Optimal. Leaf size=65 \[ \frac {\pi ^n (b x)^{1+m} (e+f x)^p \left (1+\frac {f x}{e}\right )^{-p} F_1\left (1+m;-n,-p;2+m;-\frac {d x}{\pi },-\frac {f x}{e}\right )}{b (1+m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, 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 = {140, 138}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 138
Rule 140
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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.10, size = 62, normalized size = 0.95 \begin {gather*} \frac {\pi ^n x (b x)^m (e+f x)^p \left (\frac {e+f x}{e}\right )^{-p} F_1\left (1+m;-n,-p;2+m;-\frac {d x}{\pi },-\frac {f x}{e}\right )}{1+m} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.03, 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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (e+f\,x\right )}^p\,{\left (b\,x\right )}^m\,{\left (\Pi +d\,x\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________