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

Optimal. Leaf size=55 \[ \frac {1}{6} A b^2 x^6+\frac {1}{8} c x^8 (A c+2 b B)+\frac {1}{7} b x^7 (2 A c+b B)+\frac {1}{9} B c^2 x^9 \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 55, 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}{6} A b^2 x^6+\frac {1}{8} c x^8 (A c+2 b B)+\frac {1}{7} b x^7 (2 A c+b B)+\frac {1}{9} B c^2 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 55, normalized size = 1.00 \begin {gather*} \frac {1}{6} A b^2 x^6+\frac {1}{8} c x^8 (A c+2 b B)+\frac {1}{7} b x^7 (2 A c+b B)+\frac {1}{9} B c^2 x^9 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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 )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.36, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{9} x^{9} c^{2} B + \frac {1}{4} x^{8} c b B + \frac {1}{8} x^{8} c^{2} A + \frac {1}{7} x^{7} b^{2} B + \frac {2}{7} x^{7} c b A + \frac {1}{6} x^{6} b^{2} 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)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.17, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{9} \, B c^{2} x^{9} + \frac {1}{4} \, B b c x^{8} + \frac {1}{8} \, A c^{2} x^{8} + \frac {1}{7} \, B b^{2} x^{7} + \frac {2}{7} \, A b c x^{7} + \frac {1}{6} \, A b^{2} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 52, normalized size = 0.95 \begin {gather*} \frac {B \,c^{2} x^{9}}{9}+\frac {A \,b^{2} x^{6}}{6}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{8}}{8}+\frac {\left (2 A b c +b^{2} B \right ) x^{7}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.87, size = 51, normalized size = 0.93 \begin {gather*} \frac {1}{9} \, B c^{2} x^{9} + \frac {1}{6} \, A b^{2} x^{6} + \frac {1}{8} \, {\left (2 \, B b c + A c^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B b^{2} + 2 \, A b c\right )} 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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.05, size = 51, normalized size = 0.93 \begin {gather*} x^7\,\left (\frac {B\,b^2}{7}+\frac {2\,A\,c\,b}{7}\right )+x^8\,\left (\frac {A\,c^2}{8}+\frac {B\,b\,c}{4}\right )+\frac {A\,b^2\,x^6}{6}+\frac {B\,c^2\,x^9}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x + c*x^2)^2*(A + B*x),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 54, normalized size = 0.98 \begin {gather*} \frac {A b^{2} x^{6}}{6} + \frac {B c^{2} x^{9}}{9} + x^{8} \left (\frac {A c^{2}}{8} + \frac {B b c}{4}\right ) + x^{7} \left (\frac {2 A b c}{7} + \frac {B b^{2}}{7}\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)**2,x)

[Out]

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

________________________________________________________________________________________