3.1.32 \(\int x^3 (a+b x^2)^3 \, dx\)

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

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Rubi [A]  time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \begin {gather*} \frac {\left (a+b x^2\right )^5}{10 b^2}-\frac {a \left (a+b x^2\right )^4}{8 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^2)^3,x]

[Out]

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

Rubi steps

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

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

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^2)^3,x]

[Out]

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

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^3*(a + b*x^2)^3,x]

[Out]

IntegrateAlgebraic[x^3*(a + b*x^2)^3, x]

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fricas [A]  time = 0.95, size = 35, normalized size = 1.03 \begin {gather*} \frac {1}{10} x^{10} b^{3} + \frac {3}{8} x^{8} b^{2} a + \frac {1}{2} x^{6} b a^{2} + \frac {1}{4} x^{4} a^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.95, size = 35, normalized size = 1.03 \begin {gather*} \frac {1}{10} \, b^{3} x^{10} + \frac {3}{8} \, a b^{2} x^{8} + \frac {1}{2} \, a^{2} b x^{6} + \frac {1}{4} \, a^{3} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^3,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^2+a)^3,x)

[Out]

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

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maxima [A]  time = 1.37, size = 35, normalized size = 1.03 \begin {gather*} \frac {1}{10} \, b^{3} x^{10} + \frac {3}{8} \, a b^{2} x^{8} + \frac {1}{2} \, a^{2} b x^{6} + \frac {1}{4} \, a^{3} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^3,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^2)^3,x)

[Out]

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

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sympy [A]  time = 0.07, size = 37, normalized size = 1.09 \begin {gather*} \frac {a^{3} x^{4}}{4} + \frac {a^{2} b x^{6}}{2} + \frac {3 a b^{2} x^{8}}{8} + \frac {b^{3} x^{10}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**2+a)**3,x)

[Out]

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

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