Optimal. Leaf size=153 \[ -\frac {e^{e (c+d x)^3}}{b (a+b x)}-\frac {d (b c-a d) e (c+d x) \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{b^3 \sqrt [3]{-e (c+d x)^3}}-\frac {d e (c+d x)^2 \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{b^2 \left (-e (c+d x)^3\right )^{2/3}}+\frac {3 d (b c-a d)^2 e \text {Int}\left (\frac {e^{e (c+d x)^3}}{a+b x},x\right )}{b^3} \]
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Rubi [A]
time = 0.24, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx &=-\frac {e^{e (c+d x)^3}}{b (a+b x)}+\frac {(3 d e) \int \frac {e^{e (c+d x)^3} (c+d x)^2}{a+b x} \, dx}{b}\\ &=-\frac {e^{e (c+d x)^3}}{b (a+b x)}+\frac {(3 d e) \int \left (\frac {d (b c-a d) e^{e (c+d x)^3}}{b^2}+\frac {(b c-a d)^2 e^{e (c+d x)^3}}{b^2 (a+b x)}+\frac {d e^{e (c+d x)^3} (c+d x)}{b}\right ) \, dx}{b}\\ &=-\frac {e^{e (c+d x)^3}}{b (a+b x)}+\frac {\left (3 d^2 e\right ) \int e^{e (c+d x)^3} (c+d x) \, dx}{b^2}+\frac {\left (3 d^2 (b c-a d) e\right ) \int e^{e (c+d x)^3} \, dx}{b^3}+\frac {\left (3 d (b c-a d)^2 e\right ) \int \frac {e^{e (c+d x)^3}}{a+b x} \, dx}{b^3}\\ &=-\frac {e^{e (c+d x)^3}}{b (a+b x)}-\frac {d (b c-a d) e (c+d x) \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{b^3 \sqrt [3]{-e (c+d x)^3}}-\frac {d e (c+d x)^2 \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{b^2 \left (-e (c+d x)^3\right )^{2/3}}+\frac {\left (3 d (b c-a d)^2 e\right ) \int \frac {e^{e (c+d x)^3}}{a+b x} \, dx}{b^3}\\ \end {align*}
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Mathematica [A]
time = 2.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{e (c+d x)^3}}{(a+b x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{e \left (d x +c \right )^{3}}}{\left (b x +a \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {e}}^{e\,{\left (c+d\,x\right )}^3}}{{\left (a+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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