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

Optimal. Leaf size=30 \[ \frac{a^2 x^3}{3}+\frac{1}{2} a b x^4+\frac{b^2 x^5}{5} \]

[Out]

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

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Rubi [A]  time = 0.0163855, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 43} \[ \frac{a^2 x^3}{3}+\frac{1}{2} a b x^4+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^2 + b*x^3)^2/x^2,x]

[Out]

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

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0015831, size = 30, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{1}{2} a b x^4+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^2 + b*x^3)^2/x^2,x]

[Out]

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

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Maple [A]  time = 0., size = 25, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}{a}^{2}}{3}}+{\frac{ab{x}^{4}}{2}}+{\frac{{b}^{2}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.993497, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{5} \, b^{2} 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((b*x^3+a*x^2)^2/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.71121, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{5} \, b^{2} 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((b*x^3+a*x^2)^2/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.219798, size = 24, normalized size = 0.8 \begin{align*} \frac{a^{2} x^{3}}{3} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a*x**2)**2/x**2,x)

[Out]

a**2*x**3/3 + a*b*x**4/2 + b**2*x**5/5

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Giac [A]  time = 1.20632, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{5} \, b^{2} 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((b*x^3+a*x^2)^2/x^2,x, algorithm="giac")

[Out]

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