3.1899 \(\int (a+b x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=38 \[ \frac{(a+b x)^4 (b d-a e)}{4 b^2}+\frac{e (a+b x)^5}{5 b^2} \]

[Out]

((b*d - a*e)*(a + b*x)^4)/(4*b^2) + (e*(a + b*x)^5)/(5*b^2)

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Rubi [A]  time = 0.0453112, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{(a+b x)^4 (b d-a e)}{4 b^2}+\frac{e (a+b x)^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)*(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

((b*d - a*e)*(a + b*x)^4)/(4*b^2) + (e*(a + b*x)^5)/(5*b^2)

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Rubi in Sympy [A]  time = 20.6913, size = 31, normalized size = 0.82 \[ \frac{e \left (a + b x\right )^{5}}{5 b^{2}} - \frac{\left (a + b x\right )^{4} \left (a e - b d\right )}{4 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(e*x+d)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

e*(a + b*x)**5/(5*b**2) - (a + b*x)**4*(a*e - b*d)/(4*b**2)

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Mathematica [A]  time = 0.0203039, size = 67, normalized size = 1.76 \[ a^3 d x+\frac{1}{2} a^2 x^2 (a e+3 b d)+\frac{1}{4} b^2 x^4 (3 a e+b d)+a b x^3 (a e+b d)+\frac{1}{5} b^3 e x^5 \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)*(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

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Maple [B]  time = 0.002, size = 94, normalized size = 2.5 \[{\frac{{b}^{3}e{x}^{5}}{5}}+{\frac{ \left ( \left ( ae+bd \right ){b}^{2}+2\,ae{b}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( a{b}^{2}d+2\, \left ( ae+bd \right ) ab+{a}^{2}be \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{2}bd+ \left ( ae+bd \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{3}dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(e*x+d)*(b^2*x^2+2*a*b*x+a^2),x)

[Out]

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

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Maxima [A]  time = 0.70932, size = 93, normalized size = 2.45 \[ \frac{1}{5} \, b^{3} e x^{5} + a^{3} d x + \frac{1}{4} \,{\left (b^{3} d + 3 \, a b^{2} e\right )} x^{4} +{\left (a b^{2} d + a^{2} b e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d + a^{3} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*(e*x + d),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.25916, size = 1, normalized size = 0.03 \[ \frac{1}{5} x^{5} e b^{3} + \frac{1}{4} x^{4} d b^{3} + \frac{3}{4} x^{4} e b^{2} a + x^{3} d b^{2} a + x^{3} e b a^{2} + \frac{3}{2} x^{2} d b a^{2} + \frac{1}{2} x^{2} e a^{3} + x d a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*(e*x + d),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.127732, size = 73, normalized size = 1.92 \[ a^{3} d x + \frac{b^{3} e x^{5}}{5} + x^{4} \left (\frac{3 a b^{2} e}{4} + \frac{b^{3} d}{4}\right ) + x^{3} \left (a^{2} b e + a b^{2} d\right ) + x^{2} \left (\frac{a^{3} e}{2} + \frac{3 a^{2} b d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(e*x+d)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.306851, size = 103, normalized size = 2.71 \[ \frac{1}{5} \, b^{3} x^{5} e + \frac{1}{4} \, b^{3} d x^{4} + \frac{3}{4} \, a b^{2} x^{4} e + a b^{2} d x^{3} + a^{2} b x^{3} e + \frac{3}{2} \, a^{2} b d x^{2} + \frac{1}{2} \, a^{3} x^{2} e + a^{3} d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)*(b*x + a)*(e*x + d),x, algorithm="giac")

[Out]

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