3.6 \(\int x^2 (a x^2+b x^3+c x^4)^2 \, dx\)

Optimal. Leaf size=54 \[ \frac{a^2 x^7}{7}+\frac{1}{9} x^9 \left (2 a c+b^2\right )+\frac{1}{4} a b x^8+\frac{1}{5} b c x^{10}+\frac{c^2 x^{11}}{11} \]

[Out]

(a^2*x^7)/7 + (a*b*x^8)/4 + ((b^2 + 2*a*c)*x^9)/9 + (b*c*x^10)/5 + (c^2*x^11)/11

________________________________________________________________________________________

Rubi [A]  time = 0.053594, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1585, 698} \[ \frac{a^2 x^7}{7}+\frac{1}{9} x^9 \left (2 a c+b^2\right )+\frac{1}{4} a b x^8+\frac{1}{5} b c x^{10}+\frac{c^2 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a*x^2 + b*x^3 + c*x^4)^2,x]

[Out]

(a^2*x^7)/7 + (a*b*x^8)/4 + ((b^2 + 2*a*c)*x^9)/9 + (b*c*x^10)/5 + (c^2*x^11)/11

Rule 1585

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]

Rule 698

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]))

Rubi steps

\begin{align*} \int x^2 \left (a x^2+b x^3+c x^4\right )^2 \, dx &=\int x^6 \left (a+b x+c x^2\right )^2 \, dx\\ &=\int \left (a^2 x^6+2 a b x^7+\left (b^2+2 a c\right ) x^8+2 b c x^9+c^2 x^{10}\right ) \, dx\\ &=\frac{a^2 x^7}{7}+\frac{1}{4} a b x^8+\frac{1}{9} \left (b^2+2 a c\right ) x^9+\frac{1}{5} b c x^{10}+\frac{c^2 x^{11}}{11}\\ \end{align*}

Mathematica [A]  time = 0.0101222, size = 54, normalized size = 1. \[ \frac{a^2 x^7}{7}+\frac{1}{9} x^9 \left (2 a c+b^2\right )+\frac{1}{4} a b x^8+\frac{1}{5} b c x^{10}+\frac{c^2 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x^7)/7 + (a*b*x^8)/4 + ((b^2 + 2*a*c)*x^9)/9 + (b*c*x^10)/5 + (c^2*x^11)/11

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{7}}{7}}+{\frac{ab{x}^{8}}{4}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{9}}{9}}+{\frac{bc{x}^{10}}{5}}+{\frac{{c}^{2}{x}^{11}}{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a^2*x^7+1/4*a*b*x^8+1/9*(2*a*c+b^2)*x^9+1/5*b*c*x^10+1/11*c^2*x^11

________________________________________________________________________________________

Maxima [A]  time = 1.16539, size = 59, normalized size = 1.09 \begin{align*} \frac{1}{11} \, c^{2} x^{11} + \frac{1}{5} \, b c x^{10} + \frac{1}{4} \, a b x^{8} + \frac{1}{9} \,{\left (b^{2} + 2 \, a c\right )} x^{9} + \frac{1}{7} \, a^{2} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*c^2*x^11 + 1/5*b*c*x^10 + 1/4*a*b*x^8 + 1/9*(b^2 + 2*a*c)*x^9 + 1/7*a^2*x^7

________________________________________________________________________________________

Fricas [A]  time = 1.26837, size = 116, normalized size = 2.15 \begin{align*} \frac{1}{11} x^{11} c^{2} + \frac{1}{5} x^{10} c b + \frac{1}{9} x^{9} b^{2} + \frac{2}{9} x^{9} c a + \frac{1}{4} x^{8} b a + \frac{1}{7} x^{7} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*c^2 + 1/5*x^10*c*b + 1/9*x^9*b^2 + 2/9*x^9*c*a + 1/4*x^8*b*a + 1/7*x^7*a^2

________________________________________________________________________________________

Sympy [A]  time = 0.070734, size = 48, normalized size = 0.89 \begin{align*} \frac{a^{2} x^{7}}{7} + \frac{a b x^{8}}{4} + \frac{b c x^{10}}{5} + \frac{c^{2} x^{11}}{11} + x^{9} \left (\frac{2 a c}{9} + \frac{b^{2}}{9}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**7/7 + a*b*x**8/4 + b*c*x**10/5 + c**2*x**11/11 + x**9*(2*a*c/9 + b**2/9)

________________________________________________________________________________________

Giac [A]  time = 1.07082, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{11} \, c^{2} x^{11} + \frac{1}{5} \, b c x^{10} + \frac{1}{9} \, b^{2} x^{9} + \frac{2}{9} \, a c x^{9} + \frac{1}{4} \, a b x^{8} + \frac{1}{7} \, a^{2} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*c^2*x^11 + 1/5*b*c*x^10 + 1/9*b^2*x^9 + 2/9*a*c*x^9 + 1/4*a*b*x^8 + 1/7*a^2*x^7