3.966 \(\int (b x)^m (\pi +d x)^n (e+f x)^p \, dx\)

Optimal. Leaf size=49 \[ \frac{\pi ^n e^p (b x)^{m+1} F_1\left (m+1;-n,-p;m+2;-\frac{d x}{\pi },-\frac{f x}{e}\right )}{b (m+1)} \]

[Out]

(E^p*Pi^n*(b*x)^(1 + m)*AppellF1[1 + m, -n, -p, 2 + m, -((d*x)/Pi), -((f*x)/E)])/(b*(1 + m))

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Rubi [A]  time = 0.0225794, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, 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.

[In]

Int[(b*x)^m*(Pi + d*x)^n*(E + f*x)^p,x]

[Out]

(E^p*Pi^n*(b*x)^(1 + m)*AppellF1[1 + m, -n, -p, 2 + m, -((d*x)/Pi), -((f*x)/E)])/(b*(1 + m))

Rule 133

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(c^n*e^p*(b*x)^(m +
1)*AppellF1[m + 1, -n, -p, m + 2, -((d*x)/c), -((f*x)/e)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, e, f, m, n, p},
x] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])

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*}

Mathematica [A]  time = 0.0916438, 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.

[In]

Integrate[(b*x)^m*(Pi + d*x)^n*(E + f*x)^p,x]

[Out]

(E^p*Pi^n*x*(b*x)^m*AppellF1[1 + m, -n, -p, 2 + m, -((d*x)/Pi), -((f*x)/E)])/(1 + m)

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Maple [F]  time = 0.127, size = 0, normalized size = 0. \begin{align*} \int \left ( bx \right ) ^{m} \left ( dx+\pi \right ) ^{n} \left ( fx+E \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x)^m*(d*x+Pi)^n*(f*x+E)^p,x)

[Out]

int((b*x)^m*(d*x+Pi)^n*(f*x+E)^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\pi + d x\right )}^{n} \left (b x\right )^{m}{\left (f x + E\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^m*(d*x+pi)^n*(f*x+E)^p,x, algorithm="maxima")

[Out]

integrate((pi + d*x)^n*(b*x)^m*(f*x + E)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (\pi + d x\right )}^{n} \left (b x\right )^{m}{\left (f x + E\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^m*(d*x+pi)^n*(f*x+E)^p,x, algorithm="fricas")

[Out]

integral((pi + d*x)^n*(b*x)^m*(f*x + E)^p, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)**m*(d*x+pi)**n*(f*x+E)**p,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\pi + d x\right )}^{n} \left (b x\right )^{m}{\left (f x + E\right )}^{p}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^m*(d*x+pi)^n*(f*x+E)^p,x, algorithm="giac")

[Out]

integrate((pi + d*x)^n*(b*x)^m*(f*x + E)^p, x)