3.76 \(\int x (a+b x)^3 (A+B x) \, dx\)

Optimal. Leaf size=61 \[ \frac{(a+b x)^5 (A b-2 a B)}{5 b^3}-\frac{a (a+b x)^4 (A b-a B)}{4 b^3}+\frac{B (a+b x)^6}{6 b^3} \]

[Out]

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

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Rubi [A]  time = 0.10964, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{(a+b x)^5 (A b-2 a B)}{5 b^3}-\frac{a (a+b x)^4 (A b-a B)}{4 b^3}+\frac{B (a+b x)^6}{6 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x)^3*(A + B*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 24.0858, size = 53, normalized size = 0.87 \[ \frac{B \left (a + b x\right )^{6}}{6 b^{3}} - \frac{a \left (a + b x\right )^{4} \left (A b - B a\right )}{4 b^{3}} + \frac{\left (a + b x\right )^{5} \left (A b - 2 B a\right )}{5 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.020963, size = 69, normalized size = 1.13 \[ \frac{1}{60} x^2 \left (10 a^3 (3 A+2 B x)+15 a^2 b x (4 A+3 B x)+9 a b^2 x^2 (5 A+4 B x)+2 b^3 x^3 (6 A+5 B x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(10*a^3*(3*A + 2*B*x) + 15*a^2*b*x*(4*A + 3*B*x) + 9*a*b^2*x^2*(5*A + 4*B*x
) + 2*b^3*x^3*(6*A + 5*B*x)))/60

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Maple [A]  time = 0.002, size = 76, normalized size = 1.3 \[{\frac{{b}^{3}B{x}^{6}}{6}}+{\frac{ \left ({b}^{3}A+3\,a{b}^{2}B \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{a}^{2}bA+{a}^{3}B \right ){x}^{3}}{3}}+{\frac{{a}^{3}A{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^3*(B*x+A),x)

[Out]

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

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Maxima [A]  time = 1.33746, size = 99, normalized size = 1.62 \[ \frac{1}{6} \, B b^{3} x^{6} + \frac{1}{2} \, A a^{3} x^{2} + \frac{1}{5} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{5} + \frac{3}{4} \,{\left (B a^{2} b + A a b^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.180292, size = 1, normalized size = 0.02 \[ \frac{1}{6} x^{6} b^{3} B + \frac{3}{5} x^{5} b^{2} a B + \frac{1}{5} x^{5} b^{3} A + \frac{3}{4} x^{4} b a^{2} B + \frac{3}{4} x^{4} b^{2} a A + \frac{1}{3} x^{3} a^{3} B + x^{3} b a^{2} A + \frac{1}{2} x^{2} a^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.155511, size = 80, normalized size = 1.31 \[ \frac{A a^{3} x^{2}}{2} + \frac{B b^{3} x^{6}}{6} + x^{5} \left (\frac{A b^{3}}{5} + \frac{3 B a b^{2}}{5}\right ) + x^{4} \left (\frac{3 A a b^{2}}{4} + \frac{3 B a^{2} b}{4}\right ) + x^{3} \left (A a^{2} b + \frac{B a^{3}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.262637, size = 103, normalized size = 1.69 \[ \frac{1}{6} \, B b^{3} x^{6} + \frac{3}{5} \, B a b^{2} x^{5} + \frac{1}{5} \, A b^{3} x^{5} + \frac{3}{4} \, B a^{2} b x^{4} + \frac{3}{4} \, A a b^{2} x^{4} + \frac{1}{3} \, B a^{3} x^{3} + A a^{2} b x^{3} + \frac{1}{2} \, A a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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