3.41 \(\int (e x)^m (a+b x^n)^p (A+B x^n) (c+d x^n)^q \, dx\)

Optimal. Leaf size=211 \[ \frac{A (e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} F_1\left (\frac{m+1}{n};-p,-q;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{e (m+1)}+\frac{B x^{n+1} (e x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} F_1\left (\frac{m+n+1}{n};-p,-q;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{m+n+1} \]

[Out]

(A*(e*x)^(1 + m)*(a + b*x^n)^p*(c + d*x^n)^q*AppellF1[(1 + m)/n, -p, -q, (1 + m + n)/n, -((b*x^n)/a), -((d*x^n
)/c)])/(e*(1 + m)*(1 + (b*x^n)/a)^p*(1 + (d*x^n)/c)^q) + (B*x^(1 + n)*(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*Appe
llF1[(1 + m + n)/n, -p, -q, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)])/((1 + m + n)*(1 + (b*x^n)/a)^p*(1 +
(d*x^n)/c)^q)

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Rubi [A]  time = 0.245998, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {598, 511, 510} \[ \frac{A (e x)^{m+1} \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} F_1\left (\frac{m+1}{n};-p,-q;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{e (m+1)}+\frac{B x^{n+1} (e x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} F_1\left (\frac{m+n+1}{n};-p,-q;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{m+n+1} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(a + b*x^n)^p*(A + B*x^n)*(c + d*x^n)^q,x]

[Out]

(A*(e*x)^(1 + m)*(a + b*x^n)^p*(c + d*x^n)^q*AppellF1[(1 + m)/n, -p, -q, (1 + m + n)/n, -((b*x^n)/a), -((d*x^n
)/c)])/(e*(1 + m)*(1 + (b*x^n)/a)^p*(1 + (d*x^n)/c)^q) + (B*x^(1 + n)*(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*Appe
llF1[(1 + m + n)/n, -p, -q, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)])/((1 + m + n)*(1 + (b*x^n)/a)^p*(1 +
(d*x^n)/c)^q)

Rule 598

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Dist[e, Int[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[(f*(g*x)^m)/x^m, Int[x^(m + n)*(a +
b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q}, x]

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin{align*} \int (e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right ) \left (c+d x^n\right )^q \, dx &=A \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx+\left (B x^{-m} (e x)^m\right ) \int x^{m+n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx\\ &=\left (A \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p}\right ) \int (e x)^m \left (1+\frac{b x^n}{a}\right )^p \left (c+d x^n\right )^q \, dx+\left (B x^{-m} (e x)^m \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p}\right ) \int x^{m+n} \left (1+\frac{b x^n}{a}\right )^p \left (c+d x^n\right )^q \, dx\\ &=\left (A \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac{d x^n}{c}\right )^{-q}\right ) \int (e x)^m \left (1+\frac{b x^n}{a}\right )^p \left (1+\frac{d x^n}{c}\right )^q \, dx+\left (B x^{-m} (e x)^m \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac{d x^n}{c}\right )^{-q}\right ) \int x^{m+n} \left (1+\frac{b x^n}{a}\right )^p \left (1+\frac{d x^n}{c}\right )^q \, dx\\ &=\frac{A (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac{d x^n}{c}\right )^{-q} F_1\left (\frac{1+m}{n};-p,-q;\frac{1+m+n}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{e (1+m)}+\frac{B x^{1+n} (e x)^m \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac{d x^n}{c}\right )^{-q} F_1\left (\frac{1+m+n}{n};-p,-q;\frac{1+m+2 n}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{1+m+n}\\ \end{align*}

Mathematica [A]  time = 0.439774, size = 162, normalized size = 0.77 \[ \frac{x (e x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} \left (A (m+n+1) F_1\left (\frac{m+1}{n};-p,-q;\frac{m+n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )+B (m+1) x^n F_1\left (\frac{m+n+1}{n};-p,-q;\frac{m+2 n+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )}{(m+1) (m+n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*(a + b*x^n)^p*(A + B*x^n)*(c + d*x^n)^q,x]

[Out]

(x*(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(A*(1 + m + n)*AppellF1[(1 + m)/n, -p, -q, (1 + m + n)/n, -((b*x^n)/a),
-((d*x^n)/c)] + B*(1 + m)*x^n*AppellF1[(1 + m + n)/n, -p, -q, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)]))/
((1 + m)*(1 + m + n)*(1 + (b*x^n)/a)^p*(1 + (d*x^n)/c)^q)

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Maple [F]  time = 1.056, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( A+B{x}^{n} \right ) \left ( c+d{x}^{n} \right ) ^{q}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(a+b*x^n)^p*(A+B*x^n)*(c+d*x^n)^q,x)

[Out]

int((e*x)^m*(a+b*x^n)^p*(A+B*x^n)*(c+d*x^n)^q,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{q} \left (e x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(a+b*x^n)^p*(A+B*x^n)*(c+d*x^n)^q,x, algorithm="maxima")

[Out]

integrate((B*x^n + A)*(b*x^n + a)^p*(d*x^n + c)^q*(e*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{q} \left (e x\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(a+b*x^n)^p*(A+B*x^n)*(c+d*x^n)^q,x, algorithm="fricas")

[Out]

integral((B*x^n + A)*(b*x^n + a)^p*(d*x^n + c)^q*(e*x)^m, 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((e*x)**m*(a+b*x**n)**p*(A+B*x**n)*(c+d*x**n)**q,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B x^{n} + A\right )}{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{q} \left (e x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(a+b*x^n)^p*(A+B*x^n)*(c+d*x^n)^q,x, algorithm="giac")

[Out]

integrate((B*x^n + A)*(b*x^n + a)^p*(d*x^n + c)^q*(e*x)^m, x)