3.1.1 \(\int x^4 (a+b x^2) \, dx\)

Optimal. Leaf size=17 \[ \frac {a x^5}{5}+\frac {b x^7}{7} \]

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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} \begin {gather*} \frac {a x^5}{5}+\frac {b x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^2),x]

[Out]

(a*x^5)/5 + (b*x^7)/7

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {a x^5}{5}+\frac {b x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^2),x]

[Out]

(a*x^5)/5 + (b*x^7)/7

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^4 \left (a+b x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^4*(a + b*x^2),x]

[Out]

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

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fricas [A]  time = 1.05, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{7} x^{7} b + \frac {1}{5} x^{5} a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*b + 1/5*x^5*a

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giac [A]  time = 1.09, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{7} \, b x^{7} + \frac {1}{5} \, a x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b*x^7 + 1/5*a*x^5

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \begin {gather*} \frac {1}{7} b \,x^{7}+\frac {1}{5} a \,x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^2+a),x)

[Out]

1/5*a*x^5+1/7*b*x^7

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maxima [A]  time = 1.31, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{7} \, b x^{7} + \frac {1}{5} \, a x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b*x^7 + 1/5*a*x^5

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mupad [B]  time = 0.03, size = 13, normalized size = 0.76 \begin {gather*} \frac {b\,x^7}{7}+\frac {a\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^2),x)

[Out]

(a*x^5)/5 + (b*x^7)/7

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sympy [A]  time = 0.06, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{5}}{5} + \frac {b x^{7}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**2+a),x)

[Out]

a*x**5/5 + b*x**7/7

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