+\frac {b^2 (c+d x)^3 e^{\frac {e}{c+d x}}}{3 d^3} \]
Antiderivative was successfully verified.
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Rule 2206
Rule 2210
Rule 2214
Rule 2226
Rubi steps
\begin {align*} \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx &=\int \left (\frac {(-b c+a d)^2 e^{\frac {e}{c+d x}}}{d^2}-\frac {2 b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)}{d^2}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^2}{d^2}\right ) \, dx\\ &=\frac {b^2 \int e^{\frac {e}{c+d x}} (c+d x)^2 \, dx}{d^2}-\frac {(2 b (b c-a d)) \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{d^2}+\frac {(b c-a d)^2 \int e^{\frac {e}{c+d x}} \, dx}{d^2}\\ &=\frac {(b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^3}{3 d^3}+\frac {\left (b^2 e\right ) \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{3 d^2}-\frac {(b (b c-a d) e) \int e^{\frac {e}{c+d x}} \, dx}{d^2}+\frac {\left ((b c-a d)^2 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{d^2}\\ &=\frac {(b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^2 e e^{\frac {e}{c+d x}} (c+d x)^2}{6 d^3}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}+\frac {\left (b^2 e^2\right ) \int e^{\frac {e}{c+d x}} \, dx}{6 d^2}-\frac {\left (b (b c-a d) e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{d^2}\\ &=\frac {(b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)}{d^3}+\frac {b^2 e^2 e^{\frac {e}{c+d x}} (c+d x)}{6 d^3}-\frac {b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^2 e e^{\frac {e}{c+d x}} (c+d x)^2}{6 d^3}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}+\frac {b (b c-a d) e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}+\frac {\left (b^2 e^3\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{6 d^2}\\ &=\frac {(b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)}{d^3}+\frac {b^2 e^2 e^{\frac {e}{c+d x}} (c+d x)}{6 d^3}-\frac {b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^2 e e^{\frac {e}{c+d x}} (c+d x)^2}{6 d^3}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}+\frac {b (b c-a d) e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}-\frac {b^2 e^3 \text {Ei}\left (\frac {e}{c+d x}\right )}{6 d^3}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 170, normalized size = 0.67 \[ \frac {d x e^{\frac {e}{c+d x}} \left (6 a^2 d^2+6 a b d (d x+e)+b^2 \left (-4 c e+2 d^2 x^2+d e x+e^2\right )\right )-e \left (6 a^2 d^2+6 a b d (e-2 c)+b^2 \left (6 c^2-6 c e+e^2\right )\right ) \text {Ei}\left (\frac {e}{c+d x}\right )}{6 d^3}+\frac {c e^{\frac {e}{c+d x}} \left (6 a^2 d^2+6 a b d (e-c)+b^2 \left (2 c^2-5 c e+e^2\right )\right )}{6 d^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 197, normalized size = 0.77 \[ -\frac {{\left (b^{2} e^{3} - 6 \, {\left (b^{2} c - a b d\right )} e^{2} + 6 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e\right )} {\rm Ei}\left (\frac {e}{d x + c}\right ) - {\left (2 \, b^{2} d^{3} x^{3} + 2 \, b^{2} c^{3} - 6 \, a b c^{2} d + 6 \, a^{2} c d^{2} + b^{2} c e^{2} + {\left (6 \, a b d^{3} + b^{2} d^{2} e\right )} x^{2} - {\left (5 \, b^{2} c^{2} - 6 \, a b c d\right )} e + {\left (6 \, a^{2} d^{3} + b^{2} d e^{2} - 2 \, {\left (2 \, b^{2} c d - 3 \, a b d^{2}\right )} e\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{6 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 424, normalized size = 1.66 \[ -\frac {{\left (\frac {6 \, b^{2} c^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{5}}{{\left (d x + c\right )}^{3}} - \frac {12 \, a b c d {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{5}}{{\left (d x + c\right )}^{3}} + \frac {6 \, a^{2} d^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{5}}{{\left (d x + c\right )}^{3}} - 2 \, b^{2} e^{\left (\frac {e}{d x + c} + 4\right )} + \frac {6 \, b^{2} c e^{\left (\frac {e}{d x + c} + 4\right )}}{d x + c} - \frac {6 \, b^{2} c^{2} e^{\left (\frac {e}{d x + c} + 4\right )}}{{\left (d x + c\right )}^{2}} - \frac {6 \, a b d e^{\left (\frac {e}{d x + c} + 4\right )}}{d x + c} + \frac {12 \, a b c d e^{\left (\frac {e}{d x + c} + 4\right )}}{{\left (d x + c\right )}^{2}} - \frac {6 \, a^{2} d^{2} e^{\left (\frac {e}{d x + c} + 4\right )}}{{\left (d x + c\right )}^{2}} - \frac {6 \, b^{2} c {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{6}}{{\left (d x + c\right )}^{3}} + \frac {6 \, a b d {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{6}}{{\left (d x + c\right )}^{3}} - \frac {b^{2} e^{\left (\frac {e}{d x + c} + 5\right )}}{d x + c} + \frac {6 \, b^{2} c e^{\left (\frac {e}{d x + c} + 5\right )}}{{\left (d x + c\right )}^{2}} - \frac {6 \, a b d e^{\left (\frac {e}{d x + c} + 5\right )}}{{\left (d x + c\right )}^{2}} + \frac {b^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{7}}{{\left (d x + c\right )}^{3}} - \frac {b^{2} e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{2}}\right )} {\left (d x + c\right )}^{3} e^{\left (-4\right )}}{6 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 356, normalized size = 1.40 \[ -\frac {\left (\left (-\Ei \left (1, -\frac {e}{d x +c}\right )-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}\right ) a^{2}-\frac {2 \left (-\Ei \left (1, -\frac {e}{d x +c}\right )-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}\right ) a b c}{d}+\frac {2 \left (-\frac {\Ei \left (1, -\frac {e}{d x +c}\right )}{2}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\left (d x +c \right )^{2} {\mathrm e}^{\frac {e}{d x +c}}}{2 e^{2}}\right ) a b e}{d}+\frac {\left (-\Ei \left (1, -\frac {e}{d x +c}\right )-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}\right ) b^{2} c^{2}}{d^{2}}-\frac {2 \left (-\frac {\Ei \left (1, -\frac {e}{d x +c}\right )}{2}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\left (d x +c \right )^{2} {\mathrm e}^{\frac {e}{d x +c}}}{2 e^{2}}\right ) b^{2} c e}{d^{2}}+\frac {\left (-\frac {\Ei \left (1, -\frac {e}{d x +c}\right )}{6}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{6 e}-\frac {\left (d x +c \right )^{2} {\mathrm e}^{\frac {e}{d x +c}}}{6 e^{2}}-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{3 e^{3}}\right ) b^{2} e^{2}}{d^{2}}\right ) e}{d} \]
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 \[ \frac {{\left (2 \, b^{2} d^{2} x^{3} + {\left (6 \, a b d^{2} + b^{2} d e\right )} x^{2} + {\left (6 \, a^{2} d^{2} + 6 \, a b d e - {\left (4 \, c e - e^{2}\right )} b^{2}\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{6 \, d^{2}} + \int -\frac {{\left (6 \, a b c^{2} d e - {\left (4 \, c^{3} e - c^{2} e^{2}\right )} b^{2} - {\left (6 \, a^{2} d^{3} e - 6 \, {\left (2 \, c d^{2} e - d^{2} e^{2}\right )} a b + {\left (6 \, c^{2} d e - 6 \, c d e^{2} + d e^{3}\right )} b^{2}\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{6 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.10, size = 306, normalized size = 1.20 \[ \frac {x\,{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (2\,a^2\,c+\frac {\frac {b^2\,c^3}{3}-d\,\left (a\,b\,c^2-2\,a\,b\,c\,e\right )+\frac {b^2\,c\,e^2}{3}-\frac {3\,b^2\,c^2\,e}{2}}{d^2}\right )+\frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (\frac {b^2\,c^4}{3}-d\,\left (a\,b\,c^3-a\,b\,c^2\,e\right )-\frac {5\,b^2\,c^3\,e}{6}+a^2\,c^2\,d^2+\frac {b^2\,c^2\,e^2}{6}\right )}{d^3}+x^2\,{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (\frac {\frac {b^2\,e^2}{6}-\frac {b^2\,c\,e}{2}}{d}+a^2\,d+a\,b\,c+a\,b\,e\right )+\frac {b^2\,d\,x^4\,{\mathrm {e}}^{\frac {e}{c+d\,x}}}{3}+\frac {b\,x^3\,{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (6\,a\,d+2\,b\,c+b\,e\right )}{6}}{c+d\,x}-\frac {\mathrm {ei}\left (\frac {e}{c+d\,x}\right )\,\left (\frac {b^2\,e^3}{6}+d\,\left (a\,b\,e^2-2\,a\,b\,c\,e\right )+a^2\,d^2\,e-b^2\,c\,e^2+b^2\,c^2\,e\right )}{d^3} \]
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 \[ \int \left (a + b x\right )^{2} e^{\frac {e}{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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