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

Optimal. Leaf size=54 \[ \frac{a^2 x^5}{5}+\frac{1}{7} x^7 \left (2 a c+b^2\right )+\frac{1}{3} a b x^6+\frac{1}{4} b c x^8+\frac{c^2 x^9}{9} \]

[Out]

(a^2*x^5)/5 + (a*b*x^6)/3 + ((b^2 + 2*a*c)*x^7)/7 + (b*c*x^8)/4 + (c^2*x^9)/9

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Rubi [A]  time = 0.0265261, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1594, 698} \[ \frac{a^2 x^5}{5}+\frac{1}{7} x^7 \left (2 a c+b^2\right )+\frac{1}{3} a b x^6+\frac{1}{4} b c x^8+\frac{c^2 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^2 + b*x^3 + c*x^4)^2,x]

[Out]

(a^2*x^5)/5 + (a*b*x^6)/3 + ((b^2 + 2*a*c)*x^7)/7 + (b*c*x^8)/4 + (c^2*x^9)/9

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, 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 \left (a x^2+b x^3+c x^4\right )^2 \, dx &=\int x^4 \left (a+b x+c x^2\right )^2 \, dx\\ &=\int \left (a^2 x^4+2 a b x^5+\left (b^2+2 a c\right ) x^6+2 b c x^7+c^2 x^8\right ) \, dx\\ &=\frac{a^2 x^5}{5}+\frac{1}{3} a b x^6+\frac{1}{7} \left (b^2+2 a c\right ) x^7+\frac{1}{4} b c x^8+\frac{c^2 x^9}{9}\\ \end{align*}

Mathematica [A]  time = 0.0060407, size = 54, normalized size = 1. \[ \frac{a^2 x^5}{5}+\frac{1}{7} x^7 \left (2 a c+b^2\right )+\frac{1}{3} a b x^6+\frac{1}{4} b c x^8+\frac{c^2 x^9}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x^5)/5 + (a*b*x^6)/3 + ((b^2 + 2*a*c)*x^7)/7 + (b*c*x^8)/4 + (c^2*x^9)/9

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Maple [A]  time = 0., size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{5}}{5}}+{\frac{ab{x}^{6}}{3}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{7}}{7}}+{\frac{bc{x}^{8}}{4}}+{\frac{{c}^{2}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^2*x^5+1/3*a*b*x^6+1/7*(2*a*c+b^2)*x^7+1/4*b*c*x^8+1/9*c^2*x^9

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Maxima [A]  time = 1.33673, size = 65, normalized size = 1.2 \begin{align*} \frac{1}{9} \, c^{2} x^{9} + \frac{1}{4} \, b c x^{8} + \frac{1}{7} \, b^{2} x^{7} + \frac{1}{5} \, a^{2} x^{5} + \frac{1}{21} \,{\left (6 \, c x^{7} + 7 \, b x^{6}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^2*x^9 + 1/4*b*c*x^8 + 1/7*b^2*x^7 + 1/5*a^2*x^5 + 1/21*(6*c*x^7 + 7*b*x^6)*a

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Fricas [A]  time = 1.36451, size = 112, normalized size = 2.07 \begin{align*} \frac{1}{9} x^{9} c^{2} + \frac{1}{4} x^{8} c b + \frac{1}{7} x^{7} b^{2} + \frac{2}{7} x^{7} c a + \frac{1}{3} x^{6} b a + \frac{1}{5} x^{5} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*c^2 + 1/4*x^8*c*b + 1/7*x^7*b^2 + 2/7*x^7*c*a + 1/3*x^6*b*a + 1/5*x^5*a^2

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Sympy [A]  time = 0.073278, size = 48, normalized size = 0.89 \begin{align*} \frac{a^{2} x^{5}}{5} + \frac{a b x^{6}}{3} + \frac{b c x^{8}}{4} + \frac{c^{2} x^{9}}{9} + x^{7} \left (\frac{2 a c}{7} + \frac{b^{2}}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**5/5 + a*b*x**6/3 + b*c*x**8/4 + c**2*x**9/9 + x**7*(2*a*c/7 + b**2/7)

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Giac [A]  time = 1.09585, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{9} \, c^{2} x^{9} + \frac{1}{4} \, b c x^{8} + \frac{1}{7} \, b^{2} x^{7} + \frac{2}{7} \, a c x^{7} + \frac{1}{3} \, a b x^{6} + \frac{1}{5} \, a^{2} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*c^2*x^9 + 1/4*b*c*x^8 + 1/7*b^2*x^7 + 2/7*a*c*x^7 + 1/3*a*b*x^6 + 1/5*a^2*x^5