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

Optimal. Leaf size=75 \[ \frac {1}{5} A b^3 x^5+\frac {1}{6} b^2 x^6 (3 A c+b B)+\frac {1}{8} c^2 x^8 (A c+3 b B)+\frac {3}{7} b c x^7 (A c+b B)+\frac {1}{9} B c^3 x^9 \]

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Rubi [A]  time = 0.06, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {765} \begin {gather*} \frac {1}{6} b^2 x^6 (3 A c+b B)+\frac {1}{5} A b^3 x^5+\frac {1}{8} c^2 x^8 (A c+3 b B)+\frac {3}{7} b c x^7 (A c+b B)+\frac {1}{9} B c^3 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(A + B*x)*(b*x + c*x^2)^3,x]

[Out]

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

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 (A+B x) \left (b x+c x^2\right )^3 \, dx &=\int \left (A b^3 x^4+b^2 (b B+3 A c) x^5+3 b c (b B+A c) x^6+c^2 (3 b B+A c) x^7+B c^3 x^8\right ) \, dx\\ &=\frac {1}{5} A b^3 x^5+\frac {1}{6} b^2 (b B+3 A c) x^6+\frac {3}{7} b c (b B+A c) 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 = 0.01, size = 75, normalized size = 1.00 \begin {gather*} \frac {1}{5} A b^3 x^5+\frac {1}{6} b^2 x^6 (3 A c+b B)+\frac {1}{8} c^2 x^8 (A c+3 b B)+\frac {3}{7} b c x^7 (A c+b B)+\frac {1}{9} B c^3 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.34, size = 77, normalized size = 1.03 \begin {gather*} \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} b A + \frac {1}{6} x^{6} b^{3} B + \frac {1}{2} x^{6} c b^{2} A + \frac {1}{5} x^{5} b^{3} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(B*x+A)*(c*x^2+b*x)^3,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*b*A + 1/6*x^6*b^3*B + 1/2*x^6*
c*b^2*A + 1/5*x^5*b^3*A

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giac [A]  time = 0.16, size = 77, normalized size = 1.03 \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} \, A b c^{2} x^{7} + \frac {1}{6} \, B b^{3} x^{6} + \frac {1}{2} \, A b^{2} c x^{6} + \frac {1}{5} \, A b^{3} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [A]  time = 0.04, size = 76, normalized size = 1.01 \begin {gather*} \frac {B \,c^{3} x^{9}}{9}+\frac {A \,b^{3} x^{5}}{5}+\frac {\left (A \,c^{3}+3 B b \,c^{2}\right ) x^{8}}{8}+\frac {\left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{7}}{7}+\frac {\left (3 A \,b^{2} c +b^{3} B \right ) x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.88, size = 73, normalized size = 0.97 \begin {gather*} \frac {1}{9} \, B c^{3} x^{9} + \frac {1}{5} \, A b^{3} x^{5} + \frac {1}{8} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{8} + \frac {3}{7} \, {\left (B b^{2} c + A b c^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 69, normalized size = 0.92 \begin {gather*} x^6\,\left (\frac {B\,b^3}{6}+\frac {A\,c\,b^2}{2}\right )+x^8\,\left (\frac {A\,c^3}{8}+\frac {3\,B\,b\,c^2}{8}\right )+\frac {A\,b^3\,x^5}{5}+\frac {B\,c^3\,x^9}{9}+\frac {3\,b\,c\,x^7\,\left (A\,c+B\,b\right )}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(b*x + c*x^2)^3*(A + B*x),x)

[Out]

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

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sympy [A]  time = 0.08, size = 82, normalized size = 1.09 \begin {gather*} \frac {A b^{3} x^{5}}{5} + \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 b^{2} c}{7}\right ) + x^{6} \left (\frac {A b^{2} c}{2} + \frac {B b^{3}}{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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