3.526 \(\int x (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=61 \[ \frac{(a+b x)^6 (A b-2 a B)}{6 b^3}-\frac{a (a+b x)^5 (A b-a B)}{5 b^3}+\frac{B (a+b x)^7}{7 b^3} \]

[Out]

-(a*(A*b - a*B)*(a + b*x)^5)/(5*b^3) + ((A*b - 2*a*B)*(a + b*x)^6)/(6*b^3) + (B*
(a + b*x)^7)/(7*b^3)

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Rubi [A]  time = 0.116212, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{(a+b x)^6 (A b-2 a B)}{6 b^3}-\frac{a (a+b x)^5 (A b-a B)}{5 b^3}+\frac{B (a+b x)^7}{7 b^3} \]

Antiderivative was successfully verified.

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

[Out]

-(a*(A*b - a*B)*(a + b*x)^5)/(5*b^3) + ((A*b - 2*a*B)*(a + b*x)^6)/(6*b^3) + (B*
(a + b*x)^7)/(7*b^3)

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Rubi in Sympy [A]  time = 26.8408, size = 53, normalized size = 0.87 \[ \frac{B \left (a + b x\right )^{7}}{7 b^{3}} - \frac{a \left (a + b x\right )^{5} \left (A b - B a\right )}{5 b^{3}} + \frac{\left (a + b x\right )^{6} \left (A b - 2 B a\right )}{6 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a + b*x)**7/(7*b**3) - a*(a + b*x)**5*(A*b - B*a)/(5*b**3) + (a + b*x)**6*(A*
b - 2*B*a)/(6*b**3)

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Mathematica [A]  time = 0.0485917, size = 88, normalized size = 1.44 \[ \frac{1}{210} x^2 \left (35 a^4 (3 A+2 B x)+70 a^3 b x (4 A+3 B x)+63 a^2 b^2 x^2 (5 A+4 B x)+28 a b^3 x^3 (6 A+5 B x)+5 b^4 x^4 (7 A+6 B x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(35*a^4*(3*A + 2*B*x) + 70*a^3*b*x*(4*A + 3*B*x) + 63*a^2*b^2*x^2*(5*A + 4*
B*x) + 28*a*b^3*x^3*(6*A + 5*B*x) + 5*b^4*x^4*(7*A + 6*B*x)))/210

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Maple [A]  time = 0.002, size = 100, normalized size = 1.6 \[{\frac{{b}^{4}B{x}^{7}}{7}}+{\frac{ \left ( A{b}^{4}+4\,Ba{b}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 4\,Aa{b}^{3}+6\,B{a}^{2}{b}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 6\,A{a}^{2}{b}^{2}+4\,B{a}^{3}b \right ){x}^{4}}{4}}+{\frac{ \left ( 4\,A{a}^{3}b+B{a}^{4} \right ){x}^{3}}{3}}+{\frac{{a}^{4}A{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*b^4*B*x^7+1/6*(A*b^4+4*B*a*b^3)*x^6+1/5*(4*A*a*b^3+6*B*a^2*b^2)*x^5+1/4*(6*A
*a^2*b^2+4*B*a^3*b)*x^4+1/3*(4*A*a^3*b+B*a^4)*x^3+1/2*a^4*A*x^2

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Maxima [A]  time = 0.68064, size = 134, normalized size = 2.2 \[ \frac{1}{7} \, B b^{4} x^{7} + \frac{1}{2} \, A a^{4} x^{2} + \frac{1}{6} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{6} + \frac{2}{5} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{5} + \frac{1}{2} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*b^4*x^7 + 1/2*A*a^4*x^2 + 1/6*(4*B*a*b^3 + A*b^4)*x^6 + 2/5*(3*B*a^2*b^2 +
2*A*a*b^3)*x^5 + 1/2*(2*B*a^3*b + 3*A*a^2*b^2)*x^4 + 1/3*(B*a^4 + 4*A*a^3*b)*x^
3

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Fricas [A]  time = 0.24937, size = 1, normalized size = 0.02 \[ \frac{1}{7} x^{7} b^{4} B + \frac{2}{3} x^{6} b^{3} a B + \frac{1}{6} x^{6} b^{4} A + \frac{6}{5} x^{5} b^{2} a^{2} B + \frac{4}{5} x^{5} b^{3} a A + x^{4} b a^{3} B + \frac{3}{2} x^{4} b^{2} a^{2} A + \frac{1}{3} x^{3} a^{4} B + \frac{4}{3} x^{3} b a^{3} A + \frac{1}{2} x^{2} a^{4} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*x^7*b^4*B + 2/3*x^6*b^3*a*B + 1/6*x^6*b^4*A + 6/5*x^5*b^2*a^2*B + 4/5*x^5*b^
3*a*A + x^4*b*a^3*B + 3/2*x^4*b^2*a^2*A + 1/3*x^3*a^4*B + 4/3*x^3*b*a^3*A + 1/2*
x^2*a^4*A

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Sympy [A]  time = 0.150438, size = 107, normalized size = 1.75 \[ \frac{A a^{4} x^{2}}{2} + \frac{B b^{4} x^{7}}{7} + x^{6} \left (\frac{A b^{4}}{6} + \frac{2 B a b^{3}}{3}\right ) + x^{5} \left (\frac{4 A a b^{3}}{5} + \frac{6 B a^{2} b^{2}}{5}\right ) + x^{4} \left (\frac{3 A a^{2} b^{2}}{2} + B a^{3} b\right ) + x^{3} \left (\frac{4 A a^{3} b}{3} + \frac{B a^{4}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**4*x**2/2 + B*b**4*x**7/7 + x**6*(A*b**4/6 + 2*B*a*b**3/3) + x**5*(4*A*a*b**
3/5 + 6*B*a**2*b**2/5) + x**4*(3*A*a**2*b**2/2 + B*a**3*b) + x**3*(4*A*a**3*b/3
+ B*a**4/3)

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GIAC/XCAS [A]  time = 0.27103, size = 135, normalized size = 2.21 \[ \frac{1}{7} \, B b^{4} x^{7} + \frac{2}{3} \, B a b^{3} x^{6} + \frac{1}{6} \, A b^{4} x^{6} + \frac{6}{5} \, B a^{2} b^{2} x^{5} + \frac{4}{5} \, A a b^{3} x^{5} + B a^{3} b x^{4} + \frac{3}{2} \, A a^{2} b^{2} x^{4} + \frac{1}{3} \, B a^{4} x^{3} + \frac{4}{3} \, A a^{3} b x^{3} + \frac{1}{2} \, A a^{4} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*B*b^4*x^7 + 2/3*B*a*b^3*x^6 + 1/6*A*b^4*x^6 + 6/5*B*a^2*b^2*x^5 + 4/5*A*a*b^
3*x^5 + B*a^3*b*x^4 + 3/2*A*a^2*b^2*x^4 + 1/3*B*a^4*x^3 + 4/3*A*a^3*b*x^3 + 1/2*
A*a^4*x^2