\(\int x (a x^2+b x^3+c x^4)^2 \, dx\) [7]

Optimal result

Integrand size = 20, antiderivative size = 54 \[ \int x \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {a^2 x^6}{6}+\frac {2}{7} a b x^7+\frac {1}{8} \left (b^2+2 a c\right ) x^8+\frac {2}{9} b c x^9+\frac {c^2 x^{10}}{10} \]

[Out]

1/6*a^2*x^6+2/7*a*b*x^7+1/8*(2*a*c+b^2)*x^8+2/9*b*c*x^9+1/10*c^2*x^10

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1599, 712} \[ \int x \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {a^2 x^6}{6}+\frac {1}{8} x^8 \left (2 a c+b^2\right )+\frac {2}{7} a b x^7+\frac {2}{9} b c x^9+\frac {c^2 x^{10}}{10} \]

[In]

Int[x*(a*x^2 + b*x^3 + c*x^4)^2,x]

[Out]

(a^2*x^6)/6 + (2*a*b*x^7)/7 + ((b^2 + 2*a*c)*x^8)/8 + (2*b*c*x^9)/9 + (c^2*x^10)/10

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 1599

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rubi steps \begin{align*} \text {integral}& = \int x^5 \left (a+b x+c x^2\right )^2 \, dx \\ & = \int \left (a^2 x^5+2 a b x^6+\left (b^2+2 a c\right ) x^7+2 b c x^8+c^2 x^9\right ) \, dx \\ & = \frac {a^2 x^6}{6}+\frac {2}{7} a b x^7+\frac {1}{8} \left (b^2+2 a c\right ) x^8+\frac {2}{9} b c x^9+\frac {c^2 x^{10}}{10} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int x \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {a^2 x^6}{6}+\frac {2}{7} a b x^7+\frac {1}{8} \left (b^2+2 a c\right ) x^8+\frac {2}{9} b c x^9+\frac {c^2 x^{10}}{10} \]

[In]

Integrate[x*(a*x^2 + b*x^3 + c*x^4)^2,x]

[Out]

(a^2*x^6)/6 + (2*a*b*x^7)/7 + ((b^2 + 2*a*c)*x^8)/8 + (2*b*c*x^9)/9 + (c^2*x^10)/10

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {a^{2} x^{6}}{6}+\frac {2 a b \,x^{7}}{7}+\frac {\left (2 a c +b^{2}\right ) x^{8}}{8}+\frac {2 b c \,x^{9}}{9}+\frac {c^{2} x^{10}}{10}\)	\(45\)
norman	\(\frac {c^{2} x^{10}}{10}+\frac {2 b c \,x^{9}}{9}+\left (\frac {a c}{4}+\frac {b^{2}}{8}\right ) x^{8}+\frac {2 a b \,x^{7}}{7}+\frac {a^{2} x^{6}}{6}\)	\(46\)
gosper	\(\frac {x^{6} \left (252 c^{2} x^{4}+560 b c \,x^{3}+630 a c \,x^{2}+315 b^{2} x^{2}+720 a b x +420 a^{2}\right )}{2520}\)	\(47\)
risch	\(\frac {1}{6} a^{2} x^{6}+\frac {2}{7} a b \,x^{7}+\frac {1}{4} x^{8} a c +\frac {1}{8} b^{2} x^{8}+\frac {2}{9} b c \,x^{9}+\frac {1}{10} c^{2} x^{10}\)	\(47\)
parallelrisch	\(\frac {1}{6} a^{2} x^{6}+\frac {2}{7} a b \,x^{7}+\frac {1}{4} x^{8} a c +\frac {1}{8} b^{2} x^{8}+\frac {2}{9} b c \,x^{9}+\frac {1}{10} c^{2} x^{10}\)	\(47\)

[In]

int(x*(c*x^4+b*x^3+a*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/6*a^2*x^6+2/7*a*b*x^7+1/8*(2*a*c+b^2)*x^8+2/9*b*c*x^9+1/10*c^2*x^10

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int x \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {1}{10} \, c^{2} x^{10} + \frac {2}{9} \, b c x^{9} + \frac {2}{7} \, a b x^{7} + \frac {1}{8} \, {\left (b^{2} + 2 \, a c\right )} x^{8} + \frac {1}{6} \, a^{2} x^{6} \]

[In]

integrate(x*(c*x^4+b*x^3+a*x^2)^2,x, algorithm="fricas")

[Out]

1/10*c^2*x^10 + 2/9*b*c*x^9 + 2/7*a*b*x^7 + 1/8*(b^2 + 2*a*c)*x^8 + 1/6*a^2*x^6

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int x \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {a^{2} x^{6}}{6} + \frac {2 a b x^{7}}{7} + \frac {2 b c x^{9}}{9} + \frac {c^{2} x^{10}}{10} + x^{8} \left (\frac {a c}{4} + \frac {b^{2}}{8}\right ) \]

[In]

integrate(x*(c*x**4+b*x**3+a*x**2)**2,x)

[Out]

a**2*x**6/6 + 2*a*b*x**7/7 + 2*b*c*x**9/9 + c**2*x**10/10 + x**8*(a*c/4 + b**2/8)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.81 \[ \int x \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {1}{10} \, c^{2} x^{10} + \frac {2}{9} \, b c x^{9} + \frac {2}{7} \, a b x^{7} + \frac {1}{8} \, {\left (b^{2} + 2 \, a c\right )} x^{8} + \frac {1}{6} \, a^{2} x^{6} \]

[In]

integrate(x*(c*x^4+b*x^3+a*x^2)^2,x, algorithm="maxima")

[Out]

1/10*c^2*x^10 + 2/9*b*c*x^9 + 2/7*a*b*x^7 + 1/8*(b^2 + 2*a*c)*x^8 + 1/6*a^2*x^6

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85 \[ \int x \left (a x^2+b x^3+c x^4\right )^2 \, dx=\frac {1}{10} \, c^{2} x^{10} + \frac {2}{9} \, b c x^{9} + \frac {1}{8} \, b^{2} x^{8} + \frac {1}{4} \, a c x^{8} + \frac {2}{7} \, a b x^{7} + \frac {1}{6} \, a^{2} x^{6} \]

[In]

integrate(x*(c*x^4+b*x^3+a*x^2)^2,x, algorithm="giac")

[Out]

1/10*c^2*x^10 + 2/9*b*c*x^9 + 1/8*b^2*x^8 + 1/4*a*c*x^8 + 2/7*a*b*x^7 + 1/6*a^2*x^6

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.83 \[ \int x \left (a x^2+b x^3+c x^4\right )^2 \, dx=x^8\,\left (\frac {b^2}{8}+\frac {a\,c}{4}\right )+\frac {a^2\,x^6}{6}+\frac {c^2\,x^{10}}{10}+\frac {2\,a\,b\,x^7}{7}+\frac {2\,b\,c\,x^9}{9} \]

[In]

int(x*(a*x^2 + b*x^3 + c*x^4)^2,x)

[Out]

x^8*((a*c)/4 + b^2/8) + (a^2*x^6)/6 + (c^2*x^10)/10 + (2*a*b*x^7)/7 + (2*b*c*x^9)/9