3.276 \(\int (e x)^m (c+d x^n)^q (a x^j+b x^{j+n})^p \, dx\)

Optimal. Leaf size=113 \[ \frac{x (e x)^m \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} \left (a x^j+b x^{j+n}\right )^p F_1\left (\frac{m+j p+1}{n};-p,-q;\frac{m+n+j p+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{j p+m+1} \]

[Out]

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

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Rubi [A]  time = 0.223829, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2042, 511, 510} \[ \frac{x (e x)^m \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} \left (a x^j+b x^{j+n}\right )^p F_1\left (\frac{m+j p+1}{n};-p,-q;\frac{m+n+j p+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{j p+m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2042

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

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

Mathematica [A]  time = 0.16384, size = 111, normalized size = 0.98 \[ \frac{x (e x)^m \left (\frac{b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac{d x^n}{c}+1\right )^{-q} \left (x^j \left (a+b x^n\right )\right )^p F_1\left (\frac{m+j p+1}{n};-p,-q;\frac{m+n+j p+1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{j p+m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 2.108, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( c+d{x}^{n} \right ) ^{q} \left ( a{x}^{j}+b{x}^{j+n} \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{j + n} + a x^{j}\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*(c+d*x^n)^q*(a*x^j+b*x^(j+n))^p,x, algorithm="maxima")

[Out]

integrate((b*x^(j + n) + a*x^j)^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^{j + n} + a x^{j}\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*(c+d*x^n)^q*(a*x^j+b*x^(j+n))^p,x, algorithm="fricas")

[Out]

integral((b*x^(j + n) + a*x^j)^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*(c+d*x**n)**q*(a*x**j+b*x**(j+n))**p,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x^{j + n} + a x^{j}\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*(c+d*x^n)^q*(a*x^j+b*x^(j+n))^p,x, algorithm="giac")

[Out]

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