3.11 \(\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx\)

Optimal. Leaf size=102 \[ \frac{c x^{n+1} (e x)^m (2 A d+B c)}{m+n+1}+\frac{d x^{2 n+1} (e x)^m (A d+2 B c)}{m+2 n+1}+\frac{A c^2 (e x)^{m+1}}{e (m+1)}+\frac{B d^2 x^{3 n+1} (e x)^m}{m+3 n+1} \]

[Out]

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

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Rubi [A]  time = 0.187333, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{c x^{n+1} (e x)^m (2 A d+B c)}{m+n+1}+\frac{d x^{2 n+1} (e x)^m (A d+2 B c)}{m+2 n+1}+\frac{A c^2 (e x)^{m+1}}{e (m+1)}+\frac{B d^2 x^{3 n+1} (e x)^m}{m+3 n+1} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^m*(A + B*x^n)*(c + d*x^n)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 33.3986, size = 119, normalized size = 1.17 \[ \frac{A c^{2} \left (e x\right )^{m + 1}}{e \left (m + 1\right )} + \frac{B d^{2} x^{3 n} \left (e x\right )^{- 3 n} \left (e x\right )^{m + 3 n + 1}}{e \left (m + 3 n + 1\right )} + \frac{c x^{n} \left (e x\right )^{- n} \left (e x\right )^{m + n + 1} \left (2 A d + B c\right )}{e \left (m + n + 1\right )} + \frac{d x^{- m} x^{m + 2 n + 1} \left (e x\right )^{m} \left (A d + 2 B c\right )}{m + 2 n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(A+B*x**n)*(c+d*x**n)**2,x)

[Out]

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

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Mathematica [A]  time = 0.148942, size = 78, normalized size = 0.76 \[ x (e x)^m \left (\frac{d x^{2 n} (A d+2 B c)}{m+2 n+1}+\frac{c x^n (2 A d+B c)}{m+n+1}+\frac{A c^2}{m+1}+\frac{B d^2 x^{3 n}}{m+3 n+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(e*x)^m*(A + B*x^n)*(c + d*x^n)^2,x]

[Out]

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

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Maple [C]  time = 0.084, size = 732, normalized size = 7.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(A+B*x^n)*(c+d*x^n)^2,x)

[Out]

x*(B*d^2*m^3*(x^n)^3+A*d^2*m^3*(x^n)^2+3*B*d^2*m^2*(x^n)^3+3*A*c^2*m+6*A*c^2*n+2
*B*d^2*n^2*(x^n)^3+8*B*c*d*(x^n)^2*n+6*A*c*d*x^n*m+4*A*d^2*(x^n)^2*n+3*B*c^2*m^2
*x^n+6*B*c^2*n^2*x^n+3*B*c^2*x^n*m+5*B*c^2*x^n*n+2*B*c*d*(x^n)^2+2*A*c*d*x^n+A*c
^2+B*c^2*x^n+10*A*c*d*x^n*n+3*A*d^2*m^2*(x^n)^2+3*A*d^2*n^2*(x^n)^2+B*c^2*m^3*x^
n+3*m*B*d^2*(x^n)^3+3*B*d^2*(x^n)^3*n+3*A*d^2*(x^n)^2*m+12*A*c*d*m*n^2*x^n+16*B*
c*d*m*n*(x^n)^2+8*B*c*d*m^2*n*(x^n)^2+20*A*c*d*m*n*x^n+6*B*c*d*m*n^2*(x^n)^2+10*
A*c*d*m^2*n*x^n+A*c^2*m^3+6*A*c^2*n^3+3*A*c^2*m^2+11*A*c^2*n^2+6*B*c*d*n^2*(x^n)
^2+6*A*c*d*m^2*x^n+12*A*c*d*n^2*x^n+10*B*c^2*m*n*x^n+6*B*c*d*(x^n)^2*m+3*B*d^2*m
^2*n*(x^n)^3+2*B*d^2*m*n^2*(x^n)^3+8*A*d^2*m*n*(x^n)^2+5*B*c^2*m^2*n*x^n+6*B*c^2
*m*n^2*x^n+6*B*c*d*m^2*(x^n)^2+4*A*d^2*m^2*n*(x^n)^2+3*A*d^2*m*n^2*(x^n)^2+2*B*c
*d*m^3*(x^n)^2+6*B*d^2*m*n*(x^n)^3+2*A*c*d*m^3*x^n+B*d^2*(x^n)^3+A*d^2*(x^n)^2+6
*A*c^2*m^2*n+11*A*c^2*m*n^2+12*A*c^2*m*n)/(1+m)/(1+m+n)/(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*csgn(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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(d*x^n + c)^2*(e*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23835, size = 711, normalized size = 6.97 \[ \frac{{\left (B d^{2} m^{3} + 3 \, B d^{2} m^{2} + 3 \, B d^{2} m + B d^{2} + 2 \,{\left (B d^{2} m + B d^{2}\right )} n^{2} + 3 \,{\left (B d^{2} m^{2} + 2 \, B d^{2} m + B d^{2}\right )} n\right )} x x^{3 \, n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} +{\left ({\left (2 \, B c d + A d^{2}\right )} m^{3} + 2 \, B c d + A d^{2} + 3 \,{\left (2 \, B c d + A d^{2}\right )} m^{2} + 3 \,{\left (2 \, B c d + A d^{2} +{\left (2 \, B c d + A d^{2}\right )} m\right )} n^{2} + 3 \,{\left (2 \, B c d + A d^{2}\right )} m + 4 \,{\left (2 \, B c d + A d^{2} +{\left (2 \, B c d + A d^{2}\right )} m^{2} + 2 \,{\left (2 \, B c d + A d^{2}\right )} m\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} +{\left ({\left (B c^{2} + 2 \, A c d\right )} m^{3} + B c^{2} + 2 \, A c d + 3 \,{\left (B c^{2} + 2 \, A c d\right )} m^{2} + 6 \,{\left (B c^{2} + 2 \, A c d +{\left (B c^{2} + 2 \, A c d\right )} m\right )} n^{2} + 3 \,{\left (B c^{2} + 2 \, A c d\right )} m + 5 \,{\left (B c^{2} + 2 \, A c d +{\left (B c^{2} + 2 \, A c d\right )} m^{2} + 2 \,{\left (B c^{2} + 2 \, A c d\right )} m\right )} n\right )} x x^{n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} +{\left (A c^{2} m^{3} + 6 \, A c^{2} n^{3} + 3 \, A c^{2} m^{2} + 3 \, A c^{2} m + A c^{2} + 11 \,{\left (A c^{2} m + A c^{2}\right )} n^{2} + 6 \,{\left (A c^{2} m^{2} + 2 \, A c^{2} m + A c^{2}\right )} n\right )} x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )}}{m^{4} + 6 \,{\left (m + 1\right )} n^{3} + 4 \, m^{3} + 11 \,{\left (m^{2} + 2 \, m + 1\right )} n^{2} + 6 \, m^{2} + 6 \,{\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n + 4 \, m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(d*x^n + c)^2*(e*x)^m,x, algorithm="fricas")

[Out]

((B*d^2*m^3 + 3*B*d^2*m^2 + 3*B*d^2*m + B*d^2 + 2*(B*d^2*m + B*d^2)*n^2 + 3*(B*d
^2*m^2 + 2*B*d^2*m + B*d^2)*n)*x*x^(3*n)*e^(m*log(e) + m*log(x)) + ((2*B*c*d + A
*d^2)*m^3 + 2*B*c*d + A*d^2 + 3*(2*B*c*d + A*d^2)*m^2 + 3*(2*B*c*d + A*d^2 + (2*
B*c*d + A*d^2)*m)*n^2 + 3*(2*B*c*d + A*d^2)*m + 4*(2*B*c*d + A*d^2 + (2*B*c*d +
A*d^2)*m^2 + 2*(2*B*c*d + A*d^2)*m)*n)*x*x^(2*n)*e^(m*log(e) + m*log(x)) + ((B*c
^2 + 2*A*c*d)*m^3 + B*c^2 + 2*A*c*d + 3*(B*c^2 + 2*A*c*d)*m^2 + 6*(B*c^2 + 2*A*c
*d + (B*c^2 + 2*A*c*d)*m)*n^2 + 3*(B*c^2 + 2*A*c*d)*m + 5*(B*c^2 + 2*A*c*d + (B*
c^2 + 2*A*c*d)*m^2 + 2*(B*c^2 + 2*A*c*d)*m)*n)*x*x^n*e^(m*log(e) + m*log(x)) + (
A*c^2*m^3 + 6*A*c^2*n^3 + 3*A*c^2*m^2 + 3*A*c^2*m + A*c^2 + 11*(A*c^2*m + A*c^2)
*n^2 + 6*(A*c^2*m^2 + 2*A*c^2*m + A*c^2)*n)*x*e^(m*log(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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(A+B*x**n)*(c+d*x**n)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214565, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(d*x^n + c)^2*(e*x)^m,x, algorithm="giac")

[Out]

Done