3.1008 \(\int (a+b x)^2 (A+B x) (d+e x) \, dx\)

Optimal. Leaf size=75 \[ \frac{(a+b x)^4 (-2 a B e+A b e+b B d)}{4 b^3}+\frac{(a+b x)^3 (A b-a B) (b d-a e)}{3 b^3}+\frac{B e (a+b x)^5}{5 b^3} \]

[Out]

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

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Rubi [A]  time = 0.172595, antiderivative size = 75, 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 \[ \frac{(a+b x)^4 (-2 a B e+A b e+b B d)}{4 b^3}+\frac{(a+b x)^3 (A b-a B) (b d-a e)}{3 b^3}+\frac{B e (a+b x)^5}{5 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.0623, size = 68, normalized size = 0.91 \[ \frac{B e \left (a + b x\right )^{5}}{5 b^{3}} + \frac{\left (a + b x\right )^{4} \left (A b e - 2 B a e + B b d\right )}{4 b^{3}} - \frac{\left (a + b x\right )^{3} \left (A b - B a\right ) \left (a e - b d\right )}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0450914, size = 96, normalized size = 1.28 \[ \frac{1}{3} x^3 \left (a^2 B e+2 a A b e+2 a b B d+A b^2 d\right )+a^2 A d x+\frac{1}{4} b x^4 (2 a B e+A b e+b B d)+\frac{1}{2} a x^2 (a A e+a B d+2 A b d)+\frac{1}{5} b^2 B e x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 101, normalized size = 1.4 \[{\frac{{b}^{2}Be{x}^{5}}{5}}+{\frac{ \left ( \left ({b}^{2}A+2\,Bba \right ) e+{b}^{2}Bd \right ){x}^{4}}{4}}+{\frac{ \left ( \left ( 2\,Aab+B{a}^{2} \right ) e+ \left ({b}^{2}A+2\,Bba \right ) d \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{2}Ae+ \left ( 2\,Aab+B{a}^{2} \right ) d \right ){x}^{2}}{2}}+{a}^{2}Adx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34603, size = 135, normalized size = 1.8 \[ \frac{1}{5} \, B b^{2} e x^{5} + A a^{2} d x + \frac{1}{4} \,{\left (B b^{2} d +{\left (2 \, B a b + A b^{2}\right )} e\right )} x^{4} + \frac{1}{3} \,{\left ({\left (2 \, B a b + A b^{2}\right )} d +{\left (B a^{2} + 2 \, A a b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{2} e +{\left (B a^{2} + 2 \, A a b\right )} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.182706, size = 1, normalized size = 0.01 \[ \frac{1}{5} x^{5} e b^{2} B + \frac{1}{4} x^{4} d b^{2} B + \frac{1}{2} x^{4} e b a B + \frac{1}{4} x^{4} e b^{2} A + \frac{2}{3} x^{3} d b a B + \frac{1}{3} x^{3} e a^{2} B + \frac{1}{3} x^{3} d b^{2} A + \frac{2}{3} x^{3} e b a A + \frac{1}{2} x^{2} d a^{2} B + x^{2} d b a A + \frac{1}{2} x^{2} e a^{2} A + x d a^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.131955, size = 116, normalized size = 1.55 \[ A a^{2} d x + \frac{B b^{2} e x^{5}}{5} + x^{4} \left (\frac{A b^{2} e}{4} + \frac{B a b e}{2} + \frac{B b^{2} d}{4}\right ) + x^{3} \left (\frac{2 A a b e}{3} + \frac{A b^{2} d}{3} + \frac{B a^{2} e}{3} + \frac{2 B a b d}{3}\right ) + x^{2} \left (\frac{A a^{2} e}{2} + A a b d + \frac{B a^{2} d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.216294, size = 161, normalized size = 2.15 \[ \frac{1}{5} \, B b^{2} x^{5} e + \frac{1}{4} \, B b^{2} d x^{4} + \frac{1}{2} \, B a b x^{4} e + \frac{1}{4} \, A b^{2} x^{4} e + \frac{2}{3} \, B a b d x^{3} + \frac{1}{3} \, A b^{2} d x^{3} + \frac{1}{3} \, B a^{2} x^{3} e + \frac{2}{3} \, A a b x^{3} e + \frac{1}{2} \, B a^{2} d x^{2} + A a b d x^{2} + \frac{1}{2} \, A a^{2} x^{2} e + A a^{2} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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