3.995 \(\int (a+b x) (A+B x) (d+e x)^3 \, dx\)

Optimal. Leaf size=77 \[ -\frac{(d+e x)^5 (-a B e-A b e+2 b B d)}{5 e^3}+\frac{(d+e x)^4 (b d-a e) (B d-A e)}{4 e^3}+\frac{b B (d+e x)^6}{6 e^3} \]

[Out]

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

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Rubi [A]  time = 0.221422, antiderivative size = 77, 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 \[ -\frac{(d+e x)^5 (-a B e-A b e+2 b B d)}{5 e^3}+\frac{(d+e x)^4 (b d-a e) (B d-A e)}{4 e^3}+\frac{b B (d+e x)^6}{6 e^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(A + B*x)*(d + e*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 25.103, size = 68, normalized size = 0.88 \[ \frac{B b \left (d + e x\right )^{6}}{6 e^{3}} + \frac{\left (d + e x\right )^{5} \left (A b e + B a e - 2 B b d\right )}{5 e^{3}} + \frac{\left (d + e x\right )^{4} \left (A e - B d\right ) \left (a e - b d\right )}{4 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(B*x+A)*(e*x+d)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0726969, size = 130, normalized size = 1.69 \[ \frac{1}{2} d^2 x^2 (3 a A e+a B d+A b d)+\frac{1}{5} e^2 x^5 (a B e+A b e+3 b B d)+\frac{1}{4} e x^4 (a e (A e+3 B d)+3 b d (A e+B d))+\frac{1}{3} d x^3 (3 a e (A e+B d)+b d (3 A e+B d))+a A d^3 x+\frac{1}{6} b B e^3 x^6 \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(A + B*x)*(d + e*x)^3,x]

[Out]

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

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Maple [A]  time = 0.002, size = 135, normalized size = 1.8 \[{\frac{bB{e}^{3}{x}^{6}}{6}}+{\frac{ \left ( \left ( Ab+Ba \right ){e}^{3}+3\,bBd{e}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( aA{e}^{3}+3\, \left ( Ab+Ba \right ) d{e}^{2}+3\,bB{d}^{2}e \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,aAd{e}^{2}+3\, \left ( Ab+Ba \right ){d}^{2}e+bB{d}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,aA{d}^{2}e+ \left ( Ab+Ba \right ){d}^{3} \right ){x}^{2}}{2}}+aA{d}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)*(e*x+d)^3,x)

[Out]

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

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Maxima [A]  time = 1.34176, size = 181, normalized size = 2.35 \[ \frac{1}{6} \, B b e^{3} x^{6} + A a d^{3} x + \frac{1}{5} \,{\left (3 \, B b d e^{2} +{\left (B a + A b\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, B b d^{2} e + A a e^{3} + 3 \,{\left (B a + A b\right )} d e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b d^{3} + 3 \, A a d e^{2} + 3 \,{\left (B a + A b\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, A a d^{2} e +{\left (B a + A b\right )} d^{3}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)*(e*x + d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.184749, size = 1, normalized size = 0.01 \[ \frac{1}{6} x^{6} e^{3} b B + \frac{3}{5} x^{5} e^{2} d b B + \frac{1}{5} x^{5} e^{3} a B + \frac{1}{5} x^{5} e^{3} b A + \frac{3}{4} x^{4} e d^{2} b B + \frac{3}{4} x^{4} e^{2} d a B + \frac{3}{4} x^{4} e^{2} d b A + \frac{1}{4} x^{4} e^{3} a A + \frac{1}{3} x^{3} d^{3} b B + x^{3} e d^{2} a B + x^{3} e d^{2} b A + x^{3} e^{2} d a A + \frac{1}{2} x^{2} d^{3} a B + \frac{1}{2} x^{2} d^{3} b A + \frac{3}{2} x^{2} e d^{2} a A + x d^{3} a A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)*(e*x + d)^3,x, algorithm="fricas")

[Out]

1/6*x^6*e^3*b*B + 3/5*x^5*e^2*d*b*B + 1/5*x^5*e^3*a*B + 1/5*x^5*e^3*b*A + 3/4*x^
4*e*d^2*b*B + 3/4*x^4*e^2*d*a*B + 3/4*x^4*e^2*d*b*A + 1/4*x^4*e^3*a*A + 1/3*x^3*
d^3*b*B + x^3*e*d^2*a*B + x^3*e*d^2*b*A + x^3*e^2*d*a*A + 1/2*x^2*d^3*a*B + 1/2*
x^2*d^3*b*A + 3/2*x^2*e*d^2*a*A + x*d^3*a*A

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Sympy [A]  time = 0.155829, size = 168, normalized size = 2.18 \[ A a d^{3} x + \frac{B b e^{3} x^{6}}{6} + x^{5} \left (\frac{A b e^{3}}{5} + \frac{B a e^{3}}{5} + \frac{3 B b d e^{2}}{5}\right ) + x^{4} \left (\frac{A a e^{3}}{4} + \frac{3 A b d e^{2}}{4} + \frac{3 B a d e^{2}}{4} + \frac{3 B b d^{2} e}{4}\right ) + x^{3} \left (A a d e^{2} + A b d^{2} e + B a d^{2} e + \frac{B b d^{3}}{3}\right ) + x^{2} \left (\frac{3 A a d^{2} e}{2} + \frac{A b d^{3}}{2} + \frac{B a d^{3}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(B*x+A)*(e*x+d)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.215268, size = 215, normalized size = 2.79 \[ \frac{1}{6} \, B b x^{6} e^{3} + \frac{3}{5} \, B b d x^{5} e^{2} + \frac{3}{4} \, B b d^{2} x^{4} e + \frac{1}{3} \, B b d^{3} x^{3} + \frac{1}{5} \, B a x^{5} e^{3} + \frac{1}{5} \, A b x^{5} e^{3} + \frac{3}{4} \, B a d x^{4} e^{2} + \frac{3}{4} \, A b d x^{4} e^{2} + B a d^{2} x^{3} e + A b d^{2} x^{3} e + \frac{1}{2} \, B a d^{3} x^{2} + \frac{1}{2} \, A b d^{3} x^{2} + \frac{1}{4} \, A a x^{4} e^{3} + A a d x^{3} e^{2} + \frac{3}{2} \, A a d^{2} x^{2} e + A a d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)*(e*x + d)^3,x, algorithm="giac")

[Out]

1/6*B*b*x^6*e^3 + 3/5*B*b*d*x^5*e^2 + 3/4*B*b*d^2*x^4*e + 1/3*B*b*d^3*x^3 + 1/5*
B*a*x^5*e^3 + 1/5*A*b*x^5*e^3 + 3/4*B*a*d*x^4*e^2 + 3/4*A*b*d*x^4*e^2 + B*a*d^2*
x^3*e + A*b*d^2*x^3*e + 1/2*B*a*d^3*x^2 + 1/2*A*b*d^3*x^2 + 1/4*A*a*x^4*e^3 + A*
a*d*x^3*e^2 + 3/2*A*a*d^2*x^2*e + A*a*d^3*x