3.1.43 \(\int \frac {(b x^2+c x^4)^3}{x^5} \, dx\)

Optimal. Leaf size=16 \[ \frac {\left (b+c x^2\right )^4}{8 c} \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1584, 261} \begin {gather*} \frac {\left (b+c x^2\right )^4}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^3/x^5,x]

[Out]

(b + c*x^2)^4/(8*c)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

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

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (b+c x^2\right )^4}{8 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^3/x^5,x]

[Out]

(b + c*x^2)^4/(8*c)

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IntegrateAlgebraic [B]  time = 0.01, size = 38, normalized size = 2.38 \begin {gather*} \frac {1}{8} x^2 \left (4 b^3+6 b^2 c x^2+4 b c^2 x^4+c^3 x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b*x^2 + c*x^4)^3/x^5,x]

[Out]

(x^2*(4*b^3 + 6*b^2*c*x^2 + 4*b*c^2*x^4 + c^3*x^6))/8

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fricas [B]  time = 0.40, size = 35, normalized size = 2.19 \begin {gather*} \frac {1}{8} \, c^{3} x^{8} + \frac {1}{2} \, b c^{2} x^{6} + \frac {3}{4} \, b^{2} c x^{4} + \frac {1}{2} \, b^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^5,x, algorithm="fricas")

[Out]

1/8*c^3*x^8 + 1/2*b*c^2*x^6 + 3/4*b^2*c*x^4 + 1/2*b^3*x^2

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giac [B]  time = 0.15, size = 35, normalized size = 2.19 \begin {gather*} \frac {1}{8} \, c^{3} x^{8} + \frac {1}{2} \, b c^{2} x^{6} + \frac {3}{4} \, b^{2} c x^{4} + \frac {1}{2} \, b^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^5,x, algorithm="giac")

[Out]

1/8*c^3*x^8 + 1/2*b*c^2*x^6 + 3/4*b^2*c*x^4 + 1/2*b^3*x^2

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maple [B]  time = 0.00, size = 36, normalized size = 2.25 \begin {gather*} \frac {1}{8} c^{3} x^{8}+\frac {1}{2} b \,c^{2} x^{6}+\frac {3}{4} b^{2} c \,x^{4}+\frac {1}{2} b^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^3/x^5,x)

[Out]

1/8*c^3*x^8+1/2*b*c^2*x^6+3/4*b^2*c*x^4+1/2*b^3*x^2

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maxima [B]  time = 1.28, size = 35, normalized size = 2.19 \begin {gather*} \frac {1}{8} \, c^{3} x^{8} + \frac {1}{2} \, b c^{2} x^{6} + \frac {3}{4} \, b^{2} c x^{4} + \frac {1}{2} \, b^{3} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^5,x, algorithm="maxima")

[Out]

1/8*c^3*x^8 + 1/2*b*c^2*x^6 + 3/4*b^2*c*x^4 + 1/2*b^3*x^2

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mupad [B]  time = 0.04, size = 35, normalized size = 2.19 \begin {gather*} \frac {b^3\,x^2}{2}+\frac {3\,b^2\,c\,x^4}{4}+\frac {b\,c^2\,x^6}{2}+\frac {c^3\,x^8}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^3/x^5,x)

[Out]

(b^3*x^2)/2 + (c^3*x^8)/8 + (3*b^2*c*x^4)/4 + (b*c^2*x^6)/2

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sympy [B]  time = 0.08, size = 37, normalized size = 2.31 \begin {gather*} \frac {b^{3} x^{2}}{2} + \frac {3 b^{2} c x^{4}}{4} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{8}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**3/x**5,x)

[Out]

b**3*x**2/2 + 3*b**2*c*x**4/4 + b*c**2*x**6/2 + c**3*x**8/8

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