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

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

[Out]

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

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Rubi [A]  time = 0.0324322, 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} \[ \frac{a^2 x^3}{3}+\frac{1}{5} x^5 \left (2 a c+b^2\right )+\frac{1}{2} a b x^4+\frac{1}{3} b c x^6+\frac{c^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^2 + b*x^3 + c*x^4)^2/x^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0071731, size = 54, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{1}{5} x^5 \left (2 a c+b^2\right )+\frac{1}{2} a b x^4+\frac{1}{3} b c x^6+\frac{c^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

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

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Maxima [A]  time = 1.15267, size = 59, normalized size = 1.09 \begin{align*} \frac{1}{7} \, c^{2} x^{7} + \frac{1}{3} \, b c x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{5} \,{\left (b^{2} + 2 \, a c\right )} x^{5} + \frac{1}{3} \, a^{2} x^{3} \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^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.45505, size = 107, normalized size = 1.98 \begin{align*} \frac{1}{7} \, c^{2} x^{7} + \frac{1}{3} \, b c x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{5} \,{\left (b^{2} + 2 \, a c\right )} x^{5} + \frac{1}{3} \, a^{2} x^{3} \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^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.072056, size = 48, normalized size = 0.89 \begin{align*} \frac{a^{2} x^{3}}{3} + \frac{a b x^{4}}{2} + \frac{b c x^{6}}{3} + \frac{c^{2} x^{7}}{7} + x^{5} \left (\frac{2 a c}{5} + \frac{b^{2}}{5}\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**2,x)

[Out]

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

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Giac [A]  time = 1.07228, size = 62, normalized size = 1.15 \begin{align*} \frac{1}{7} \, c^{2} x^{7} + \frac{1}{3} \, b c x^{6} + \frac{1}{5} \, b^{2} x^{5} + \frac{2}{5} \, a c x^{5} + \frac{1}{2} \, a b x^{4} + \frac{1}{3} \, a^{2} x^{3} \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^2,x, algorithm="giac")

[Out]

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