3.1.48 \(\int \frac {(b x^2+c x^4)^3}{x^{10}} \, dx\)

Optimal. Leaf size=37 \[ -\frac {b^3}{3 x^3}-\frac {3 b^2 c}{x}+3 b c^2 x+\frac {c^3 x^3}{3} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 37, 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 {3 b^2 c}{x}-\frac {b^3}{3 x^3}+3 b c^2 x+\frac {c^3 x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^3/x^10,x]

[Out]

-b^3/(3*x^3) - (3*b^2*c)/x + 3*b*c^2*x + (c^3*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 )^3}{x^{10}} \, dx &=\int \frac {\left (b+c x^2\right )^3}{x^4} \, dx\\ &=\int \left (3 b c^2+\frac {b^3}{x^4}+\frac {3 b^2 c}{x^2}+c^3 x^2\right ) \, dx\\ &=-\frac {b^3}{3 x^3}-\frac {3 b^2 c}{x}+3 b c^2 x+\frac {c^3 x^3}{3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 37, normalized size = 1.00 \begin {gather*} -\frac {b^3}{3 x^3}-\frac {3 b^2 c}{x}+3 b c^2 x+\frac {c^3 x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^3/x^10,x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b x^2+c x^4\right )^3}{x^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b*x^2 + c*x^4)^3/x^10,x]

[Out]

IntegrateAlgebraic[(b*x^2 + c*x^4)^3/x^10, x]

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 36, normalized size = 0.97 \begin {gather*} \frac {c^{3} x^{6} + 9 \, b c^{2} x^{4} - 9 \, b^{2} c x^{2} - b^{3}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^10,x, algorithm="fricas")

[Out]

1/3*(c^3*x^6 + 9*b*c^2*x^4 - 9*b^2*c*x^2 - b^3)/x^3

________________________________________________________________________________________

giac [A]  time = 0.17, size = 34, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, c^{3} x^{3} + 3 \, b c^{2} x - \frac {9 \, b^{2} c x^{2} + b^{3}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^10,x, algorithm="giac")

[Out]

1/3*c^3*x^3 + 3*b*c^2*x - 1/3*(9*b^2*c*x^2 + b^3)/x^3

________________________________________________________________________________________

maple [A]  time = 0.01, size = 34, normalized size = 0.92 \begin {gather*} \frac {c^{3} x^{3}}{3}+3 b \,c^{2} x -\frac {3 b^{2} c}{x}-\frac {b^{3}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^3/x^10,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.29, size = 34, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, c^{3} x^{3} + 3 \, b c^{2} x - \frac {9 \, b^{2} c x^{2} + b^{3}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^10,x, algorithm="maxima")

[Out]

1/3*c^3*x^3 + 3*b*c^2*x - 1/3*(9*b^2*c*x^2 + b^3)/x^3

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 36, normalized size = 0.97 \begin {gather*} \frac {c^3\,x^3}{3}-\frac {\frac {b^3}{3}+3\,c\,b^2\,x^2}{x^3}+3\,b\,c^2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^3/x^10,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.17, size = 36, normalized size = 0.97 \begin {gather*} 3 b c^{2} x + \frac {c^{3} x^{3}}{3} + \frac {- b^{3} - 9 b^{2} c x^{2}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**3/x**10,x)

[Out]

3*b*c**2*x + c**3*x**3/3 + (-b**3 - 9*b**2*c*x**2)/(3*x**3)

________________________________________________________________________________________