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

Optimal. Leaf size=30 \[ \frac {b^2 x^4}{4}+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 30, 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*} \frac {b^2 x^4}{4}+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/x,x]

[Out]

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

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} \, dx &=\int x^3 \left (b+c x^2\right )^2 \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int x (b+c x)^2 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (b^2 x+2 b c x^2+c^2 x^3\right ) \, dx,x,x^2\right )\\ &=\frac {b^2 x^4}{4}+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \frac {b^2 x^4}{4}+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x,x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.02, size = 28, normalized size = 0.93 \begin {gather*} \frac {1}{24} x^4 \left (6 b^2+8 b c x^2+3 c^2 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*(6*b^2 + 8*b*c*x^2 + 3*c^2*x^4))/24

________________________________________________________________________________________

fricas [A]  time = 2.02, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{8} \, c^{2} x^{8} + \frac {1}{3} \, b c x^{6} + \frac {1}{4} \, b^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.19, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{8} \, c^{2} x^{8} + \frac {1}{3} \, b c x^{6} + \frac {1}{4} \, b^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.00, size = 25, normalized size = 0.83 \begin {gather*} \frac {1}{8} c^{2} x^{8}+\frac {1}{3} b c \,x^{6}+\frac {1}{4} b^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.35, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{8} \, c^{2} x^{8} + \frac {1}{3} \, b c x^{6} + \frac {1}{4} \, b^{2} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 24, normalized size = 0.80 \begin {gather*} \frac {b^2\,x^4}{4}+\frac {b\,c\,x^6}{3}+\frac {c^2\,x^8}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^2/x,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 24, normalized size = 0.80 \begin {gather*} \frac {b^{2} x^{4}}{4} + \frac {b c x^{6}}{3} + \frac {c^{2} x^{8}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________