3.544 \(\int (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx\)

Optimal. Leaf size=38 \[ \frac{(a+b x)^7 (A b-a B)}{7 b^2}+\frac{B (a+b x)^8}{8 b^2} \]

[Out]

((A*b - a*B)*(a + b*x)^7)/(7*b^2) + (B*(a + b*x)^8)/(8*b^2)

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Rubi [A]  time = 0.0480752, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(a+b x)^7 (A b-a B)}{7 b^2}+\frac{B (a+b x)^8}{8 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((A*b - a*B)*(a + b*x)^7)/(7*b^2) + (B*(a + b*x)^8)/(8*b^2)

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Rubi in Sympy [A]  time = 28.1931, size = 31, normalized size = 0.82 \[ \frac{B \left (a + b x\right )^{8}}{8 b^{2}} + \frac{\left (a + b x\right )^{7} \left (A b - B a\right )}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

B*(a + b*x)**8/(8*b**2) + (a + b*x)**7*(A*b - B*a)/(7*b**2)

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Mathematica [B]  time = 0.0633394, size = 122, normalized size = 3.21 \[ \frac{1}{56} x \left (28 a^6 (2 A+B x)+56 a^5 b x (3 A+2 B x)+70 a^4 b^2 x^2 (4 A+3 B x)+56 a^3 b^3 x^3 (5 A+4 B x)+28 a^2 b^4 x^4 (6 A+5 B x)+8 a b^5 x^5 (7 A+6 B x)+b^6 x^6 (8 A+7 B x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(x*(28*a^6*(2*A + B*x) + 56*a^5*b*x*(3*A + 2*B*x) + 70*a^4*b^2*x^2*(4*A + 3*B*x)
+ 56*a^3*b^3*x^3*(5*A + 4*B*x) + 28*a^2*b^4*x^4*(6*A + 5*B*x) + 8*a*b^5*x^5*(7*
A + 6*B*x) + b^6*x^6*(8*A + 7*B*x)))/56

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Maple [B]  time = 0.001, size = 145, normalized size = 3.8 \[{\frac{B{b}^{6}{x}^{8}}{8}}+{\frac{ \left ( A{b}^{6}+6\,Ba{b}^{5} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,Aa{b}^{5}+15\,B{a}^{2}{b}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( 15\,A{a}^{2}{b}^{4}+20\,B{a}^{3}{b}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 20\,A{a}^{3}{b}^{3}+15\,B{b}^{2}{a}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 15\,A{b}^{2}{a}^{4}+6\,B{a}^{5}b \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,A{a}^{5}b+B{a}^{6} \right ){x}^{2}}{2}}+A{a}^{6}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

1/8*B*b^6*x^8+1/7*(A*b^6+6*B*a*b^5)*x^7+1/6*(6*A*a*b^5+15*B*a^2*b^4)*x^6+1/5*(15
*A*a^2*b^4+20*B*a^3*b^3)*x^5+1/4*(20*A*a^3*b^3+15*B*a^4*b^2)*x^4+1/3*(15*A*a^4*b
^2+6*B*a^5*b)*x^3+1/2*(6*A*a^5*b+B*a^6)*x^2+A*a^6*x

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Maxima [A]  time = 0.678586, size = 192, normalized size = 5.05 \[ \frac{1}{8} \, B b^{6} x^{8} + A a^{6} x + \frac{1}{7} \,{\left (6 \, B a b^{5} + A b^{6}\right )} x^{7} + \frac{1}{2} \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{6} +{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{5} + \frac{5}{4} \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{4} +{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{6} + 6 \, A a^{5} b\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A),x, algorithm="maxima")

[Out]

1/8*B*b^6*x^8 + A*a^6*x + 1/7*(6*B*a*b^5 + A*b^6)*x^7 + 1/2*(5*B*a^2*b^4 + 2*A*a
*b^5)*x^6 + (4*B*a^3*b^3 + 3*A*a^2*b^4)*x^5 + 5/4*(3*B*a^4*b^2 + 4*A*a^3*b^3)*x^
4 + (2*B*a^5*b + 5*A*a^4*b^2)*x^3 + 1/2*(B*a^6 + 6*A*a^5*b)*x^2

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Fricas [A]  time = 0.247156, size = 1, normalized size = 0.03 \[ \frac{1}{8} x^{8} b^{6} B + \frac{6}{7} x^{7} b^{5} a B + \frac{1}{7} x^{7} b^{6} A + \frac{5}{2} x^{6} b^{4} a^{2} B + x^{6} b^{5} a A + 4 x^{5} b^{3} a^{3} B + 3 x^{5} b^{4} a^{2} A + \frac{15}{4} x^{4} b^{2} a^{4} B + 5 x^{4} b^{3} a^{3} A + 2 x^{3} b a^{5} B + 5 x^{3} b^{2} a^{4} A + \frac{1}{2} x^{2} a^{6} B + 3 x^{2} b a^{5} A + x a^{6} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A),x, algorithm="fricas")

[Out]

1/8*x^8*b^6*B + 6/7*x^7*b^5*a*B + 1/7*x^7*b^6*A + 5/2*x^6*b^4*a^2*B + x^6*b^5*a*
A + 4*x^5*b^3*a^3*B + 3*x^5*b^4*a^2*A + 15/4*x^4*b^2*a^4*B + 5*x^4*b^3*a^3*A + 2
*x^3*b*a^5*B + 5*x^3*b^2*a^4*A + 1/2*x^2*a^6*B + 3*x^2*b*a^5*A + x*a^6*A

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Sympy [A]  time = 0.191714, size = 148, normalized size = 3.89 \[ A a^{6} x + \frac{B b^{6} x^{8}}{8} + x^{7} \left (\frac{A b^{6}}{7} + \frac{6 B a b^{5}}{7}\right ) + x^{6} \left (A a b^{5} + \frac{5 B a^{2} b^{4}}{2}\right ) + x^{5} \left (3 A a^{2} b^{4} + 4 B a^{3} b^{3}\right ) + x^{4} \left (5 A a^{3} b^{3} + \frac{15 B a^{4} b^{2}}{4}\right ) + x^{3} \left (5 A a^{4} b^{2} + 2 B a^{5} b\right ) + x^{2} \left (3 A a^{5} b + \frac{B a^{6}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

A*a**6*x + B*b**6*x**8/8 + x**7*(A*b**6/7 + 6*B*a*b**5/7) + x**6*(A*a*b**5 + 5*B
*a**2*b**4/2) + x**5*(3*A*a**2*b**4 + 4*B*a**3*b**3) + x**4*(5*A*a**3*b**3 + 15*
B*a**4*b**2/4) + x**3*(5*A*a**4*b**2 + 2*B*a**5*b) + x**2*(3*A*a**5*b + B*a**6/2
)

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GIAC/XCAS [A]  time = 0.267234, size = 196, normalized size = 5.16 \[ \frac{1}{8} \, B b^{6} x^{8} + \frac{6}{7} \, B a b^{5} x^{7} + \frac{1}{7} \, A b^{6} x^{7} + \frac{5}{2} \, B a^{2} b^{4} x^{6} + A a b^{5} x^{6} + 4 \, B a^{3} b^{3} x^{5} + 3 \, A a^{2} b^{4} x^{5} + \frac{15}{4} \, B a^{4} b^{2} x^{4} + 5 \, A a^{3} b^{3} x^{4} + 2 \, B a^{5} b x^{3} + 5 \, A a^{4} b^{2} x^{3} + \frac{1}{2} \, B a^{6} x^{2} + 3 \, A a^{5} b x^{2} + A a^{6} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3*(B*x + A),x, algorithm="giac")

[Out]

1/8*B*b^6*x^8 + 6/7*B*a*b^5*x^7 + 1/7*A*b^6*x^7 + 5/2*B*a^2*b^4*x^6 + A*a*b^5*x^
6 + 4*B*a^3*b^3*x^5 + 3*A*a^2*b^4*x^5 + 15/4*B*a^4*b^2*x^4 + 5*A*a^3*b^3*x^4 + 2
*B*a^5*b*x^3 + 5*A*a^4*b^2*x^3 + 1/2*B*a^6*x^2 + 3*A*a^5*b*x^2 + A*a^6*x