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

Optimal. Leaf size=40 \[ -\frac {a^3}{2 x^2}+3 a^2 b \log (x)+\frac {3}{2} a b^2 x^2+\frac {b^3 x^4}{4} \]

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Rubi [A]  time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, 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} \begin {gather*} 3 a^2 b \log (x)-\frac {a^3}{2 x^2}+\frac {3}{2} a b^2 x^2+\frac {b^3 x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^3,x]

[Out]

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

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])

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]]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 40, normalized size = 1.00 \begin {gather*} -\frac {a^3}{2 x^2}+3 a^2 b \log (x)+\frac {3}{2} a b^2 x^2+\frac {b^3 x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^3,x]

[Out]

-1/2*a^3/x^2 + (3*a*b^2*x^2)/2 + (b^3*x^4)/4 + 3*a^2*b*Log[x]

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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^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.93, size = 38, normalized size = 0.95 \begin {gather*} \frac {b^{3} x^{6} + 6 \, a b^{2} x^{4} + 12 \, a^{2} b x^{2} \log \relax (x) - 2 \, a^{3}}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(b^3*x^6 + 6*a*b^2*x^4 + 12*a^2*b*x^2*log(x) - 2*a^3)/x^2

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giac [A]  time = 1.08, size = 46, normalized size = 1.15 \begin {gather*} \frac {1}{4} \, b^{3} x^{4} + \frac {3}{2} \, a b^{2} x^{2} + \frac {3}{2} \, a^{2} b \log \left (x^{2}\right ) - \frac {3 \, a^{2} b x^{2} + a^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 35, normalized size = 0.88 \begin {gather*} \frac {b^{3} x^{4}}{4}+\frac {3 a \,b^{2} x^{2}}{2}+3 a^{2} b \ln \relax (x )-\frac {a^{3}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.38, size = 36, normalized size = 0.90 \begin {gather*} \frac {1}{4} \, b^{3} x^{4} + \frac {3}{2} \, a b^{2} x^{2} + \frac {3}{2} \, a^{2} b \log \left (x^{2}\right ) - \frac {a^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 34, normalized size = 0.85 \begin {gather*} \frac {b^3\,x^4}{4}-\frac {a^3}{2\,x^2}+\frac {3\,a\,b^2\,x^2}{2}+3\,a^2\,b\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3/x^3,x)

[Out]

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

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sympy [A]  time = 0.14, size = 37, normalized size = 0.92 \begin {gather*} - \frac {a^{3}}{2 x^{2}} + 3 a^{2} b \log {\relax (x )} + \frac {3 a b^{2} x^{2}}{2} + \frac {b^{3} x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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