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

Optimal. Leaf size=34 \[ \frac {\left (b+c x^2\right )^5}{10 c^2}-\frac {b \left (b+c x^2\right )^4}{8 c^2} \]

[Out]

-1/8*b*(c*x^2+b)^4/c^2+1/10*(c*x^2+b)^5/c^2

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Rubi [A]  time = 0.04, antiderivative size = 34, 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} \[ \frac {\left (b+c x^2\right )^5}{10 c^2}-\frac {b \left (b+c x^2\right )^4}{8 c^2} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^3/x^3,x]

[Out]

-(b*(b + c*x^2)^4)/(8*c^2) + (b + c*x^2)^5/(10*c^2)

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 )^3}{x^3} \, dx &=\int x^3 \left (b+c x^2\right )^3 \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int x (b+c x)^3 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {b (b+c x)^3}{c}+\frac {(b+c x)^4}{c}\right ) \, dx,x,x^2\right )\\ &=-\frac {b \left (b+c x^2\right )^4}{8 c^2}+\frac {\left (b+c x^2\right )^5}{10 c^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 43, normalized size = 1.26 \[ \frac {b^3 x^4}{4}+\frac {1}{2} b^2 c x^6+\frac {3}{8} b c^2 x^8+\frac {c^3 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^3/x^3,x]

[Out]

(b^3*x^4)/4 + (b^2*c*x^6)/2 + (3*b*c^2*x^8)/8 + (c^3*x^10)/10

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fricas [A]  time = 0.79, size = 35, normalized size = 1.03 \[ \frac {1}{10} \, c^{3} x^{10} + \frac {3}{8} \, b c^{2} x^{8} + \frac {1}{2} \, b^{2} c x^{6} + \frac {1}{4} \, b^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^3*x^10 + 3/8*b*c^2*x^8 + 1/2*b^2*c*x^6 + 1/4*b^3*x^4

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giac [A]  time = 0.18, size = 35, normalized size = 1.03 \[ \frac {1}{10} \, c^{3} x^{10} + \frac {3}{8} \, b c^{2} x^{8} + \frac {1}{2} \, b^{2} c x^{6} + \frac {1}{4} \, b^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^3*x^10 + 3/8*b*c^2*x^8 + 1/2*b^2*c*x^6 + 1/4*b^3*x^4

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maple [A]  time = 0.00, size = 36, normalized size = 1.06 \[ \frac {1}{10} c^{3} x^{10}+\frac {3}{8} b \,c^{2} x^{8}+\frac {1}{2} b^{2} c \,x^{6}+\frac {1}{4} b^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^3*x^10+3/8*b*c^2*x^8+1/2*b^2*c*x^6+1/4*b^3*x^4

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maxima [A]  time = 1.32, size = 35, normalized size = 1.03 \[ \frac {1}{10} \, c^{3} x^{10} + \frac {3}{8} \, b c^{2} x^{8} + \frac {1}{2} \, b^{2} c x^{6} + \frac {1}{4} \, b^{3} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*c^3*x^10 + 3/8*b*c^2*x^8 + 1/2*b^2*c*x^6 + 1/4*b^3*x^4

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mupad [B]  time = 0.04, size = 35, normalized size = 1.03 \[ \frac {b^3\,x^4}{4}+\frac {b^2\,c\,x^6}{2}+\frac {3\,b\,c^2\,x^8}{8}+\frac {c^3\,x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2 + c*x^4)^3/x^3,x)

[Out]

(b^3*x^4)/4 + (c^3*x^10)/10 + (b^2*c*x^6)/2 + (3*b*c^2*x^8)/8

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sympy [A]  time = 0.08, size = 37, normalized size = 1.09 \[ \frac {b^{3} x^{4}}{4} + \frac {b^{2} c x^{6}}{2} + \frac {3 b c^{2} x^{8}}{8} + \frac {c^{3} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**3*x**4/4 + b**2*c*x**6/2 + 3*b*c**2*x**8/8 + c**3*x**10/10

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