Optimal. Leaf size=240 \[ \frac {b d^2 e^2 e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{2 (b c-a d)^4}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {d^2 e e^{\frac {e}{c+d x}}}{2 (b c-a d)^3}+\frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (a+b x) (b c-a d)^2}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2} \]
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Rubi [A] time = 1.04, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2223, 6742, 2222, 2210, 2228, 2178, 2209} \[ \frac {b d^2 e^2 e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{2 (b c-a d)^4}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {d^2 e e^{\frac {e}{c+d x}}}{2 (b c-a d)^3}+\frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (a+b x) (b c-a d)^2}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 2178
Rule 2209
Rule 2210
Rule 2222
Rule 2223
Rule 2228
Rule 6742
Rubi steps
\begin {align*} \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx &=-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}-\frac {(d e) \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2 (c+d x)^2} \, dx}{2 b}\\ &=-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}-\frac {(d e) \int \left (\frac {b^2 e^{\frac {e}{c+d x}}}{(b c-a d)^2 (a+b x)^2}-\frac {2 b^2 d e^{\frac {e}{c+d x}}}{(b c-a d)^3 (a+b x)}+\frac {d^2 e^{\frac {e}{c+d x}}}{(b c-a d)^2 (c+d x)^2}+\frac {2 b d^2 e^{\frac {e}{c+d x}}}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b}\\ &=-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {\left (b d^2 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx}{(b c-a d)^3}-\frac {\left (d^3 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{(b c-a d)^3}-\frac {(b d e) \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^2} \, dx}{2 (b c-a d)^2}-\frac {\left (d^3 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{(c+d x)^2} \, dx}{2 b (b c-a d)^2}\\ &=\frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {d^2 e \text {Ei}\left (\frac {e}{c+d x}\right )}{(b c-a d)^3}+\frac {\left (d^3 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{(b c-a d)^3}+\frac {\left (d^2 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{(a+b x) (c+d x)} \, dx}{(b c-a d)^2}+\frac {\left (d^2 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{(a+b x) (c+d x)^2} \, dx}{2 (b c-a d)^2}\\ &=\frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {\left (d^2 e\right ) \operatorname {Subst}\left (\int \frac {\exp \left (-\frac {b e}{-b c+a d}+\frac {d e x}{-b c+a d}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3}+\frac {\left (d^2 e^2\right ) \int \left (\frac {b^2 e^{\frac {e}{c+d x}}}{(b c-a d)^2 (a+b x)}-\frac {d e^{\frac {e}{c+d x}}}{(b c-a d) (c+d x)^2}-\frac {b d e^{\frac {e}{c+d x}}}{(b c-a d)^2 (c+d x)}\right ) \, dx}{2 (b c-a d)^2}\\ &=\frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {\left (b^2 d^2 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{a+b x} \, dx}{2 (b c-a d)^4}-\frac {\left (b d^3 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{2 (b c-a d)^4}-\frac {\left (d^3 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{(c+d x)^2} \, dx}{2 (b c-a d)^3}\\ &=\frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}+\frac {d^2 e e^{\frac {e}{c+d x}}}{2 (b c-a d)^3}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {b d^2 e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 (b c-a d)^4}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {\left (b d^3 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{2 (b c-a d)^4}+\frac {\left (b d^2 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{(a+b x) (c+d x)} \, dx}{2 (b c-a d)^3}\\ &=\frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}+\frac {d^2 e e^{\frac {e}{c+d x}}}{2 (b c-a d)^3}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {\left (b d^2 e^2\right ) \operatorname {Subst}\left (\int \frac {\exp \left (-\frac {b e}{-b c+a d}+\frac {d e x}{-b c+a d}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4}\\ &=\frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}+\frac {d^2 e e^{\frac {e}{c+d x}}}{2 (b c-a d)^3}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (b c-a d)^2 (a+b x)}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {b d^2 e^2 e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{2 (b c-a d)^4}\\ \end {align*}
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Mathematica [F] time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.42, size = 517, normalized size = 2.15 \[ \frac {{\left (a^{2} b d^{2} e^{2} + {\left (b^{3} d^{2} e^{2} + 2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e\right )} x^{2} + 2 \, {\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} e + 2 \, {\left (a b^{2} d^{2} e^{2} + 2 \, {\left (a b^{2} c d^{2} - a^{2} b d^{3}\right )} e\right )} x\right )} {\rm Ei}\left (-\frac {b d e x + a d e}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x}\right ) e^{\left (\frac {b e}{b c - a d}\right )} - {\left (b^{3} c^{4} - 4 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} - 2 \, a^{3} c d^{3} - {\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4} + {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e\right )} x^{2} - {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e - {\left (2 \, a b^{2} c^{2} d^{2} - 4 \, a^{2} b c d^{3} + 2 \, a^{3} d^{4} + {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} e\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{2 \, {\left (a^{2} b^{4} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + a^{6} d^{4} + {\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 6 \, a^{3} b^{3} c^{2} d^{2} - 4 \, a^{4} b^{2} c d^{3} + a^{5} b d^{4}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.53, size = 1759, normalized size = 7.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 240, normalized size = 1.00 \[ -\frac {\left (-\frac {\left (-\frac {\Ei \left (1, -\frac {b e}{a d -b c}-\frac {e}{d x +c}\right ) {\mathrm e}^{-\frac {b e}{a d -b c}}}{2}-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {b e}{a d -b c}+\frac {e}{d x +c}\right )^{2}}-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {b e}{a d -b c}+\frac {e}{d x +c}\right )}\right ) b \,d^{3} e}{\left (a d -b c \right )^{4}}+\frac {\left (-\Ei \left (1, -\frac {b e}{a d -b c}-\frac {e}{d x +c}\right ) {\mathrm e}^{-\frac {b e}{a d -b c}}-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{\frac {b e}{a d -b c}+\frac {e}{d x +c}}\right ) d^{3}}{\left (a d -b c \right )^{3}}\right ) e}{d} \]
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 \[ \int \frac {e^{\left (\frac {e}{d x + c}\right )}}{{\left (b x + a\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}}{{\left (a+b\,x\right )}^3} \,d x \]
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 \[ \int \frac {e^{\frac {e}{c + d x}}}{\left (a + b x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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