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

Optimal. Leaf size=39 \[ \frac{3}{2} a^2 b x^2+a^3 \log (x)+\frac{3}{4} a b^2 x^4+\frac{b^3 x^6}{6} \]

[Out]

(3*a^2*b*x^2)/2 + (3*a*b^2*x^4)/4 + (b^3*x^6)/6 + a^3*Log[x]

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Rubi [A]  time = 0.0184897, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{3}{2} a^2 b x^2+a^3 \log (x)+\frac{3}{4} a b^2 x^4+\frac{b^3 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x,x]

[Out]

(3*a^2*b*x^2)/2 + (3*a*b^2*x^4)/4 + (b^3*x^6)/6 + a^3*Log[x]

Rule 266

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

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

Mathematica [A]  time = 0.0034781, size = 39, normalized size = 1. \[ \frac{3}{2} a^2 b x^2+a^3 \log (x)+\frac{3}{4} a b^2 x^4+\frac{b^3 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x,x]

[Out]

(3*a^2*b*x^2)/2 + (3*a*b^2*x^4)/4 + (b^3*x^6)/6 + a^3*Log[x]

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Maple [A]  time = 0.002, size = 34, normalized size = 0.9 \begin{align*}{\frac{3\,{a}^{2}b{x}^{2}}{2}}+{\frac{3\,a{b}^{2}{x}^{4}}{4}}+{\frac{{b}^{3}{x}^{6}}{6}}+{a}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a^2*b*x^2+3/4*a*b^2*x^4+1/6*b^3*x^6+a^3*ln(x)

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Maxima [A]  time = 2.0592, size = 49, normalized size = 1.26 \begin{align*} \frac{1}{6} \, b^{3} x^{6} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{2} \, a^{2} b x^{2} + \frac{1}{2} \, a^{3} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^3*x^6 + 3/4*a*b^2*x^4 + 3/2*a^2*b*x^2 + 1/2*a^3*log(x^2)

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Fricas [A]  time = 1.98528, size = 78, normalized size = 2. \begin{align*} \frac{1}{6} \, b^{3} x^{6} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^3*x^6 + 3/4*a*b^2*x^4 + 3/2*a^2*b*x^2 + a^3*log(x)

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Sympy [A]  time = 0.251546, size = 37, normalized size = 0.95 \begin{align*} a^{3} \log{\left (x \right )} + \frac{3 a^{2} b x^{2}}{2} + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*log(x) + 3*a**2*b*x**2/2 + 3*a*b**2*x**4/4 + b**3*x**6/6

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Giac [A]  time = 1.90448, size = 49, normalized size = 1.26 \begin{align*} \frac{1}{6} \, b^{3} x^{6} + \frac{3}{4} \, a b^{2} x^{4} + \frac{3}{2} \, a^{2} b x^{2} + \frac{1}{2} \, a^{3} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*b^3*x^6 + 3/4*a*b^2*x^4 + 3/2*a^2*b*x^2 + 1/2*a^3*log(x^2)