Optimal. Leaf size=125 \[ \frac {x (e x)^m \left (c x^n\right )^{-\frac {m+1}{n}} \exp \left (\frac {(m+1) \left (-4 a b d^2 n+m+1\right )}{4 b^2 d^2 n^2}\right ) \text {erf}\left (\frac {-2 a b d^2 n-2 b^2 d^2 n \log \left (c x^n\right )+m+1}{2 b d n}\right )}{m+1}+\frac {(e x)^{m+1} \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)} \]
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Rubi [A] time = 0.31, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {6401, 2278, 2274, 15, 20, 2276, 2234, 2205} \[ \frac {x (e x)^m \left (c x^n\right )^{-\frac {m+1}{n}} \exp \left (\frac {(m+1) \left (-4 a b d^2 n+m+1\right )}{4 b^2 d^2 n^2}\right ) \text {Erf}\left (\frac {-2 a b d^2 n-2 b^2 d^2 n \log \left (c x^n\right )+m+1}{2 b d n}\right )}{m+1}+\frac {(e x)^{m+1} \text {Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 20
Rule 2205
Rule 2234
Rule 2274
Rule 2276
Rule 2278
Rule 6401
Rubi steps
\begin {align*} \int (e x)^m \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac {(e x)^{1+m} \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(2 b d n) \int e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2} (e x)^m \, dx}{(1+m) \sqrt {\pi }}\\ &=\frac {(e x)^{1+m} \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(2 b d n) \int \exp \left (-a^2 d^2-2 a b d^2 \log \left (c x^n\right )-b^2 d^2 \log ^2\left (c x^n\right )\right ) (e x)^m \, dx}{(1+m) \sqrt {\pi }}\\ &=\frac {(e x)^{1+m} \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(2 b d n) \int e^{-a^2 d^2-b^2 d^2 \log ^2\left (c x^n\right )} (e x)^m \left (c x^n\right )^{-2 a b d^2} \, dx}{(1+m) \sqrt {\pi }}\\ &=\frac {(e x)^{1+m} \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (2 b d n x^{2 a b d^2 n} \left (c x^n\right )^{-2 a b d^2}\right ) \int e^{-a^2 d^2-b^2 d^2 \log ^2\left (c x^n\right )} x^{-2 a b d^2 n} (e x)^m \, dx}{(1+m) \sqrt {\pi }}\\ &=\frac {(e x)^{1+m} \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (2 b d n x^{-m+2 a b d^2 n} (e x)^m \left (c x^n\right )^{-2 a b d^2}\right ) \int e^{-a^2 d^2-b^2 d^2 \log ^2\left (c x^n\right )} x^{m-2 a b d^2 n} \, dx}{(1+m) \sqrt {\pi }}\\ &=\frac {(e x)^{1+m} \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (2 b d x (e x)^m \left (c x^n\right )^{-2 a b d^2-\frac {1+m-2 a b d^2 n}{n}}\right ) \operatorname {Subst}\left (\int \exp \left (-a^2 d^2+\frac {\left (1+m-2 a b d^2 n\right ) x}{n}-b^2 d^2 x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{(1+m) \sqrt {\pi }}\\ &=\frac {(e x)^{1+m} \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (2 b d \exp \left (\frac {(1+m) \left (1+m-4 a b d^2 n\right )}{4 b^2 d^2 n^2}\right ) x (e x)^m \left (c x^n\right )^{-2 a b d^2-\frac {1+m-2 a b d^2 n}{n}}\right ) \operatorname {Subst}\left (\int \exp \left (-\frac {\left (\frac {1+m-2 a b d^2 n}{n}-2 b^2 d^2 x\right )^2}{4 b^2 d^2}\right ) \, dx,x,\log \left (c x^n\right )\right )}{(1+m) \sqrt {\pi }}\\ &=\frac {(e x)^{1+m} \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}+\frac {\exp \left (\frac {(1+m) \left (1+m-4 a b d^2 n\right )}{4 b^2 d^2 n^2}\right ) x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {erf}\left (\frac {1+m-2 a b d^2 n-2 b^2 d^2 n \log \left (c x^n\right )}{2 b d n}\right )}{1+m}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 127, normalized size = 1.02 \[ \frac {(e x)^m \left (x \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x^{-m} \text {erf}\left (a d-\frac {-2 b^2 d^2 n \log \left (c x^n\right )+m+1}{2 b d n}\right ) \exp \left (\frac {(m+1) \left (-4 a b d^2 n-4 b^2 d^2 n \log \left (c x^n\right )+4 b^2 d^2 n^2 \log (x)+m+1\right )}{4 b^2 d^2 n^2}\right )\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 180, normalized size = 1.44 \[ \frac {x \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) e^{\left (m \log \relax (e) + m \log \relax (x)\right )} - \sqrt {b^{2} d^{2} n^{2}} \operatorname {erf}\left (\frac {{\left (2 \, b^{2} d^{2} n^{2} \log \relax (x) + 2 \, b^{2} d^{2} n \log \relax (c) + 2 \, a b d^{2} n - m - 1\right )} \sqrt {b^{2} d^{2} n^{2}}}{2 \, b^{2} d^{2} n^{2}}\right ) e^{\left (\frac {4 \, b^{2} d^{2} m n^{2} \log \relax (e) - 4 \, {\left (b^{2} d^{2} m + b^{2} d^{2}\right )} n \log \relax (c) + m^{2} - 4 \, {\left (a b d^{2} m + a b d^{2}\right )} n + 2 \, m + 1}{4 \, b^{2} d^{2} n^{2}}\right )}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.74, size = 156, normalized size = 1.25 \[ \frac {x^{m + 1} \operatorname {erf}\left (b d n \log \relax (x) + b d \log \relax (c) + a d\right ) e^{m}}{m + 1} + \frac {\pi \operatorname {erf}\left (-b d n \log \relax (x) - b d \log \relax (c) - a d + \frac {m}{2 \, b d n} + \frac {1}{2 \, b d n}\right ) e^{\left (m - \frac {a m}{b n} - \frac {a}{b n} + \frac {m^{2}}{4 \, b^{2} d^{2} n^{2}} + \frac {m}{2 \, b^{2} d^{2} n^{2}} + \frac {1}{4 \, b^{2} d^{2} n^{2}}\right )}}{{\left (\pi + \pi m\right )} c^{\frac {m}{n}} c^{\left (\frac {1}{n}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \erf \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
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 \[ \frac {e^{m} x x^{m} \operatorname {erf}\left (b d \log \left (x^{n}\right ) + {\left (b \log \relax (c) + a\right )} d\right )}{m + 1} - \frac {-\frac {\sqrt {\pi } c^{2 \, a b d^{2}} e^{m} \operatorname {erf}\left (-b d n \log \relax (x) - b d \log \relax (c) - a d + \frac {m}{2 \, b d n} + \frac {1}{2 \, b d n}\right ) e^{\left (-\frac {a m}{b n} - \frac {a}{b n} + \frac {m^{2}}{4 \, b^{2} d^{2} n^{2}} + \frac {m}{2 \, b^{2} d^{2} n^{2}} + \frac {1}{4 \, b^{2} d^{2} n^{2}}\right )}}{c^{\frac {m}{n}} c^{\left (\frac {1}{n}\right )}}}{\sqrt {\pi } c^{2 \, a b d^{2}} {\left (m + 1\right )}} \]
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 \[ \int \mathrm {erf}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \]
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 \[ \int \left (e x\right )^{m} \operatorname {erf}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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