Optimal. Leaf size=76 \[ \frac {e^{-\frac {(1+m)^2}{4 n^2}} \sqrt {\pi } (d+e x)^{1+m} \left ((d+e x)^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {1+m+2 n \log \left ((d+e x)^n\right )}{2 n}\right )}{2 e n} \]
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Rubi [A]
time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2308, 2266,
2235} \begin {gather*} \frac {\sqrt {\pi } e^{-\frac {(m+1)^2}{4 n^2}} (d+e x)^{m+1} \left ((d+e x)^n\right )^{-\frac {m+1}{n}} \text {Erfi}\left (\frac {2 n \log \left ((d+e x)^n\right )+m+1}{2 n}\right )}{2 e n} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2266
Rule 2308
Rubi steps
\begin {align*} \int e^{\log ^2\left ((d+e x)^n\right )} (d+e x)^m \, dx &=\frac {\text {Subst}\left (\int e^{\log ^2\left (x^n\right )} x^m \, dx,x,d+e x\right )}{e}\\ &=\frac {\left ((d+e x)^{1+m} \left ((d+e x)^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int e^{\frac {(1+m) x}{n}+x^2} \, dx,x,\log \left ((d+e x)^n\right )\right )}{e n}\\ &=\frac {\left (e^{-\frac {(1+m)^2}{4 n^2}} (d+e x)^{1+m} \left ((d+e x)^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int e^{\frac {1}{4} \left (\frac {1+m}{n}+2 x\right )^2} \, dx,x,\log \left ((d+e x)^n\right )\right )}{e n}\\ &=\frac {e^{-\frac {(1+m)^2}{4 n^2}} \sqrt {\pi } (d+e x)^{1+m} \left ((d+e x)^n\right )^{-\frac {1+m}{n}} \text {erfi}\left (\frac {1+m+2 n \log \left ((d+e x)^n\right )}{2 n}\right )}{2 e n}\\ \end {align*}
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Mathematica [F]
time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int e^{\log ^2\left ((d+e x)^n\right )} (d+e x)^m \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{\ln \left (\left (e x +d \right )^{n}\right )^{2}} \left (e x +d \right )^{m}\, dx\]
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 = 0.42, size = 55, normalized size = 0.72 \begin {gather*} \frac {\sqrt {\pi } \sqrt {n^{2}} \operatorname {erfi}\left (\frac {{\left (2 \, n^{2} \log \left (x e + d\right ) + m + 1\right )} \sqrt {n^{2}}}{2 \, n^{2}}\right ) e^{\left (-\frac {m^{2} + 2 \, m + 1}{4 \, n^{2}} - 1\right )}}{2 \, n} \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 (d + e x\right )^{m} e^{\log {\left (\left (d + e x\right )^{n} \right )}^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 5.00, size = 53, normalized size = 0.70 \begin {gather*} -\frac {i \, \sqrt {\pi } \operatorname {erf}\left (i \, n \log \left (x e + d\right ) + \frac {i \, m}{2 \, n} + \frac {i}{2 \, n}\right ) e^{\left (-\frac {m^{2}}{4 \, n^{2}} - \frac {m}{2 \, n^{2}} - \frac {1}{4 \, n^{2}} - 1\right )}}{2 \, n} \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 = -0.01 \begin {gather*} \int {\mathrm {e}}^{{\ln \left ({\left (d+e\,x\right )}^n\right )}^2}\,{\left (d+e\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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