3.1.28 \(\int x^3 (A+B x) (b x+c x^2)^3 \, dx\)

Optimal. Leaf size=75 \[ \frac {1}{7} A b^3 x^7+\frac {1}{8} b^2 x^8 (3 A c+b B)+\frac {1}{10} c^2 x^{10} (A c+3 b B)+\frac {1}{3} b c x^9 (A c+b B)+\frac {1}{11} B c^3 x^{11} \]

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Rubi [A]  time = 0.08, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {765} \begin {gather*} \frac {1}{8} b^2 x^8 (3 A c+b B)+\frac {1}{7} A b^3 x^7+\frac {1}{10} c^2 x^{10} (A c+3 b B)+\frac {1}{3} b c x^9 (A c+b B)+\frac {1}{11} B c^3 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

(A*b^3*x^7)/7 + (b^2*(b*B + 3*A*c)*x^8)/8 + (b*c*(b*B + A*c)*x^9)/3 + (c^2*(3*b*B + A*c)*x^10)/10 + (B*c^3*x^1
1)/11

Rule 765

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^3 (A+B x) \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 x^6+b^2 (b B+3 A c) x^7+3 b c (b B+A c) x^8+c^2 (3 b B+A c) x^9+B c^3 x^{10}\right ) \, dx\\ &=\frac {1}{7} A b^3 x^7+\frac {1}{8} b^2 (b B+3 A c) x^8+\frac {1}{3} b c (b B+A c) x^9+\frac {1}{10} c^2 (3 b B+A c) x^{10}+\frac {1}{11} B c^3 x^{11}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 75, normalized size = 1.00 \begin {gather*} \frac {1}{7} A b^3 x^7+\frac {1}{8} b^2 x^8 (3 A c+b B)+\frac {1}{10} c^2 x^{10} (A c+3 b B)+\frac {1}{3} b c x^9 (A c+b B)+\frac {1}{11} B c^3 x^{11} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b^3*x^7)/7 + (b^2*(b*B + 3*A*c)*x^8)/8 + (b*c*(b*B + A*c)*x^9)/3 + (c^2*(3*b*B + A*c)*x^10)/10 + (B*c^3*x^1
1)/11

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^3 (A+B x) \left (b x+c x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^3*(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

IntegrateAlgebraic[x^3*(A + B*x)*(b*x + c*x^2)^3, x]

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fricas [A]  time = 0.35, size = 77, normalized size = 1.03 \begin {gather*} \frac {1}{11} x^{11} c^{3} B + \frac {3}{10} x^{10} c^{2} b B + \frac {1}{10} x^{10} c^{3} A + \frac {1}{3} x^{9} c b^{2} B + \frac {1}{3} x^{9} c^{2} b A + \frac {1}{8} x^{8} b^{3} B + \frac {3}{8} x^{8} c b^{2} A + \frac {1}{7} x^{7} b^{3} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*c^3*B + 3/10*x^10*c^2*b*B + 1/10*x^10*c^3*A + 1/3*x^9*c*b^2*B + 1/3*x^9*c^2*b*A + 1/8*x^8*b^3*B + 3/
8*x^8*c*b^2*A + 1/7*x^7*b^3*A

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giac [A]  time = 0.15, size = 77, normalized size = 1.03 \begin {gather*} \frac {1}{11} \, B c^{3} x^{11} + \frac {3}{10} \, B b c^{2} x^{10} + \frac {1}{10} \, A c^{3} x^{10} + \frac {1}{3} \, B b^{2} c x^{9} + \frac {1}{3} \, A b c^{2} x^{9} + \frac {1}{8} \, B b^{3} x^{8} + \frac {3}{8} \, A b^{2} c x^{8} + \frac {1}{7} \, A b^{3} x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*c^3*x^11 + 3/10*B*b*c^2*x^10 + 1/10*A*c^3*x^10 + 1/3*B*b^2*c*x^9 + 1/3*A*b*c^2*x^9 + 1/8*B*b^3*x^8 + 3/
8*A*b^2*c*x^8 + 1/7*A*b^3*x^7

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maple [A]  time = 0.04, size = 76, normalized size = 1.01 \begin {gather*} \frac {B \,c^{3} x^{11}}{11}+\frac {A \,b^{3} x^{7}}{7}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{10}}{10}+\frac {\left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{9}}{9}+\frac {\left (3 A \,b^{2} c +b^{3} B \right ) x^{8}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*c^3*x^11+1/10*(A*c^3+3*B*b*c^2)*x^10+1/9*(3*A*b*c^2+3*B*b^2*c)*x^9+1/8*(3*A*b^2*c+B*b^3)*x^8+1/7*A*b^3*
x^7

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maxima [A]  time = 0.90, size = 73, normalized size = 0.97 \begin {gather*} \frac {1}{11} \, B c^{3} x^{11} + \frac {1}{7} \, A b^{3} x^{7} + \frac {1}{10} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + \frac {1}{3} \, {\left (B b^{2} c + A b c^{2}\right )} x^{9} + \frac {1}{8} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*c^3*x^11 + 1/7*A*b^3*x^7 + 1/10*(3*B*b*c^2 + A*c^3)*x^10 + 1/3*(B*b^2*c + A*b*c^2)*x^9 + 1/8*(B*b^3 + 3
*A*b^2*c)*x^8

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mupad [B]  time = 0.04, size = 69, normalized size = 0.92 \begin {gather*} x^8\,\left (\frac {B\,b^3}{8}+\frac {3\,A\,c\,b^2}{8}\right )+x^{10}\,\left (\frac {A\,c^3}{10}+\frac {3\,B\,b\,c^2}{10}\right )+\frac {A\,b^3\,x^7}{7}+\frac {B\,c^3\,x^{11}}{11}+\frac {b\,c\,x^9\,\left (A\,c+B\,b\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x + c*x^2)^3*(A + B*x),x)

[Out]

x^8*((B*b^3)/8 + (3*A*b^2*c)/8) + x^10*((A*c^3)/10 + (3*B*b*c^2)/10) + (A*b^3*x^7)/7 + (B*c^3*x^11)/11 + (b*c*
x^9*(A*c + B*b))/3

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sympy [A]  time = 0.08, size = 80, normalized size = 1.07 \begin {gather*} \frac {A b^{3} x^{7}}{7} + \frac {B c^{3} x^{11}}{11} + x^{10} \left (\frac {A c^{3}}{10} + \frac {3 B b c^{2}}{10}\right ) + x^{9} \left (\frac {A b c^{2}}{3} + \frac {B b^{2} c}{3}\right ) + x^{8} \left (\frac {3 A b^{2} c}{8} + \frac {B b^{3}}{8}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b**3*x**7/7 + B*c**3*x**11/11 + x**10*(A*c**3/10 + 3*B*b*c**2/10) + x**9*(A*b*c**2/3 + B*b**2*c/3) + x**8*(3
*A*b**2*c/8 + B*b**3/8)

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