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

Optimal. Leaf size=16 \[ \frac {\left (a+b x^2\right )^3}{6 b} \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1584, 261} \begin {gather*} \frac {\left (a+b x^2\right )^3}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*x + b*x^3)^2/x,x]

[Out]

(a + b*x^2)^3/(6*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

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]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (a+b x^2\right )^3}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x + b*x^3)^2/x,x]

[Out]

(a + b*x^2)^3/(6*b)

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IntegrateAlgebraic [A]  time = 0.01, size = 27, normalized size = 1.69 \begin {gather*} \frac {1}{6} x^2 \left (3 a^2+3 a b x^2+b^2 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a*x + b*x^3)^2/x,x]

[Out]

(x^2*(3*a^2 + 3*a*b*x^2 + b^2*x^4))/6

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fricas [A]  time = 0.37, size = 24, normalized size = 1.50 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, a^{2} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*x^6 + 1/2*a*b*x^4 + 1/2*a^2*x^2

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giac [A]  time = 0.20, size = 24, normalized size = 1.50 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, a^{2} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*x^6 + 1/2*a*b*x^4 + 1/2*a^2*x^2

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maple [A]  time = 0.04, size = 25, normalized size = 1.56 \begin {gather*} \frac {1}{6} b^{2} x^{6}+\frac {1}{2} a b \,x^{4}+\frac {1}{2} a^{2} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*x^6+1/2*a*b*x^4+1/2*a^2*x^2

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maxima [A]  time = 1.29, size = 24, normalized size = 1.50 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} + \frac {1}{2} \, a b x^{4} + \frac {1}{2} \, a^{2} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^2*x^6 + 1/2*a*b*x^4 + 1/2*a^2*x^2

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mupad [B]  time = 0.03, size = 24, normalized size = 1.50 \begin {gather*} \frac {a^2\,x^2}{2}+\frac {a\,b\,x^4}{2}+\frac {b^2\,x^6}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + b*x^3)^2/x,x)

[Out]

(a^2*x^2)/2 + (b^2*x^6)/6 + (a*b*x^4)/2

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sympy [B]  time = 0.07, size = 24, normalized size = 1.50 \begin {gather*} \frac {a^{2} x^{2}}{2} + \frac {a b x^{4}}{2} + \frac {b^{2} x^{6}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**2/2 + a*b*x**4/2 + b**2*x**6/6

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