3.3155 \(\int (a+b x) (A+B x) (d+e x)^m \, dx\)

Optimal. Leaf size=90 \[ \frac{(b d-a e) (B d-A e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{(d+e x)^{m+2} (-a B e-A b e+2 b B d)}{e^3 (m+2)}+\frac{b B (d+e x)^{m+3}}{e^3 (m+3)} \]

[Out]

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

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Rubi [A]  time = 0.138235, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{(b d-a e) (B d-A e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac{(d+e x)^{m+2} (-a B e-A b e+2 b B d)}{e^3 (m+2)}+\frac{b B (d+e x)^{m+3}}{e^3 (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.6966, size = 78, normalized size = 0.87 \[ \frac{B b \left (d + e x\right )^{m + 3}}{e^{3} \left (m + 3\right )} + \frac{\left (d + e x\right )^{m + 1} \left (A e - B d\right ) \left (a e - b d\right )}{e^{3} \left (m + 1\right )} + \frac{\left (d + e x\right )^{m + 2} \left (A b e + B a e - 2 B b d\right )}{e^{3} \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(B*x+A)*(e*x+d)**m,x)

[Out]

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

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Mathematica [A]  time = 0.13304, size = 103, normalized size = 1.14 \[ \frac{(d+e x)^{m+1} \left (a e (m+3) (A e (m+2)-B d+B e (m+1) x)+b \left (A e (m+3) (e (m+1) x-d)+B \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )\right )}{e^3 (m+1) (m+2) (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.008, size = 189, normalized size = 2.1 \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ( Bb{e}^{2}{m}^{2}{x}^{2}+Ab{e}^{2}{m}^{2}x+Ba{e}^{2}{m}^{2}x+3\,Bb{e}^{2}m{x}^{2}+Aa{e}^{2}{m}^{2}+4\,Ab{e}^{2}mx+4\,Ba{e}^{2}mx-2\,Bbdemx+2\,bB{x}^{2}{e}^{2}+5\,Aa{e}^{2}m-Abdem+3\,Ab{e}^{2}x-Badem+3\,Ba{e}^{2}x-2\,Bbdex+6\,Aa{e}^{2}-3\,Abde-3\,Bade+2\,Bb{d}^{2} \right ) }{{e}^{3} \left ({m}^{3}+6\,{m}^{2}+11\,m+6 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)*(e*x+d)^m,x)

[Out]

(e*x+d)^(1+m)*(B*b*e^2*m^2*x^2+A*b*e^2*m^2*x+B*a*e^2*m^2*x+3*B*b*e^2*m*x^2+A*a*e
^2*m^2+4*A*b*e^2*m*x+4*B*a*e^2*m*x-2*B*b*d*e*m*x+2*B*b*e^2*x^2+5*A*a*e^2*m-A*b*d
*e*m+3*A*b*e^2*x-B*a*d*e*m+3*B*a*e^2*x-2*B*b*d*e*x+6*A*a*e^2-3*A*b*d*e-3*B*a*d*e
+2*B*b*d^2)/e^3/(m^3+6*m^2+11*m+6)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266113, size = 346, normalized size = 3.84 \[ \frac{{\left (A a d e^{2} m^{2} + 2 \, B b d^{3} + 6 \, A a d e^{2} - 3 \,{\left (B a + A b\right )} d^{2} e +{\left (B b e^{3} m^{2} + 3 \, B b e^{3} m + 2 \, B b e^{3}\right )} x^{3} +{\left (3 \,{\left (B a + A b\right )} e^{3} +{\left (B b d e^{2} +{\left (B a + A b\right )} e^{3}\right )} m^{2} +{\left (B b d e^{2} + 4 \,{\left (B a + A b\right )} e^{3}\right )} m\right )} x^{2} +{\left (5 \, A a d e^{2} -{\left (B a + A b\right )} d^{2} e\right )} m +{\left (6 \, A a e^{3} +{\left (A a e^{3} +{\left (B a + A b\right )} d e^{2}\right )} m^{2} -{\left (2 \, B b d^{2} e - 5 \, A a e^{3} - 3 \,{\left (B a + A b\right )} d e^{2}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*a*d*e^2*m^2 + 2*B*b*d^3 + 6*A*a*d*e^2 - 3*(B*a + A*b)*d^2*e + (B*b*e^3*m^2 +
3*B*b*e^3*m + 2*B*b*e^3)*x^3 + (3*(B*a + A*b)*e^3 + (B*b*d*e^2 + (B*a + A*b)*e^3
)*m^2 + (B*b*d*e^2 + 4*(B*a + A*b)*e^3)*m)*x^2 + (5*A*a*d*e^2 - (B*a + A*b)*d^2*
e)*m + (6*A*a*e^3 + (A*a*e^3 + (B*a + A*b)*d*e^2)*m^2 - (2*B*b*d^2*e - 5*A*a*e^3
- 3*(B*a + A*b)*d*e^2)*m)*x)*(e*x + d)^m/(e^3*m^3 + 6*e^3*m^2 + 11*e^3*m + 6*e^
3)

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Sympy [A]  time = 3.96574, size = 1952, normalized size = 21.69 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(B*x+A)*(e*x+d)**m,x)

[Out]

Piecewise((d**m*(A*a*x + A*b*x**2/2 + B*a*x**2/2 + B*b*x**3/3), Eq(e, 0)), (-A*a
*d*e**2/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2) + A*b*e**3*x**2/(2*d**3*e*
*3 + 4*d**2*e**4*x + 2*d*e**5*x**2) + B*a*e**3*x**2/(2*d**3*e**3 + 4*d**2*e**4*x
+ 2*d*e**5*x**2) + 2*B*b*d**3*log(d/e + x)/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e
**5*x**2) + B*b*d**3/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2) + 4*B*b*d**2*
e*x*log(d/e + x)/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2) + 2*B*b*d*e**2*x*
*2*log(d/e + x)/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2) - 2*B*b*d*e**2*x**
2/(2*d**3*e**3 + 4*d**2*e**4*x + 2*d*e**5*x**2), Eq(m, -3)), (-A*a*e**2/(d*e**3
+ e**4*x) + A*b*d*e*log(d/e + x)/(d*e**3 + e**4*x) + A*b*d*e/(d*e**3 + e**4*x) +
A*b*e**2*x*log(d/e + x)/(d*e**3 + e**4*x) + B*a*d*e*log(d/e + x)/(d*e**3 + e**4
*x) + B*a*d*e/(d*e**3 + e**4*x) + B*a*e**2*x*log(d/e + x)/(d*e**3 + e**4*x) - 2*
B*b*d**2*log(d/e + x)/(d*e**3 + e**4*x) - 2*B*b*d**2/(d*e**3 + e**4*x) - 2*B*b*d
*e*x*log(d/e + x)/(d*e**3 + e**4*x) + B*b*e**2*x**2/(d*e**3 + e**4*x), Eq(m, -2)
), (A*a*log(d/e + x)/e - A*b*d*log(d/e + x)/e**2 + A*b*x/e - B*a*d*log(d/e + x)/
e**2 + B*a*x/e + B*b*d**2*log(d/e + x)/e**3 - B*b*d*x/e**2 + B*b*x**2/(2*e), Eq(
m, -1)), (A*a*d*e**2*m**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*
e**3) + 5*A*a*d*e**2*m*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**
3) + 6*A*a*d*e**2*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) +
A*a*e**3*m**2*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 5*
A*a*e**3*m*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 6*A*a
*e**3*x*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - A*b*d**2*e
*m*(d + e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - 3*A*b*d**2*e*(d
+ e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + A*b*d*e**2*m**2*x*(d
+ e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 3*A*b*d*e**2*m*x*(d
+ e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + A*b*e**3*m**2*x**2*(d
+ e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 4*A*b*e**3*m*x**2*(d
+ e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 3*A*b*e**3*x**2*(d +
e*x)**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - B*a*d**2*e*m*(d + e*x)
**m/(e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) - 3*B*a*d**2*e*(d + e*x)**m/(
e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + B*a*d*e**2*m**2*x*(d + e*x)**m/(
e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 3*B*a*d*e**2*m*x*(d + e*x)**m/(e
**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + B*a*e**3*m**2*x**2*(d + e*x)**m/(
e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 4*B*a*e**3*m*x**2*(d + e*x)**m/(
e**3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 3*B*a*e**3*x**2*(d + e*x)**m/(e*
*3*m**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 2*B*b*d**3*(d + e*x)**m/(e**3*m**3
+ 6*e**3*m**2 + 11*e**3*m + 6*e**3) - 2*B*b*d**2*e*m*x*(d + e*x)**m/(e**3*m**3
+ 6*e**3*m**2 + 11*e**3*m + 6*e**3) + B*b*d*e**2*m**2*x**2*(d + e*x)**m/(e**3*m*
*3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + B*b*d*e**2*m*x**2*(d + e*x)**m/(e**3*m*
*3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + B*b*e**3*m**2*x**3*(d + e*x)**m/(e**3*m
**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 3*B*b*e**3*m*x**3*(d + e*x)**m/(e**3*m
**3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3) + 2*B*b*e**3*x**3*(d + e*x)**m/(e**3*m**
3 + 6*e**3*m**2 + 11*e**3*m + 6*e**3), True))

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GIAC/XCAS [A]  time = 0.235472, size = 744, normalized size = 8.27 \[ \frac{B b m^{2} x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + B b d m^{2} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + B a m^{2} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + A b m^{2} x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 3 \, B b m x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + B a d m^{2} x e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + A b d m^{2} x e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + B b d m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - 2 \, B b d^{2} m x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + A a m^{2} x e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 4 \, B a m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 4 \, A b m x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 2 \, B b x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + A a d m^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 3 \, B a d m x e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 3 \, A b d m x e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - B a d^{2} m e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} - A b d^{2} m e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + 2 \, B b d^{3} e^{\left (m{\rm ln}\left (x e + d\right )\right )} + 5 \, A a m x e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 3 \, B a x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 3 \, A b x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 5 \, A a d m e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} - 3 \, B a d^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} - 3 \, A b d^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + 6 \, A a x e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 6 \, A a d e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )}}{m^{3} e^{3} + 6 \, m^{2} e^{3} + 11 \, m e^{3} + 6 \, e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(B*b*m^2*x^3*e^(m*ln(x*e + d) + 3) + B*b*d*m^2*x^2*e^(m*ln(x*e + d) + 2) + B*a*m
^2*x^2*e^(m*ln(x*e + d) + 3) + A*b*m^2*x^2*e^(m*ln(x*e + d) + 3) + 3*B*b*m*x^3*e
^(m*ln(x*e + d) + 3) + B*a*d*m^2*x*e^(m*ln(x*e + d) + 2) + A*b*d*m^2*x*e^(m*ln(x
*e + d) + 2) + B*b*d*m*x^2*e^(m*ln(x*e + d) + 2) - 2*B*b*d^2*m*x*e^(m*ln(x*e + d
) + 1) + A*a*m^2*x*e^(m*ln(x*e + d) + 3) + 4*B*a*m*x^2*e^(m*ln(x*e + d) + 3) + 4
*A*b*m*x^2*e^(m*ln(x*e + d) + 3) + 2*B*b*x^3*e^(m*ln(x*e + d) + 3) + A*a*d*m^2*e
^(m*ln(x*e + d) + 2) + 3*B*a*d*m*x*e^(m*ln(x*e + d) + 2) + 3*A*b*d*m*x*e^(m*ln(x
*e + d) + 2) - B*a*d^2*m*e^(m*ln(x*e + d) + 1) - A*b*d^2*m*e^(m*ln(x*e + d) + 1)
+ 2*B*b*d^3*e^(m*ln(x*e + d)) + 5*A*a*m*x*e^(m*ln(x*e + d) + 3) + 3*B*a*x^2*e^(
m*ln(x*e + d) + 3) + 3*A*b*x^2*e^(m*ln(x*e + d) + 3) + 5*A*a*d*m*e^(m*ln(x*e + d
) + 2) - 3*B*a*d^2*e^(m*ln(x*e + d) + 1) - 3*A*b*d^2*e^(m*ln(x*e + d) + 1) + 6*A
*a*x*e^(m*ln(x*e + d) + 3) + 6*A*a*d*e^(m*ln(x*e + d) + 2))/(m^3*e^3 + 6*m^2*e^3
+ 11*m*e^3 + 6*e^3)