3.1.41 \(\int (e x)^m (a+b x^n)^p (A+B x^n) (c+d x^n)^q \, dx\) [41]

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

[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*AppellF1((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 = 0.16, antiderivative size = 211, normalized size of antiderivative = 1.00, 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 = {612, 525, 524} \begin {gather*} \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} \end {gather*}

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 524

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

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[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 612

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]

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*}

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Mathematica [A]
time = 0.31, size = 162, normalized size = 0.77 \begin {gather*} \frac {x (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} \left (A (1+m+n) 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 )+B (1+m) x^n 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 )\right )}{(1+m) (1+m+n)} \end {gather*}

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 = 0.33, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (a +b \,x^{n}\right )^{p} \left (A +B \,x^{n}\right ) \left (c +d \,x^{n}\right )^{q}\, dx\]

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 = 0.00, 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((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*(x*e)^m, 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((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*(x*e)^m, x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

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]

Exception raised: HeuristicGCDFailed >> no luck

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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((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*(x*e)^m, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,{\left (a+b\,x^n\right )}^p\,{\left (c+d\,x^n\right )}^q \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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