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

Optimal. Leaf size=34 \[ -\frac {a^3}{x}+3 a^2 b x+a b^2 x^3+\frac {b^3 x^5}{5} \]

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \begin {gather*} 3 a^2 b x-\frac {a^3}{x}+a b^2 x^3+\frac {b^3 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^2,x]

[Out]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 34, normalized size = 1.00 \begin {gather*} -\frac {a^3}{x}+3 a^2 b x+a b^2 x^3+\frac {b^3 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^2,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^3}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.73, size = 36, normalized size = 1.06 \begin {gather*} \frac {b^{3} x^{6} + 5 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} - 5 \, a^{3}}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*(b^3*x^6 + 5*a*b^2*x^4 + 15*a^2*b*x^2 - 5*a^3)/x

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giac [A]  time = 1.04, size = 32, normalized size = 0.94 \begin {gather*} \frac {1}{5} \, b^{3} x^{5} + a b^{2} x^{3} + 3 \, a^{2} b x - \frac {a^{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 33, normalized size = 0.97 \begin {gather*} \frac {b^{3} x^{5}}{5}+a \,b^{2} x^{3}+3 a^{2} b x -\frac {a^{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.38, size = 32, normalized size = 0.94 \begin {gather*} \frac {1}{5} \, b^{3} x^{5} + a b^{2} x^{3} + 3 \, a^{2} b x - \frac {a^{3}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 32, normalized size = 0.94 \begin {gather*} \frac {b^3\,x^5}{5}-\frac {a^3}{x}+a\,b^2\,x^3+3\,a^2\,b\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3/x^2,x)

[Out]

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

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sympy [A]  time = 0.11, size = 29, normalized size = 0.85 \begin {gather*} - \frac {a^{3}}{x} + 3 a^{2} b x + a b^{2} x^{3} + \frac {b^{3} x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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