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

Optimal. Leaf size=43 \[ \frac {a^3 x^3}{3}+\frac {3}{5} a^2 b x^5+\frac {3}{7} a b^2 x^7+\frac {b^3 x^9}{9} \]

[Out]

1/3*a^3*x^3+3/5*a^2*b*x^5+3/7*a*b^2*x^7+1/9*b^3*x^9

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Rubi [A]  time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \frac {3}{5} a^2 b x^5+\frac {a^3 x^3}{3}+\frac {3}{7} a b^2 x^7+\frac {b^3 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2)^3,x]

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^5)/5 + (3*a*b^2*x^7)/7 + (b^3*x^9)/9

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]

Rubi steps

\begin {align*} \int x^2 \left (a+b x^2\right )^3 \, dx &=\int \left (a^3 x^2+3 a^2 b x^4+3 a b^2 x^6+b^3 x^8\right ) \, dx\\ &=\frac {a^3 x^3}{3}+\frac {3}{5} a^2 b x^5+\frac {3}{7} a b^2 x^7+\frac {b^3 x^9}{9}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 43, normalized size = 1.00 \[ \frac {a^3 x^3}{3}+\frac {3}{5} a^2 b x^5+\frac {3}{7} a b^2 x^7+\frac {b^3 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^2)^3,x]

[Out]

(a^3*x^3)/3 + (3*a^2*b*x^5)/5 + (3*a*b^2*x^7)/7 + (b^3*x^9)/9

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fricas [A]  time = 0.75, size = 35, normalized size = 0.81 \[ \frac {1}{9} x^{9} b^{3} + \frac {3}{7} x^{7} b^{2} a + \frac {3}{5} x^{5} b a^{2} + \frac {1}{3} x^{3} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*b^3 + 3/7*x^7*b^2*a + 3/5*x^5*b*a^2 + 1/3*x^3*a^3

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giac [A]  time = 1.03, size = 35, normalized size = 0.81 \[ \frac {1}{9} \, b^{3} x^{9} + \frac {3}{7} \, a b^{2} x^{7} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{3} \, a^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^3*x^9 + 3/7*a*b^2*x^7 + 3/5*a^2*b*x^5 + 1/3*a^3*x^3

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maple [A]  time = 0.00, size = 36, normalized size = 0.84 \[ \frac {1}{9} b^{3} x^{9}+\frac {3}{7} a \,b^{2} x^{7}+\frac {3}{5} a^{2} b \,x^{5}+\frac {1}{3} a^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^3*x^3+3/5*a^2*b*x^5+3/7*a*b^2*x^7+1/9*b^3*x^9

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maxima [A]  time = 1.32, size = 35, normalized size = 0.81 \[ \frac {1}{9} \, b^{3} x^{9} + \frac {3}{7} \, a b^{2} x^{7} + \frac {3}{5} \, a^{2} b x^{5} + \frac {1}{3} \, a^{3} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b^3*x^9 + 3/7*a*b^2*x^7 + 3/5*a^2*b*x^5 + 1/3*a^3*x^3

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mupad [B]  time = 0.04, size = 35, normalized size = 0.81 \[ \frac {a^3\,x^3}{3}+\frac {3\,a^2\,b\,x^5}{5}+\frac {3\,a\,b^2\,x^7}{7}+\frac {b^3\,x^9}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^2)^3,x)

[Out]

(a^3*x^3)/3 + (b^3*x^9)/9 + (3*a^2*b*x^5)/5 + (3*a*b^2*x^7)/7

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sympy [A]  time = 0.07, size = 39, normalized size = 0.91 \[ \frac {a^{3} x^{3}}{3} + \frac {3 a^{2} b x^{5}}{5} + \frac {3 a b^{2} x^{7}}{7} + \frac {b^{3} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**3/3 + 3*a**2*b*x**5/5 + 3*a*b**2*x**7/7 + b**3*x**9/9

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