Optimal. Leaf size=65 \[ \frac {\pi ^n (b x)^{m+1} (e+f x)^p \left (\frac {f x}{e}+1\right )^{-p} F_1\left (m+1;-n,-p;m+2;-\frac {d x}{\pi },-\frac {f x}{e}\right )}{b (m+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {135, 133} \[ \frac {\pi ^n (b x)^{m+1} (e+f x)^p \left (\frac {f x}{e}+1\right )^{-p} F_1\left (m+1;-n,-p;m+2;-\frac {d x}{\pi },-\frac {f x}{e}\right )}{b (m+1)} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rubi steps
\begin {align*} \int (b x)^m (\pi +d x)^n (e+f x)^p \, dx &=\left ((e+f x)^p \left (1+\frac {f x}{e}\right )^{-p}\right ) \int (b x)^m (\pi +d x)^n \left (1+\frac {f x}{e}\right )^p \, dx\\ &=\frac {\pi ^n (b x)^{1+m} (e+f x)^p \left (1+\frac {f x}{e}\right )^{-p} 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 = 62, normalized size = 0.95 \[ \frac {\pi ^n x (b x)^m (e+f x)^p \left (\frac {e+f x}{e}\right )^{-p} F_1\left (m+1;-n,-p;m+2;-\frac {d x}{\pi },-\frac {f x}{e}\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, 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 = 0.14, 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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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 \[ \int {\left (e+f\,x\right )}^p\,{\left (b\,x\right )}^m\,{\left (\Pi +d\,x\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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