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

Optimal. Leaf size=54 \[ \frac{a^2 x^6}{6}+\frac{1}{8} x^8 \left (2 a c+b^2\right )+\frac{2}{7} a b x^7+\frac{2}{9} b c x^9+\frac{c^2 x^{10}}{10} \]

[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

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Rubi [A]  time = 0.028877, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1585, 698} \[ \frac{a^2 x^6}{6}+\frac{1}{8} x^8 \left (2 a c+b^2\right )+\frac{2}{7} a b x^7+\frac{2}{9} b c x^9+\frac{c^2 x^{10}}{10} \]

Antiderivative was successfully verified.

[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 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 \left (a x^2+b x^3+c x^4\right )^2 \, dx &=\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]  time = 0.0068741, size = 54, normalized size = 1. \[ \frac{a^2 x^6}{6}+\frac{1}{8} x^8 \left (2 a c+b^2\right )+\frac{2}{7} a b x^7+\frac{2}{9} b c x^9+\frac{c^2 x^{10}}{10} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0., size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{6}}{6}}+{\frac{2\,ab{x}^{7}}{7}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{8}}{8}}+{\frac{2\,bc{x}^{9}}{9}}+{\frac{{c}^{2}{x}^{10}}{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [A]  time = 1.12733, size = 59, normalized size = 1.09 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.28945, size = 115, normalized size = 2.13 \begin{align*} \frac{1}{10} x^{10} c^{2} + \frac{2}{9} x^{9} c b + \frac{1}{8} x^{8} b^{2} + \frac{1}{4} x^{8} c a + \frac{2}{7} x^{7} b a + \frac{1}{6} x^{6} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.069998, size = 49, normalized size = 0.91 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.08124, size = 62, normalized size = 1.15 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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