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

Optimal. Leaf size=166 \[ \frac{1}{2} a^3 A x^2+\frac{1}{3} a^2 x^3 (a B+3 A b)+\frac{3}{7} c x^7 \left (a B c+A b c+b^2 B\right )+\frac{3}{4} a x^4 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{6} x^6 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{5} x^5 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{8} c^2 x^8 (A c+3 b B)+\frac{1}{9} B c^3 x^9 \]

[Out]

(a^3*A*x^2)/2 + (a^2*(3*A*b + a*B)*x^3)/3 + (3*a*(a*b*B + A*(b^2 + a*c))*x^4)/4
+ ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^5)/5 + ((b^3*B + 3*A*b^2*c + 6*a*b*
B*c + 3*a*A*c^2)*x^6)/6 + (3*c*(b^2*B + A*b*c + a*B*c)*x^7)/7 + (c^2*(3*b*B + A*
c)*x^8)/8 + (B*c^3*x^9)/9

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Rubi [A]  time = 0.475433, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053 \[ \frac{1}{2} a^3 A x^2+\frac{1}{3} a^2 x^3 (a B+3 A b)+\frac{3}{7} c x^7 \left (a B c+A b c+b^2 B\right )+\frac{3}{4} a x^4 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{6} x^6 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{5} x^5 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{8} c^2 x^8 (A c+3 b B)+\frac{1}{9} B c^3 x^9 \]

Antiderivative was successfully verified.

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

[Out]

(a^3*A*x^2)/2 + (a^2*(3*A*b + a*B)*x^3)/3 + (3*a*(a*b*B + A*(b^2 + a*c))*x^4)/4
+ ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^5)/5 + ((b^3*B + 3*A*b^2*c + 6*a*b*
B*c + 3*a*A*c^2)*x^6)/6 + (3*c*(b^2*B + A*b*c + a*B*c)*x^7)/7 + (c^2*(3*b*B + A*
c)*x^8)/8 + (B*c^3*x^9)/9

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A a^{3} \int x\, dx + \frac{B c^{3} x^{9}}{9} + \frac{a^{2} x^{3} \left (3 A b + B a\right )}{3} + \frac{3 a x^{4} \left (A a c + A b^{2} + B a b\right )}{4} + \frac{c^{2} x^{8} \left (A c + 3 B b\right )}{8} + \frac{3 c x^{7} \left (A b c + B a c + B b^{2}\right )}{7} + x^{6} \left (\frac{A a c^{2}}{2} + \frac{A b^{2} c}{2} + B a b c + \frac{B b^{3}}{6}\right ) + x^{5} \left (\frac{6 A a b c}{5} + \frac{A b^{3}}{5} + \frac{3 B a^{2} c}{5} + \frac{3 B a b^{2}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**3*Integral(x, x) + B*c**3*x**9/9 + a**2*x**3*(3*A*b + B*a)/3 + 3*a*x**4*(A*
a*c + A*b**2 + B*a*b)/4 + c**2*x**8*(A*c + 3*B*b)/8 + 3*c*x**7*(A*b*c + B*a*c +
B*b**2)/7 + x**6*(A*a*c**2/2 + A*b**2*c/2 + B*a*b*c + B*b**3/6) + x**5*(6*A*a*b*
c/5 + A*b**3/5 + 3*B*a**2*c/5 + 3*B*a*b**2/5)

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Mathematica [A]  time = 0.0850515, size = 166, normalized size = 1. \[ \frac{1}{2} a^3 A x^2+\frac{1}{3} a^2 x^3 (a B+3 A b)+\frac{3}{7} c x^7 \left (a B c+A b c+b^2 B\right )+\frac{3}{4} a x^4 \left (A \left (a c+b^2\right )+a b B\right )+\frac{1}{6} x^6 \left (3 a A c^2+6 a b B c+3 A b^2 c+b^3 B\right )+\frac{1}{5} x^5 \left (A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )\right )+\frac{1}{8} c^2 x^8 (A c+3 b B)+\frac{1}{9} B c^3 x^9 \]

Antiderivative was successfully verified.

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

[Out]

(a^3*A*x^2)/2 + (a^2*(3*A*b + a*B)*x^3)/3 + (3*a*(a*b*B + A*(b^2 + a*c))*x^4)/4
+ ((3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))*x^5)/5 + ((b^3*B + 3*A*b^2*c + 6*a*b*
B*c + 3*a*A*c^2)*x^6)/6 + (3*c*(b^2*B + A*b*c + a*B*c)*x^7)/7 + (c^2*(3*b*B + A*
c)*x^8)/8 + (B*c^3*x^9)/9

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Maple [A]  time = 0.002, size = 226, normalized size = 1.4 \[{\frac{B{c}^{3}{x}^{9}}{9}}+{\frac{ \left ( A{c}^{3}+3\,Bb{c}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( 3\,Ab{c}^{2}+B \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( A \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( A \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +B \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( A \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,a{b}^{2}+{a}^{2}c \right ) +3\,B{a}^{2}b \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,A{a}^{2}b+B{a}^{3} \right ){x}^{3}}{3}}+{\frac{{a}^{3}A{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*B*c^3*x^9+1/8*(A*c^3+3*B*b*c^2)*x^8+1/7*(3*A*b*c^2+B*(a*c^2+2*b^2*c+c*(2*a*c
+b^2)))*x^7+1/6*(A*(a*c^2+2*b^2*c+c*(2*a*c+b^2))+B*(4*a*b*c+b*(2*a*c+b^2)))*x^6+
1/5*(A*(4*a*b*c+b*(2*a*c+b^2))+B*(a*(2*a*c+b^2)+2*a*b^2+a^2*c))*x^5+1/4*(A*(a*(2
*a*c+b^2)+2*a*b^2+a^2*c)+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 = 0.694808, size = 224, normalized size = 1.35 \[ \frac{1}{9} \, B c^{3} x^{9} + \frac{1}{8} \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{8} + \frac{3}{7} \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{6} + \frac{1}{2} \, A a^{3} x^{2} + \frac{1}{5} \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{5} + \frac{3}{4} \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{4} + \frac{1}{3} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*B*c^3*x^9 + 1/8*(3*B*b*c^2 + A*c^3)*x^8 + 3/7*(B*b^2*c + (B*a + A*b)*c^2)*x^
7 + 1/6*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^6 + 1/2*A*a^3*x^2 + 1/5*(3
*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^5 + 3/4*(B*a^2*b + A*a*b^2 + A*a^2*c
)*x^4 + 1/3*(B*a^3 + 3*A*a^2*b)*x^3

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Fricas [A]  time = 0.256529, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} c^{3} B + \frac{3}{8} x^{8} c^{2} b B + \frac{1}{8} x^{8} c^{3} A + \frac{3}{7} x^{7} c b^{2} B + \frac{3}{7} x^{7} c^{2} a B + \frac{3}{7} x^{7} c^{2} b A + \frac{1}{6} x^{6} b^{3} B + x^{6} c b a B + \frac{1}{2} x^{6} c b^{2} A + \frac{1}{2} x^{6} c^{2} a A + \frac{3}{5} x^{5} b^{2} a B + \frac{3}{5} x^{5} c a^{2} B + \frac{1}{5} x^{5} b^{3} A + \frac{6}{5} x^{5} c b a A + \frac{3}{4} x^{4} b a^{2} B + \frac{3}{4} x^{4} b^{2} a A + \frac{3}{4} x^{4} c a^{2} A + \frac{1}{3} x^{3} a^{3} B + x^{3} b a^{2} A + \frac{1}{2} x^{2} a^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*x^9*c^3*B + 3/8*x^8*c^2*b*B + 1/8*x^8*c^3*A + 3/7*x^7*c*b^2*B + 3/7*x^7*c^2*
a*B + 3/7*x^7*c^2*b*A + 1/6*x^6*b^3*B + x^6*c*b*a*B + 1/2*x^6*c*b^2*A + 1/2*x^6*
c^2*a*A + 3/5*x^5*b^2*a*B + 3/5*x^5*c*a^2*B + 1/5*x^5*b^3*A + 6/5*x^5*c*b*a*A +
3/4*x^4*b*a^2*B + 3/4*x^4*b^2*a*A + 3/4*x^4*c*a^2*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 = 0.195751, size = 199, normalized size = 1.2 \[ \frac{A a^{3} x^{2}}{2} + \frac{B c^{3} x^{9}}{9} + x^{8} \left (\frac{A c^{3}}{8} + \frac{3 B b c^{2}}{8}\right ) + x^{7} \left (\frac{3 A b c^{2}}{7} + \frac{3 B a c^{2}}{7} + \frac{3 B b^{2} c}{7}\right ) + x^{6} \left (\frac{A a c^{2}}{2} + \frac{A b^{2} c}{2} + B a b c + \frac{B b^{3}}{6}\right ) + x^{5} \left (\frac{6 A a b c}{5} + \frac{A b^{3}}{5} + \frac{3 B a^{2} c}{5} + \frac{3 B a b^{2}}{5}\right ) + x^{4} \left (\frac{3 A a^{2} c}{4} + \frac{3 A a b^{2}}{4} + \frac{3 B a^{2} b}{4}\right ) + x^{3} \left (A a^{2} b + \frac{B a^{3}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**3*x**2/2 + B*c**3*x**9/9 + x**8*(A*c**3/8 + 3*B*b*c**2/8) + x**7*(3*A*b*c**
2/7 + 3*B*a*c**2/7 + 3*B*b**2*c/7) + x**6*(A*a*c**2/2 + A*b**2*c/2 + B*a*b*c + B
*b**3/6) + x**5*(6*A*a*b*c/5 + A*b**3/5 + 3*B*a**2*c/5 + 3*B*a*b**2/5) + x**4*(3
*A*a**2*c/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 = 0.267737, size = 258, normalized size = 1.55 \[ \frac{1}{9} \, B c^{3} x^{9} + \frac{3}{8} \, B b c^{2} x^{8} + \frac{1}{8} \, A c^{3} x^{8} + \frac{3}{7} \, B b^{2} c x^{7} + \frac{3}{7} \, B a c^{2} x^{7} + \frac{3}{7} \, A b c^{2} x^{7} + \frac{1}{6} \, B b^{3} x^{6} + B a b c x^{6} + \frac{1}{2} \, A b^{2} c x^{6} + \frac{1}{2} \, A a c^{2} x^{6} + \frac{3}{5} \, B a b^{2} x^{5} + \frac{1}{5} \, A b^{3} x^{5} + \frac{3}{5} \, B a^{2} c x^{5} + \frac{6}{5} \, A a b c x^{5} + \frac{3}{4} \, B a^{2} b x^{4} + \frac{3}{4} \, A a b^{2} x^{4} + \frac{3}{4} \, A a^{2} c x^{4} + \frac{1}{3} \, B a^{3} x^{3} + A a^{2} b x^{3} + \frac{1}{2} \, A a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*B*c^3*x^9 + 3/8*B*b*c^2*x^8 + 1/8*A*c^3*x^8 + 3/7*B*b^2*c*x^7 + 3/7*B*a*c^2*
x^7 + 3/7*A*b*c^2*x^7 + 1/6*B*b^3*x^6 + B*a*b*c*x^6 + 1/2*A*b^2*c*x^6 + 1/2*A*a*
c^2*x^6 + 3/5*B*a*b^2*x^5 + 1/5*A*b^3*x^5 + 3/5*B*a^2*c*x^5 + 6/5*A*a*b*c*x^5 +
3/4*B*a^2*b*x^4 + 3/4*A*a*b^2*x^4 + 3/4*A*a^2*c*x^4 + 1/3*B*a^3*x^3 + A*a^2*b*x^
3 + 1/2*A*a^3*x^2