3.10.66 \(\int (b x)^m (\pi +d x)^n (e+f x)^p \, dx\) [966]

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

[Out]

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

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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 = {138} \begin {gather*} \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)} \end {gather*}

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 138

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

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Mathematica [A]
time = 0.10, size = 45, normalized size = 0.92 \begin {gather*} \frac {e^p \pi ^n x (b x)^m 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]

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 = 0.04, size = 0, normalized size = 0.00 \[\int \left (b x \right )^{m} \left (d x +\pi \right )^{n} \left ({\mathrm e}+f x \right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x)^m*(d*x+Pi)^n*(exp(1)+f*x)^p,x)

[Out]

int((b*x)^m*(d*x+Pi)^n*(exp(1)+f*x)^p,x)

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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]

integrate((b*x)^m*(d*x+pi)^n*(exp(1)+f*x)^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 = 0.00, 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]

integrate((b*x)^m*(d*x+pi)^n*(exp(1)+f*x)^p,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b x\right )^{m} \left (d x + \pi \right )^{n} \left (f x + e\right )^{p}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)**m*(d*x+pi)**n*(exp(1)+f*x)**p,x)

[Out]

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

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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]

integrate((b*x)^m*(d*x+pi)^n*(exp(1)+f*x)^p,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (\mathrm {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]

int((exp(1) + f*x)^p*(b*x)^m*(Pi + d*x)^n,x)

[Out]

int((exp(1) + f*x)^p*(b*x)^m*(Pi + d*x)^n, x)

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