3.1.33 \(\int \frac {(b x^2+c x^4)^2}{x^6} \, dx\)

Optimal. Leaf size=24 \[ -\frac {b^2}{x}+2 b c x+\frac {c^2 x^3}{3} \]

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, 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, 270} \begin {gather*} -\frac {b^2}{x}+2 b c x+\frac {c^2 x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/x^6,x]

[Out]

-(b^2/x) + 2*b*c*x + (c^2*x^3)/3

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]

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 (b x^2+c x^4\right )^2}{x^6} \, dx &=\int \frac {\left (b+c x^2\right )^2}{x^2} \, dx\\ &=\int \left (2 b c+\frac {b^2}{x^2}+c^2 x^2\right ) \, dx\\ &=-\frac {b^2}{x}+2 b c x+\frac {c^2 x^3}{3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} -\frac {b^2}{x}+2 b c x+\frac {c^2 x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x^6,x]

[Out]

-(b^2/x) + 2*b*c*x + (c^2*x^3)/3

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b x^2+c x^4\right )^2}{x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b*x^2 + c*x^4)^2/x^6,x]

[Out]

IntegrateAlgebraic[(b*x^2 + c*x^4)^2/x^6, x]

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fricas [A]  time = 0.59, size = 25, normalized size = 1.04 \begin {gather*} \frac {c^{2} x^{4} + 6 \, b c x^{2} - 3 \, b^{2}}{3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^6,x, algorithm="fricas")

[Out]

1/3*(c^2*x^4 + 6*b*c*x^2 - 3*b^2)/x

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giac [A]  time = 0.15, size = 22, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, c^{2} x^{3} + 2 \, b c x - \frac {b^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^6,x, algorithm="giac")

[Out]

1/3*c^2*x^3 + 2*b*c*x - b^2/x

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maple [A]  time = 0.00, size = 23, normalized size = 0.96 \begin {gather*} \frac {c^{2} x^{3}}{3}+2 b c x -\frac {b^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^2/x^6,x)

[Out]

-b^2/x+2*b*c*x+1/3*c^2*x^3

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maxima [A]  time = 1.31, size = 22, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, c^{2} x^{3} + 2 \, b c x - \frac {b^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^6,x, algorithm="maxima")

[Out]

1/3*c^2*x^3 + 2*b*c*x - b^2/x

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mupad [B]  time = 0.04, size = 22, normalized size = 0.92 \begin {gather*} \frac {c^2\,x^3}{3}-\frac {b^2}{x}+2\,b\,c\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^2/x^6,x)

[Out]

(c^2*x^3)/3 - b^2/x + 2*b*c*x

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sympy [A]  time = 0.11, size = 19, normalized size = 0.79 \begin {gather*} - \frac {b^{2}}{x} + 2 b c x + \frac {c^{2} x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**2/x**6,x)

[Out]

-b**2/x + 2*b*c*x + c**2*x**3/3

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