3.1.32 \(\int \frac {(b x^2+c x^4)^2}{x^5} \, dx\)

Optimal. Leaf size=23 \[ b^2 \log (x)+b c x^2+\frac {c^2 x^4}{4} \]

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Rubi [A]  time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1584, 266, 43} \begin {gather*} b^2 \log (x)+b c x^2+\frac {c^2 x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/x^5,x]

[Out]

b*c*x^2 + (c^2*x^4)/4 + b^2*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]]

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^5} \, dx &=\int \frac {\left (b+c x^2\right )^2}{x} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(b+c x)^2}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (2 b c+\frac {b^2}{x}+c^2 x\right ) \, dx,x,x^2\right )\\ &=b c x^2+\frac {c^2 x^4}{4}+b^2 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} b^2 \log (x)+b c x^2+\frac {c^2 x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x^5,x]

[Out]

b*c*x^2 + (c^2*x^4)/4 + b^2*Log[x]

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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^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b*x^2 + c*x^4)^2/x^5,x]

[Out]

IntegrateAlgebraic[(b*x^2 + c*x^4)^2/x^5, x]

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fricas [A]  time = 0.48, size = 21, normalized size = 0.91 \begin {gather*} \frac {1}{4} \, c^{2} x^{4} + b c x^{2} + b^{2} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^5,x, algorithm="fricas")

[Out]

1/4*c^2*x^4 + b*c*x^2 + b^2*log(x)

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giac [A]  time = 0.15, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{4} \, c^{2} x^{4} + b c x^{2} + \frac {1}{2} \, b^{2} \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^5,x, algorithm="giac")

[Out]

1/4*c^2*x^4 + b*c*x^2 + 1/2*b^2*log(x^2)

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maple [A]  time = 0.00, size = 22, normalized size = 0.96 \begin {gather*} \frac {c^{2} x^{4}}{4}+b c \,x^{2}+b^{2} \ln \relax (x ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^2/x^5,x)

[Out]

b*c*x^2+1/4*c^2*x^4+b^2*ln(x)

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maxima [A]  time = 1.35, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{4} \, c^{2} x^{4} + b c x^{2} + \frac {1}{2} \, b^{2} \log \left (x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^5,x, algorithm="maxima")

[Out]

1/4*c^2*x^4 + b*c*x^2 + 1/2*b^2*log(x^2)

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mupad [B]  time = 0.03, size = 21, normalized size = 0.91 \begin {gather*} b^2\,\ln \relax (x)+\frac {c^2\,x^4}{4}+b\,c\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^2/x^5,x)

[Out]

b^2*log(x) + (c^2*x^4)/4 + b*c*x^2

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sympy [A]  time = 0.11, size = 20, normalized size = 0.87 \begin {gather*} b^{2} \log {\relax (x )} + b c x^{2} + \frac {c^{2} x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**2/x**5,x)

[Out]

b**2*log(x) + b*c*x**2 + c**2*x**4/4

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