3.3.76 \(\int (e x)^m (c+d x^n)^q (a x^j+b x^{j+n})^p \, dx\) [276]

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

[Out]

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

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Rubi [A]
time = 0.15, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2067, 525, 524} \begin {gather*} \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} \end {gather*}

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 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 2067

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])

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*}

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Mathematica [A]
time = 0.15, size = 111, normalized size = 0.98 \begin {gather*} \frac {x (e x)^m \left (x^j \left (a+b x^n\right )\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+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 {gather*}

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 = 1.55, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (c +d \,x^{n}\right )^{q} \left (a \,x^{j}+b \,x^{j +n}\right )^{p}\, dx\]

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 = 0.00, 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*(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*(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*(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*(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: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 6438 deep

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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*(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*(x*e)^m, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a\,x^j+b\,x^{j+n}\right )}^p\,{\left (e\,x\right )}^m\,{\left (c+d\,x^n\right )}^q \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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