3.9 \(\int \frac{(a x^2+b x^3+c x^4)^2}{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0320555, 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} \[ \frac{a^2 x^4}{4}+\frac{1}{6} x^6 \left (2 a c+b^2\right )+\frac{2}{5} a b x^5+\frac{2}{7} b c x^7+\frac{c^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^2 + b*x^3 + c*x^4)^2/x,x]

[Out]

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

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

Mathematica [A]  time = 0.0062127, size = 54, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{1}{6} x^6 \left (2 a c+b^2\right )+\frac{2}{5} a b x^5+\frac{2}{7} b c x^7+\frac{c^2 x^8}{8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 45, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{2\,ab{x}^{5}}{5}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{6}}{6}}+{\frac{2\,bc{x}^{7}}{7}}+{\frac{{c}^{2}{x}^{8}}{8}} \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,x)

[Out]

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

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Maxima [A]  time = 1.12473, size = 59, normalized size = 1.09 \begin{align*} \frac{1}{8} \, c^{2} x^{8} + \frac{2}{7} \, b c x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{6} \,{\left (b^{2} + 2 \, a c\right )} x^{6} + \frac{1}{4} \, a^{2} x^{4} \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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.42506, size = 107, normalized size = 1.98 \begin{align*} \frac{1}{8} \, c^{2} x^{8} + \frac{2}{7} \, b c x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{6} \,{\left (b^{2} + 2 \, a c\right )} x^{6} + \frac{1}{4} \, a^{2} x^{4} \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,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.070307, size = 49, normalized size = 0.91 \begin{align*} \frac{a^{2} x^{4}}{4} + \frac{2 a b x^{5}}{5} + \frac{2 b c x^{7}}{7} + \frac{c^{2} x^{8}}{8} + x^{6} \left (\frac{a c}{3} + \frac{b^{2}}{6}\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,x)

[Out]

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

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Giac [A]  time = 1.09907, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{8} \, c^{2} x^{8} + \frac{2}{7} \, b c x^{7} + \frac{1}{6} \, b^{2} x^{6} + \frac{1}{3} \, a c x^{6} + \frac{2}{5} \, a b x^{5} + \frac{1}{4} \, a^{2} x^{4} \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,x, algorithm="giac")

[Out]

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