3.9.68 \(\int x (A+B x) (a+b x+c x^2)^3 \, dx\) [868]

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

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {779} \begin {gather*} \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 \end {gather*}

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

Rule 779

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int x (A+B x) \left (a+b x+c x^2\right )^3 \, dx &=\int \left (a^3 A x+a^2 (3 A b+a B) x^2+3 a \left (a b B+A \left (b^2+a c\right )\right ) x^3+\left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^4+\left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^5+3 c \left (b^2 B+A b c+a B c\right ) x^6+c^2 (3 b B+A c) x^7+B c^3 x^8\right ) \, dx\\ &=\frac {1}{2} a^3 A x^2+\frac {1}{3} a^2 (3 A b+a B) x^3+\frac {3}{4} a \left (a b B+A \left (b^2+a c\right )\right ) x^4+\frac {1}{5} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^5+\frac {1}{6} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^6+\frac {3}{7} c \left (b^2 B+A b c+a B c\right ) x^7+\frac {1}{8} c^2 (3 b B+A c) x^8+\frac {1}{9} B c^3 x^9\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 166, normalized size = 1.00 \begin {gather*} \frac {1}{2} a^3 A x^2+\frac {1}{3} a^2 (3 A b+a B) x^3+\frac {3}{4} a \left (a b B+A \left (b^2+a c\right )\right ) x^4+\frac {1}{5} \left (3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )\right ) x^5+\frac {1}{6} \left (b^3 B+3 A b^2 c+6 a b B c+3 a A c^2\right ) x^6+\frac {3}{7} c \left (b^2 B+A b c+a B c\right ) x^7+\frac {1}{8} c^2 (3 b B+A c) x^8+\frac {1}{9} B c^3 x^9 \end {gather*}

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 = 0.74, size = 226, normalized size = 1.36

method	result	size
norman	\(\frac {B \,c^{3} x^{9}}{9}+\left (\frac {1}{8} A \,c^{3}+\frac {3}{8} B b \,c^{2}\right ) x^{8}+\left (\frac {3}{7} A b \,c^{2}+\frac {3}{7} B a \,c^{2}+\frac {3}{7} B \,b^{2} c \right ) x^{7}+\left (\frac {1}{2} A a \,c^{2}+\frac {1}{2} A \,b^{2} c +a b B c +\frac {1}{6} B \,b^{3}\right ) x^{6}+\left (\frac {6}{5} A a b c +\frac {1}{5} A \,b^{3}+\frac {3}{5} B \,a^{2} c +\frac {3}{5} B a \,b^{2}\right ) x^{5}+\left (\frac {3}{4} a^{2} A c +\frac {3}{4} A a \,b^{2}+\frac {3}{4} B \,a^{2} b \right ) x^{4}+\left (A \,a^{2} b +\frac {1}{3} B \,a^{3}\right ) x^{3}+\frac {a^{3} A \,x^{2}}{2}\)	\(168\)
gosper	\(\frac {1}{9} B \,c^{3} x^{9}+\frac {1}{8} A \,c^{3} x^{8}+\frac {3}{8} x^{8} B b \,c^{2}+\frac {3}{7} x^{7} A b \,c^{2}+\frac {3}{7} a B \,c^{2} x^{7}+\frac {3}{7} x^{7} B \,b^{2} c +\frac {1}{2} a A \,c^{2} x^{6}+\frac {1}{2} x^{6} A \,b^{2} c +x^{6} a b B c +\frac {1}{6} b^{3} B \,x^{6}+\frac {6}{5} x^{5} A a b c +\frac {1}{5} A \,b^{3} x^{5}+\frac {3}{5} a^{2} B c \,x^{5}+\frac {3}{5} B a \,b^{2} x^{5}+\frac {3}{4} a^{2} A c \,x^{4}+\frac {3}{4} a A \,b^{2} x^{4}+\frac {3}{4} B \,a^{2} b \,x^{4}+x^{3} A \,a^{2} b +\frac {1}{3} a^{3} B \,x^{3}+\frac {1}{2} a^{3} A \,x^{2}\)	\(192\)
risch	\(\frac {1}{9} B \,c^{3} x^{9}+\frac {1}{8} A \,c^{3} x^{8}+\frac {3}{8} x^{8} B b \,c^{2}+\frac {3}{7} x^{7} A b \,c^{2}+\frac {3}{7} a B \,c^{2} x^{7}+\frac {3}{7} x^{7} B \,b^{2} c +\frac {1}{2} a A \,c^{2} x^{6}+\frac {1}{2} x^{6} A \,b^{2} c +x^{6} a b B c +\frac {1}{6} b^{3} B \,x^{6}+\frac {6}{5} x^{5} A a b c +\frac {1}{5} A \,b^{3} x^{5}+\frac {3}{5} a^{2} B c \,x^{5}+\frac {3}{5} B a \,b^{2} x^{5}+\frac {3}{4} a^{2} A c \,x^{4}+\frac {3}{4} a A \,b^{2} x^{4}+\frac {3}{4} B \,a^{2} b \,x^{4}+x^{3} A \,a^{2} b +\frac {1}{3} a^{3} B \,x^{3}+\frac {1}{2} a^{3} A \,x^{2}\)	\(192\)
default	\(\frac {B \,c^{3} x^{9}}{9}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{8}}{8}+\frac {\left (3 A b \,c^{2}+B \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )\right ) x^{7}}{7}+\frac {\left (A \left (c^{2} a +2 b^{2} c +c \left (2 a c +b^{2}\right )\right )+B \left (4 a b c +b \left (2 a c +b^{2}\right )\right )\right ) x^{6}}{6}+\frac {\left (A \left (4 a b c +b \left (2 a c +b^{2}\right )\right )+B \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )\right ) x^{5}}{5}+\frac {\left (A \left (a \left (2 a c +b^{2}\right )+2 a \,b^{2}+a^{2} c \right )+3 B \,a^{2} b \right ) x^{4}}{4}+\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) x^{3}}{3}+\frac {a^{3} A \,x^{2}}{2}\)	\(226\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(B*x+A)*(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[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*(c^2*a+2*b^2*c+c*(2*a*c+b^2)))*x^7+1/6*(A*(c^2*a+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 = 0.28, size = 166, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)*(c*x^2+b*x+a)^3,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 = 5.49, size = 166, normalized size = 1.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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Sympy [A]
time = 0.02, size = 199, normalized size = 1.20 \begin {gather*} \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} \cdot \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} \cdot \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} \cdot \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 ) \end {gather*}

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 [A]
time = 1.84, size = 191, normalized size = 1.15 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)*(c*x^2+b*x+a)^3,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

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Mupad [B]
time = 1.16, size = 167, normalized size = 1.01 \begin {gather*} x^5\,\left (\frac {3\,B\,c\,a^2}{5}+\frac {3\,B\,a\,b^2}{5}+\frac {6\,A\,c\,a\,b}{5}+\frac {A\,b^3}{5}\right )+x^6\,\left (\frac {B\,b^3}{6}+\frac {A\,b^2\,c}{2}+B\,a\,b\,c+\frac {A\,a\,c^2}{2}\right )+x^3\,\left (\frac {B\,a^3}{3}+A\,b\,a^2\right )+x^8\,\left (\frac {A\,c^3}{8}+\frac {3\,B\,b\,c^2}{8}\right )+x^4\,\left (\frac {3\,B\,a^2\,b}{4}+\frac {3\,A\,c\,a^2}{4}+\frac {3\,A\,a\,b^2}{4}\right )+x^7\,\left (\frac {3\,B\,b^2\,c}{7}+\frac {3\,A\,b\,c^2}{7}+\frac {3\,B\,a\,c^2}{7}\right )+\frac {A\,a^3\,x^2}{2}+\frac {B\,c^3\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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