Optimal. Leaf size=56 \[ -\frac {(a+b x)^{m+1} \left (\log (f) \left (-(a+b x)^n\right )\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-(a+b x)^n \log (f)\right )}{b n} \]
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Rubi [A] time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2218} \[ -\frac {(a+b x)^{m+1} \left (\log (f) \left (-(a+b x)^n\right )\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},\log (f) \left (-(a+b x)^n\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 2218
Rubi steps
\begin {align*} \int f^{(a+b x)^n} (a+b x)^m \, dx &=-\frac {(a+b x)^{1+m} \Gamma \left (\frac {1+m}{n},-(a+b x)^n \log (f)\right ) \left (-(a+b x)^n \log (f)\right )^{-\frac {1+m}{n}}}{b n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 56, normalized size = 1.00 \[ -\frac {(a+b x)^{m+1} \left (\log (f) \left (-(a+b x)^n\right )\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-(a+b x)^n \log (f)\right )}{b n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{m} f^{\left ({\left (b x + a\right )}^{n}\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{m} f^{\left ({\left (b x + a\right )}^{n}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int f^{\left (b x +a \right )^{n}} \left (b x +a \right )^{m}\, dx \]
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 \[ \int {\left (b x + a\right )}^{m} f^{\left ({\left (b x + a\right )}^{n}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.01, size = 94, normalized size = 1.68 \[ \frac {f^{{\left (a+b\,x\right )}^n}\,{\mathrm {e}}^{-\frac {\ln \relax (f)\,{\left (a+b\,x\right )}^n}{2}}\,{\left (a+b\,x\right )}^{m+1}\,{\mathrm {M}}_{1-\frac {m+n+1}{2\,n},\frac {m+n+1}{2\,n}-\frac {1}{2}}\left (\ln \relax (f)\,{\left (a+b\,x\right )}^n\right )}{b\,\left (m+1\right )\,{\left (\ln \relax (f)\,{\left (a+b\,x\right )}^n\right )}^{\frac {m+n+1}{2\,n}}} \]
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 \[ \int f^{\left (a + b x\right )^{n}} \left (a + b x\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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