3.1022 \(\int (a+b x)^3 (A+B x) (d+e x) \, dx\)

Optimal. Leaf size=75 \[ \frac{(a+b x)^5 (-2 a B e+A b e+b B d)}{5 b^3}+\frac{(a+b x)^4 (A b-a B) (b d-a e)}{4 b^3}+\frac{B e (a+b x)^6}{6 b^3} \]

[Out]

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

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Rubi [A]  time = 0.240965, 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 \[ \frac{(a+b x)^5 (-2 a B e+A b e+b B d)}{5 b^3}+\frac{(a+b x)^4 (A b-a B) (b d-a e)}{4 b^3}+\frac{B e (a+b x)^6}{6 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.3431, size = 68, normalized size = 0.91 \[ \frac{B e \left (a + b x\right )^{6}}{6 b^{3}} + \frac{\left (a + b x\right )^{5} \left (A b e - 2 B a e + B b d\right )}{5 b^{3}} - \frac{\left (a + b x\right )^{4} \left (A b - B a\right ) \left (a e - b d\right )}{4 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0661501, size = 130, normalized size = 1.73 \[ a^3 A d x+\frac{1}{2} a^2 x^2 (a A e+a B d+3 A b d)+\frac{1}{5} b^2 x^5 (3 a B e+A b e+b B d)+\frac{1}{4} b x^4 (A b (3 a e+b d)+3 a B (a e+b d))+\frac{1}{3} a x^3 (3 A b (a e+b d)+a B (a e+3 b d))+\frac{1}{6} b^3 B e x^6 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.001, size = 149, normalized size = 2. \[{\frac{{b}^{3}Be{x}^{6}}{6}}+{\frac{ \left ( \left ({b}^{3}A+3\,a{b}^{2}B \right ) e+{b}^{3}Bd \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ) e+ \left ({b}^{3}A+3\,a{b}^{2}B \right ) d \right ){x}^{4}}{4}}+{\frac{ \left ( \left ( 3\,A{a}^{2}b+B{a}^{3} \right ) e+ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ) d \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{3}Ae+ \left ( 3\,A{a}^{2}b+B{a}^{3} \right ) d \right ){x}^{2}}{2}}+{a}^{3}Adx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.3522, size = 197, normalized size = 2.63 \[ \frac{1}{6} \, B b^{3} e x^{6} + A a^{3} d x + \frac{1}{5} \,{\left (B b^{3} d +{\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} x^{5} + \frac{1}{4} \,{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} d + 3 \,{\left (B a^{2} b + A a b^{2}\right )} e\right )} x^{4} + \frac{1}{3} \,{\left (3 \,{\left (B a^{2} b + A a b^{2}\right )} d +{\left (B a^{3} + 3 \, A a^{2} b\right )} e\right )} x^{3} + \frac{1}{2} \,{\left (A a^{3} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.185269, size = 1, normalized size = 0.01 \[ \frac{1}{6} x^{6} e b^{3} B + \frac{1}{5} x^{5} d b^{3} B + \frac{3}{5} x^{5} e b^{2} a B + \frac{1}{5} x^{5} e b^{3} A + \frac{3}{4} x^{4} d b^{2} a B + \frac{3}{4} x^{4} e b a^{2} B + \frac{1}{4} x^{4} d b^{3} A + \frac{3}{4} x^{4} e b^{2} a A + x^{3} d b a^{2} B + \frac{1}{3} x^{3} e a^{3} B + x^{3} d b^{2} a A + x^{3} e b a^{2} A + \frac{1}{2} x^{2} d a^{3} B + \frac{3}{2} x^{2} d b a^{2} A + \frac{1}{2} x^{2} e a^{3} A + x d a^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.171503, size = 168, normalized size = 2.24 \[ A a^{3} d x + \frac{B b^{3} e x^{6}}{6} + x^{5} \left (\frac{A b^{3} e}{5} + \frac{3 B a b^{2} e}{5} + \frac{B b^{3} d}{5}\right ) + x^{4} \left (\frac{3 A a b^{2} e}{4} + \frac{A b^{3} d}{4} + \frac{3 B a^{2} b e}{4} + \frac{3 B a b^{2} d}{4}\right ) + x^{3} \left (A a^{2} b e + A a b^{2} d + \frac{B a^{3} e}{3} + B a^{2} b d\right ) + x^{2} \left (\frac{A a^{3} e}{2} + \frac{3 A a^{2} b d}{2} + \frac{B a^{3} d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217882, size = 231, normalized size = 3.08 \[ \frac{1}{6} \, B b^{3} x^{6} e + \frac{1}{5} \, B b^{3} d x^{5} + \frac{3}{5} \, B a b^{2} x^{5} e + \frac{1}{5} \, A b^{3} x^{5} e + \frac{3}{4} \, B a b^{2} d x^{4} + \frac{1}{4} \, A b^{3} d x^{4} + \frac{3}{4} \, B a^{2} b x^{4} e + \frac{3}{4} \, A a b^{2} x^{4} e + B a^{2} b d x^{3} + A a b^{2} d x^{3} + \frac{1}{3} \, B a^{3} x^{3} e + A a^{2} b x^{3} e + \frac{1}{2} \, B a^{3} d x^{2} + \frac{3}{2} \, A a^{2} b d x^{2} + \frac{1}{2} \, A a^{3} x^{2} e + A a^{3} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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