3.1.16 \(\int x^2 (A+B x) (b x+c x^2)^2 \, dx\) [16]

Optimal. Leaf size=55 \[ \frac {1}{5} A b^2 x^5+\frac {1}{6} b (b B+2 A c) x^6+\frac {1}{7} c (2 b B+A c) x^7+\frac {1}{8} B c^2 x^8 \]

[Out]

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

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Rubi [A]
time = 0.04, 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 = {779} \begin {gather*} \frac {1}{5} A b^2 x^5+\frac {1}{7} c x^7 (A c+2 b B)+\frac {1}{6} b x^6 (2 A c+b B)+\frac {1}{8} B c^2 x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(A + B*x)*(b*x + c*x^2)^2,x]

[Out]

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

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

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Mathematica [A]
time = 0.01, size = 55, normalized size = 1.00 \begin {gather*} \frac {1}{5} A b^2 x^5+\frac {1}{6} b (b B+2 A c) x^6+\frac {1}{7} c (2 b B+A c) x^7+\frac {1}{8} B c^2 x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.51, size = 52, normalized size = 0.95

method	result	size
gosper	\(\frac {x^{5} \left (105 B \,c^{2} x^{3}+120 A \,c^{2} x^{2}+240 b B \,x^{2} c +280 A b c x +140 b^{2} B x +168 b^{2} A \right )}{840}\)	\(52\)
default	\(\frac {B \,c^{2} x^{8}}{8}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{7}}{7}+\frac {\left (2 A b c +b^{2} B \right ) x^{6}}{6}+\frac {A \,b^{2} x^{5}}{5}\)	\(52\)
norman	\(\frac {B \,c^{2} x^{8}}{8}+\left (\frac {1}{7} A \,c^{2}+\frac {2}{7} b B c \right ) x^{7}+\left (\frac {1}{3} A b c +\frac {1}{6} b^{2} B \right ) x^{6}+\frac {A \,b^{2} x^{5}}{5}\)	\(52\)
risch	\(\frac {1}{8} B \,c^{2} x^{8}+\frac {1}{7} A \,c^{2} x^{7}+\frac {2}{7} x^{7} b B c +\frac {1}{3} x^{6} A b c +\frac {1}{6} b^{2} B \,x^{6}+\frac {1}{5} A \,b^{2} x^{5}\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 51, normalized size = 0.93 \begin {gather*} \frac {1}{8} \, B c^{2} x^{8} + \frac {1}{5} \, A b^{2} x^{5} + \frac {1}{7} \, {\left (2 \, B b c + A c^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B b^{2} + 2 \, A b c\right )} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.63, size = 51, normalized size = 0.93 \begin {gather*} \frac {1}{8} \, B c^{2} x^{8} + \frac {1}{5} \, A b^{2} x^{5} + \frac {1}{7} \, {\left (2 \, B b c + A c^{2}\right )} x^{7} + \frac {1}{6} \, {\left (B b^{2} + 2 \, A b c\right )} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.02, size = 54, normalized size = 0.98 \begin {gather*} \frac {A b^{2} x^{5}}{5} + \frac {B c^{2} x^{8}}{8} + x^{7} \left (\frac {A c^{2}}{7} + \frac {2 B b c}{7}\right ) + x^{6} \left (\frac {A b c}{3} + \frac {B b^{2}}{6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.99, size = 53, normalized size = 0.96 \begin {gather*} \frac {1}{8} \, B c^{2} x^{8} + \frac {2}{7} \, B b c x^{7} + \frac {1}{7} \, A c^{2} x^{7} + \frac {1}{6} \, B b^{2} x^{6} + \frac {1}{3} \, A b c x^{6} + \frac {1}{5} \, A b^{2} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.05, size = 51, normalized size = 0.93 \begin {gather*} x^6\,\left (\frac {B\,b^2}{6}+\frac {A\,c\,b}{3}\right )+x^7\,\left (\frac {A\,c^2}{7}+\frac {2\,B\,b\,c}{7}\right )+\frac {A\,b^2\,x^5}{5}+\frac {B\,c^2\,x^8}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(b*x + c*x^2)^2*(A + B*x),x)

[Out]

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

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