Optimal. Leaf size=49 \[ \frac {\pi ^n e^p (b x)^{m+1} F_1\left (m+1;-n,-p;m+2;-\frac {d x}{\pi },-\frac {f x}{e}\right )}{b (m+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {133} \[ \frac {\pi ^n e^p (b x)^{m+1} F_1\left (m+1;-n,-p;m+2;-\frac {d x}{\pi },-\frac {f x}{e}\right )}{b (m+1)} \]
Antiderivative was successfully verified.
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Rule 133
Rubi steps
\begin {align*} \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx &=\frac {e^p \pi ^n (b x)^{1+m} F_1\left (1+m;-n,-p;2+m;-\frac {d x}{\pi },-\frac {f x}{e}\right )}{b (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 45, normalized size = 0.92 \[ \frac {\pi ^n e^p x (b x)^m F_1\left (m+1;-n,-p;m+2;-\frac {d x}{\pi },-\frac {f x}{e}\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\pi + d x\right )}^{n} \left (b x\right )^{m} {\left (f x + E\right )}^{p}, x\right ) \]
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 \[ \int {\left (\pi + d x\right )}^{n} \left (b x\right )^{m} {\left (f x + E\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \left (b x \right )^{m} \left (f x +E \right )^{p} \left (d x +\pi \right )^{n}\, dx \]
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 \[ \int {\left (\pi + d x\right )}^{n} \left (b x\right )^{m} {\left (f x + E\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\mathrm {e}+f\,x\right )}^p\,{\left (b\,x\right )}^m\,{\left (\Pi +d\,x\right )}^n \,d x \]
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 \[ \int \left (b x\right )^{m} \left (d x + \pi \right )^{n} \left (f x + e\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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