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

Optimal. Leaf size=17 \[ \frac {a x^3}{3}+\frac {b x^5}{5} \]

[Out]

1/3*a*x^3+1/5*b*x^5

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Rubi [A]  time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ \frac {a x^3}{3}+\frac {b x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2),x]

[Out]

(a*x^3)/3 + (b*x^5)/5

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ \frac {a x^3}{3}+\frac {b x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^2),x]

[Out]

(a*x^3)/3 + (b*x^5)/5

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fricas [A]  time = 0.74, size = 13, normalized size = 0.76 \[ \frac {1}{5} x^{5} b + \frac {1}{3} x^{3} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*b + 1/3*x^3*a

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giac [A]  time = 1.11, size = 13, normalized size = 0.76 \[ \frac {1}{5} \, b x^{5} + \frac {1}{3} \, a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5 + 1/3*a*x^3

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \[ \frac {1}{5} b \,x^{5}+\frac {1}{3} a \,x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3+1/5*b*x^5

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maxima [A]  time = 1.34, size = 13, normalized size = 0.76 \[ \frac {1}{5} \, b x^{5} + \frac {1}{3} \, a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5 + 1/3*a*x^3

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mupad [B]  time = 0.02, size = 13, normalized size = 0.76 \[ \frac {b\,x^5}{5}+\frac {a\,x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x^2),x)

[Out]

(a*x^3)/3 + (b*x^5)/5

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sympy [A]  time = 0.06, size = 12, normalized size = 0.71 \[ \frac {a x^{3}}{3} + \frac {b x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**3/3 + b*x**5/5

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