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

Optimal. Leaf size=108 \[ \frac{x^{n+1} (e x)^m (a A d+a B c+A b c)}{m+n+1}+\frac{x^{2 n+1} (e x)^m (a B d+A b d+b B c)}{m+2 n+1}+\frac{a A c (e x)^{m+1}}{e (m+1)}+\frac{b B d x^{3 n+1} (e x)^m}{m+3 n+1} \]

[Out]

((A*b*c + a*B*c + a*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + ((b*B*c + A*b*d + a*B*d)*x^(1 + 2*n)*(e*x)^m)/(1 + m
+ 2*n) + (b*B*d*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (a*A*c*(e*x)^(1 + m))/(e*(1 + m))

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Rubi [A]  time = 0.0838475, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {570, 20, 30} \[ \frac{x^{n+1} (e x)^m (a A d+a B c+A b c)}{m+n+1}+\frac{x^{2 n+1} (e x)^m (a B d+A b d+b B c)}{m+2 n+1}+\frac{a A c (e x)^{m+1}}{e (m+1)}+\frac{b B d x^{3 n+1} (e x)^m}{m+3 n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b*c + a*B*c + a*A*d)*x^(1 + n)*(e*x)^m)/(1 + m + n) + ((b*B*c + A*b*d + a*B*d)*x^(1 + 2*n)*(e*x)^m)/(1 + m
+ 2*n) + (b*B*d*x^(1 + 3*n)*(e*x)^m)/(1 + m + 3*n) + (a*A*c*(e*x)^(1 + m))/(e*(1 + m))

Rule 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int (e x)^m \left (a+b x^n\right ) \left (A+B x^n\right ) \left (c+d x^n\right ) \, dx &=\int \left (a A c (e x)^m+(A b c+a B c+a A d) x^n (e x)^m+(b B c+A b d+a B d) x^{2 n} (e x)^m+b B d x^{3 n} (e x)^m\right ) \, dx\\ &=\frac{a A c (e x)^{1+m}}{e (1+m)}+(b B d) \int x^{3 n} (e x)^m \, dx+(A b c+a B c+a A d) \int x^n (e x)^m \, dx+(b B c+A b d+a B d) \int x^{2 n} (e x)^m \, dx\\ &=\frac{a A c (e x)^{1+m}}{e (1+m)}+\left (b B d x^{-m} (e x)^m\right ) \int x^{m+3 n} \, dx+\left ((A b c+a B c+a A d) x^{-m} (e x)^m\right ) \int x^{m+n} \, dx+\left ((b B c+A b d+a B d) x^{-m} (e x)^m\right ) \int x^{m+2 n} \, dx\\ &=\frac{(A b c+a B c+a A d) x^{1+n} (e x)^m}{1+m+n}+\frac{(b B c+A b d+a B d) x^{1+2 n} (e x)^m}{1+m+2 n}+\frac{b B d x^{1+3 n} (e x)^m}{1+m+3 n}+\frac{a A c (e x)^{1+m}}{e (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.146838, size = 84, normalized size = 0.78 \[ x (e x)^m \left (\frac{x^n (a A d+a B c+A b c)}{m+n+1}+\frac{x^{2 n} (a B d+A b d+b B c)}{m+2 n+1}+\frac{a A c}{m+1}+\frac{b B d x^{3 n}}{m+3 n+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(e*x)^m*((a*A*c)/(1 + m) + ((A*b*c + a*B*c + a*A*d)*x^n)/(1 + m + n) + ((b*B*c + A*b*d + a*B*d)*x^(2*n))/(1
+ m + 2*n) + (b*B*d*x^(3*n))/(1 + m + 3*n))

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Maple [C]  time = 0.056, size = 891, normalized size = 8.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(3*A*a*c*m+d*b*(x^n)^2*A+d*a*(x^n)^2*B+c*b*(x^n)^2*B+d*a*x^n*A+c*b*x^n*A+3*B*a*d*m^2*(x^n)^2+3*B*a*d*n^2*(x^
n)^2+3*B*b*c*m^2*(x^n)^2+3*B*b*c*n^2*(x^n)^2+6*A*b*c*m*n^2*x^n+5*A*a*d*m^2*n*x^n+6*A*a*d*m*n^2*x^n+5*A*b*c*m^2
*n*x^n+c*a*B*x^n+d*b*(x^n)^3*B+10*A*a*d*m*n*x^n+10*A*b*c*m*n*x^n+B*b*d*m^3*(x^n)^3+A*b*d*m^3*(x^n)^2+B*a*d*m^3
*(x^n)^2+B*b*c*m^3*(x^n)^2+3*A*x^n*b*c*m+5*A*x^n*b*c*n+3*B*x^n*a*c*m+5*B*x^n*a*c*n+4*B*(x^n)^2*a*d*n+3*B*(x^n)
^2*b*c*m+4*B*(x^n)^2*b*c*n+3*A*x^n*a*d*m+5*A*x^n*a*d*n+3*A*(x^n)^2*b*d*m+4*A*(x^n)^2*b*d*n+3*B*(x^n)^2*a*d*m+3
*B*(x^n)^3*b*d*m+3*B*(x^n)^3*b*d*n+B*a*c*m^3*x^n+a*A*c+3*B*a*c*m^2*x^n+6*B*a*c*n^2*x^n+10*B*a*c*m*n*x^n+8*B*b*
c*m*n*(x^n)^2+6*B*b*d*m*n*(x^n)^3+4*B*b*c*m^2*n*(x^n)^2+3*B*b*c*m*n^2*(x^n)^2+3*B*a*d*m*n^2*(x^n)^2+3*A*b*d*m*
n^2*(x^n)^2+4*B*a*d*m^2*n*(x^n)^2+4*A*b*d*m^2*n*(x^n)^2+2*B*b*d*m*n^2*(x^n)^3+3*B*b*d*m^2*n*(x^n)^3+3*A*a*d*m^
2*x^n+6*A*a*d*n^2*x^n+3*A*b*c*m^2*x^n+6*A*b*c*n^2*x^n+6*A*a*c*m^2*n+11*A*a*c*m*n^2+12*A*a*c*m*n+3*B*b*d*m^2*(x
^n)^3+2*B*b*d*n^2*(x^n)^3+A*a*d*m^3*x^n+A*b*c*m^3*x^n+3*A*b*d*m^2*(x^n)^2+3*A*b*d*n^2*(x^n)^2+6*A*a*c*n^3+A*a*
c*m^3+3*A*a*c*m^2+11*A*a*c*n^2+6*a*A*c*n+5*B*a*c*m^2*n*x^n+6*B*a*c*m*n^2*x^n+8*B*a*d*m*n*(x^n)^2+8*A*b*d*m*n*(
x^n)^2)/(1+m)/(m+n+1)/(1+m+2*n)/(1+m+3*n)*exp(1/2*m*(-I*Pi*csgn(I*e*x)^3+I*Pi*csgn(I*e*x)^2*csgn(I*e)+I*Pi*csg
n(I*e*x)^2*csgn(I*x)-I*Pi*csgn(I*e*x)*csgn(I*e)*csgn(I*x)+2*ln(e)+2*ln(x)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.13548, size = 1354, normalized size = 12.54 \begin{align*} \frac{{\left (B b d m^{3} + 3 \, B b d m^{2} + 3 \, B b d m + B b d + 2 \,{\left (B b d m + B b d\right )} n^{2} + 3 \,{\left (B b d m^{2} + 2 \, B b d m + B b d\right )} n\right )} x x^{3 \, n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} +{\left ({\left (B b c +{\left (B a + A b\right )} d\right )} m^{3} + B b c + 3 \,{\left (B b c +{\left (B a + A b\right )} d\right )} m^{2} + 3 \,{\left (B b c +{\left (B a + A b\right )} d +{\left (B b c +{\left (B a + A b\right )} d\right )} m\right )} n^{2} +{\left (B a + A b\right )} d + 3 \,{\left (B b c +{\left (B a + A b\right )} d\right )} m + 4 \,{\left (B b c +{\left (B b c +{\left (B a + A b\right )} d\right )} m^{2} +{\left (B a + A b\right )} d + 2 \,{\left (B b c +{\left (B a + A b\right )} d\right )} m\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} +{\left ({\left (A a d +{\left (B a + A b\right )} c\right )} m^{3} + A a d + 3 \,{\left (A a d +{\left (B a + A b\right )} c\right )} m^{2} + 6 \,{\left (A a d +{\left (B a + A b\right )} c +{\left (A a d +{\left (B a + A b\right )} c\right )} m\right )} n^{2} +{\left (B a + A b\right )} c + 3 \,{\left (A a d +{\left (B a + A b\right )} c\right )} m + 5 \,{\left (A a d +{\left (A a d +{\left (B a + A b\right )} c\right )} m^{2} +{\left (B a + A b\right )} c + 2 \,{\left (A a d +{\left (B a + A b\right )} c\right )} m\right )} n\right )} x x^{n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} +{\left (A a c m^{3} + 6 \, A a c n^{3} + 3 \, A a c m^{2} + 3 \, A a c m + A a c + 11 \,{\left (A a c m + A a c\right )} n^{2} + 6 \,{\left (A a c m^{2} + 2 \, A a c m + A a c\right )} n\right )} x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )}}{m^{4} + 6 \,{\left (m + 1\right )} n^{3} + 4 \, m^{3} + 11 \,{\left (m^{2} + 2 \, m + 1\right )} n^{2} + 6 \, m^{2} + 6 \,{\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n + 4 \, m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*b*d*m^3 + 3*B*b*d*m^2 + 3*B*b*d*m + B*b*d + 2*(B*b*d*m + B*b*d)*n^2 + 3*(B*b*d*m^2 + 2*B*b*d*m + B*b*d)*n)
*x*x^(3*n)*e^(m*log(e) + m*log(x)) + ((B*b*c + (B*a + A*b)*d)*m^3 + B*b*c + 3*(B*b*c + (B*a + A*b)*d)*m^2 + 3*
(B*b*c + (B*a + A*b)*d + (B*b*c + (B*a + A*b)*d)*m)*n^2 + (B*a + A*b)*d + 3*(B*b*c + (B*a + A*b)*d)*m + 4*(B*b
*c + (B*b*c + (B*a + A*b)*d)*m^2 + (B*a + A*b)*d + 2*(B*b*c + (B*a + A*b)*d)*m)*n)*x*x^(2*n)*e^(m*log(e) + m*l
og(x)) + ((A*a*d + (B*a + A*b)*c)*m^3 + A*a*d + 3*(A*a*d + (B*a + A*b)*c)*m^2 + 6*(A*a*d + (B*a + A*b)*c + (A*
a*d + (B*a + A*b)*c)*m)*n^2 + (B*a + A*b)*c + 3*(A*a*d + (B*a + A*b)*c)*m + 5*(A*a*d + (A*a*d + (B*a + A*b)*c)
*m^2 + (B*a + A*b)*c + 2*(A*a*d + (B*a + A*b)*c)*m)*n)*x*x^n*e^(m*log(e) + m*log(x)) + (A*a*c*m^3 + 6*A*a*c*n^
3 + 3*A*a*c*m^2 + 3*A*a*c*m + A*a*c + 11*(A*a*c*m + A*a*c)*n^2 + 6*(A*a*c*m^2 + 2*A*a*c*m + A*a*c)*n)*x*e^(m*l
og(e) + m*log(x)))/(m^4 + 6*(m + 1)*n^3 + 4*m^3 + 11*(m^2 + 2*m + 1)*n^2 + 6*m^2 + 6*(m^3 + 3*m^2 + 3*m + 1)*n
+ 4*m + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.12256, size = 1742, normalized size = 16.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*b*d*m^3*x*x^m*x^(3*n)*e^m + 3*B*b*d*m^2*n*x*x^m*x^(3*n)*e^m + 2*B*b*d*m*n^2*x*x^m*x^(3*n)*e^m + B*b*c*m^3*x
*x^m*x^(2*n)*e^m + B*a*d*m^3*x*x^m*x^(2*n)*e^m + A*b*d*m^3*x*x^m*x^(2*n)*e^m + 4*B*b*c*m^2*n*x*x^m*x^(2*n)*e^m
+ 4*B*a*d*m^2*n*x*x^m*x^(2*n)*e^m + 4*A*b*d*m^2*n*x*x^m*x^(2*n)*e^m + 3*B*b*c*m*n^2*x*x^m*x^(2*n)*e^m + 3*B*a
*d*m*n^2*x*x^m*x^(2*n)*e^m + 3*A*b*d*m*n^2*x*x^m*x^(2*n)*e^m + B*a*c*m^3*x*x^m*x^n*e^m + A*b*c*m^3*x*x^m*x^n*e
^m + A*a*d*m^3*x*x^m*x^n*e^m + 5*B*a*c*m^2*n*x*x^m*x^n*e^m + 5*A*b*c*m^2*n*x*x^m*x^n*e^m + 5*A*a*d*m^2*n*x*x^m
*x^n*e^m + 6*B*a*c*m*n^2*x*x^m*x^n*e^m + 6*A*b*c*m*n^2*x*x^m*x^n*e^m + 6*A*a*d*m*n^2*x*x^m*x^n*e^m + A*a*c*m^3
*x*x^m*e^m + 6*A*a*c*m^2*n*x*x^m*e^m + 11*A*a*c*m*n^2*x*x^m*e^m + 6*A*a*c*n^3*x*x^m*e^m + 3*B*b*d*m^2*x*x^m*x^
(3*n)*e^m + 6*B*b*d*m*n*x*x^m*x^(3*n)*e^m + 2*B*b*d*n^2*x*x^m*x^(3*n)*e^m + 3*B*b*c*m^2*x*x^m*x^(2*n)*e^m + 3*
B*a*d*m^2*x*x^m*x^(2*n)*e^m + 3*A*b*d*m^2*x*x^m*x^(2*n)*e^m + 8*B*b*c*m*n*x*x^m*x^(2*n)*e^m + 8*B*a*d*m*n*x*x^
m*x^(2*n)*e^m + 8*A*b*d*m*n*x*x^m*x^(2*n)*e^m + 3*B*b*c*n^2*x*x^m*x^(2*n)*e^m + 3*B*a*d*n^2*x*x^m*x^(2*n)*e^m
+ 3*A*b*d*n^2*x*x^m*x^(2*n)*e^m + 3*B*a*c*m^2*x*x^m*x^n*e^m + 3*A*b*c*m^2*x*x^m*x^n*e^m + 3*A*a*d*m^2*x*x^m*x^
n*e^m + 10*B*a*c*m*n*x*x^m*x^n*e^m + 10*A*b*c*m*n*x*x^m*x^n*e^m + 10*A*a*d*m*n*x*x^m*x^n*e^m + 6*B*a*c*n^2*x*x
^m*x^n*e^m + 6*A*b*c*n^2*x*x^m*x^n*e^m + 6*A*a*d*n^2*x*x^m*x^n*e^m + 3*A*a*c*m^2*x*x^m*e^m + 12*A*a*c*m*n*x*x^
m*e^m + 11*A*a*c*n^2*x*x^m*e^m + 3*B*b*d*m*x*x^m*x^(3*n)*e^m + 3*B*b*d*n*x*x^m*x^(3*n)*e^m + 3*B*b*c*m*x*x^m*x
^(2*n)*e^m + 3*B*a*d*m*x*x^m*x^(2*n)*e^m + 3*A*b*d*m*x*x^m*x^(2*n)*e^m + 4*B*b*c*n*x*x^m*x^(2*n)*e^m + 4*B*a*d
*n*x*x^m*x^(2*n)*e^m + 4*A*b*d*n*x*x^m*x^(2*n)*e^m + 3*B*a*c*m*x*x^m*x^n*e^m + 3*A*b*c*m*x*x^m*x^n*e^m + 3*A*a
*d*m*x*x^m*x^n*e^m + 5*B*a*c*n*x*x^m*x^n*e^m + 5*A*b*c*n*x*x^m*x^n*e^m + 5*A*a*d*n*x*x^m*x^n*e^m + 3*A*a*c*m*x
*x^m*e^m + 6*A*a*c*n*x*x^m*e^m + B*b*d*x*x^m*x^(3*n)*e^m + B*b*c*x*x^m*x^(2*n)*e^m + B*a*d*x*x^m*x^(2*n)*e^m +
A*b*d*x*x^m*x^(2*n)*e^m + B*a*c*x*x^m*x^n*e^m + A*b*c*x*x^m*x^n*e^m + A*a*d*x*x^m*x^n*e^m + A*a*c*x*x^m*e^m)/
(m^4 + 6*m^3*n + 11*m^2*n^2 + 6*m*n^3 + 4*m^3 + 18*m^2*n + 22*m*n^2 + 6*n^3 + 6*m^2 + 18*m*n + 11*n^2 + 4*m +
6*n + 1)