3.1.39 \(\int \frac {(b x^2+c x^4)^2}{x^{12}} \, dx\)

Optimal. Leaf size=30 \[ -\frac {b^2}{7 x^7}-\frac {2 b c}{5 x^5}-\frac {c^2}{3 x^3} \]

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Rubi [A]  time = 0.02, antiderivative size = 30, 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}{7 x^7}-\frac {2 b c}{5 x^5}-\frac {c^2}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/x^12,x]

[Out]

-b^2/(7*x^7) - (2*b*c)/(5*x^5) - c^2/(3*x^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^{12}} \, dx &=\int \frac {\left (b+c x^2\right )^2}{x^8} \, dx\\ &=\int \left (\frac {b^2}{x^8}+\frac {2 b c}{x^6}+\frac {c^2}{x^4}\right ) \, dx\\ &=-\frac {b^2}{7 x^7}-\frac {2 b c}{5 x^5}-\frac {c^2}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} -\frac {b^2}{7 x^7}-\frac {2 b c}{5 x^5}-\frac {c^2}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x^12,x]

[Out]

-1/7*b^2/x^7 - (2*b*c)/(5*x^5) - c^2/(3*x^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^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b*x^2 + c*x^4)^2/x^12,x]

[Out]

IntegrateAlgebraic[(b*x^2 + c*x^4)^2/x^12, x]

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fricas [A]  time = 0.68, size = 26, normalized size = 0.87 \begin {gather*} -\frac {35 \, c^{2} x^{4} + 42 \, b c x^{2} + 15 \, b^{2}}{105 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^12,x, algorithm="fricas")

[Out]

-1/105*(35*c^2*x^4 + 42*b*c*x^2 + 15*b^2)/x^7

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giac [A]  time = 0.15, size = 26, normalized size = 0.87 \begin {gather*} -\frac {35 \, c^{2} x^{4} + 42 \, b c x^{2} + 15 \, b^{2}}{105 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^12,x, algorithm="giac")

[Out]

-1/105*(35*c^2*x^4 + 42*b*c*x^2 + 15*b^2)/x^7

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maple [A]  time = 0.00, size = 25, normalized size = 0.83 \begin {gather*} -\frac {c^{2}}{3 x^{3}}-\frac {2 b c}{5 x^{5}}-\frac {b^{2}}{7 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^2/x^12,x)

[Out]

-1/7*b^2/x^7-2/5*b*c/x^5-1/3*c^2/x^3

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maxima [A]  time = 1.37, size = 26, normalized size = 0.87 \begin {gather*} -\frac {35 \, c^{2} x^{4} + 42 \, b c x^{2} + 15 \, b^{2}}{105 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^12,x, algorithm="maxima")

[Out]

-1/105*(35*c^2*x^4 + 42*b*c*x^2 + 15*b^2)/x^7

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mupad [B]  time = 0.04, size = 26, normalized size = 0.87 \begin {gather*} -\frac {\frac {b^2}{7}+\frac {2\,b\,c\,x^2}{5}+\frac {c^2\,x^4}{3}}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^2/x^12,x)

[Out]

-(b^2/7 + (c^2*x^4)/3 + (2*b*c*x^2)/5)/x^7

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sympy [A]  time = 0.23, size = 27, normalized size = 0.90 \begin {gather*} \frac {- 15 b^{2} - 42 b c x^{2} - 35 c^{2} x^{4}}{105 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**2/x**12,x)

[Out]

(-15*b**2 - 42*b*c*x**2 - 35*c**2*x**4)/(105*x**7)

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