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

Optimal. Leaf size=25 \[ b^2 x+\frac {2}{3} b c x^3+\frac {c^2 x^5}{5} \]

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Rubi [A]  time = 0.01, antiderivative size = 25, 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, 194} \begin {gather*} b^2 x+\frac {2}{3} b c x^3+\frac {c^2 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/x^4,x]

[Out]

b^2*x + (2*b*c*x^3)/3 + (c^2*x^5)/5

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && 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^4} \, dx &=\int \left (b+c x^2\right )^2 \, dx\\ &=\int \left (b^2+2 b c x^2+c^2 x^4\right ) \, dx\\ &=b^2 x+\frac {2}{3} b c x^3+\frac {c^2 x^5}{5}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} b^2 x+\frac {2}{3} b c x^3+\frac {c^2 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x^4,x]

[Out]

b^2*x + (2*b*c*x^3)/3 + (c^2*x^5)/5

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IntegrateAlgebraic [A]  time = 0.02, size = 25, normalized size = 1.00 \begin {gather*} b^2 x+\frac {2}{3} b c x^3+\frac {c^2 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

b^2*x + (2*b*c*x^3)/3 + (c^2*x^5)/5

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fricas [A]  time = 0.74, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} + \frac {2}{3} \, b c x^{3} + 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^4,x, algorithm="fricas")

[Out]

1/5*c^2*x^5 + 2/3*b*c*x^3 + b^2*x

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giac [A]  time = 0.18, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} + \frac {2}{3} \, b c x^{3} + 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^4,x, algorithm="giac")

[Out]

1/5*c^2*x^5 + 2/3*b*c*x^3 + b^2*x

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maple [A]  time = 0.00, size = 22, normalized size = 0.88 \begin {gather*} \frac {1}{5} c^{2} x^{5}+\frac {2}{3} b c \,x^{3}+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^4,x)

[Out]

b^2*x+2/3*b*c*x^3+1/5*c^2*x^5

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maxima [A]  time = 1.35, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{5} \, c^{2} x^{5} + \frac {2}{3} \, b c x^{3} + 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^4,x, algorithm="maxima")

[Out]

1/5*c^2*x^5 + 2/3*b*c*x^3 + b^2*x

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mupad [B]  time = 0.03, size = 21, normalized size = 0.84 \begin {gather*} b^2\,x+\frac {2\,b\,c\,x^3}{3}+\frac {c^2\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^2/x^4,x)

[Out]

b^2*x + (c^2*x^5)/5 + (2*b*c*x^3)/3

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sympy [A]  time = 0.08, size = 22, normalized size = 0.88 \begin {gather*} b^{2} x + \frac {2 b c x^{3}}{3} + \frac {c^{2} x^{5}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**2*x + 2*b*c*x**3/3 + c**2*x**5/5

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