3.606 \(\int x^3 (a+b x^4) \, dx\)

Optimal. Leaf size=17 \[ \frac {a x^4}{4}+\frac {b x^8}{8} \]

[Out]

1/4*a*x^4+1/8*b*x^8

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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} \[ \frac {a x^4}{4}+\frac {b x^8}{8} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^4),x]

[Out]

(a*x^4)/4 + (b*x^8)/8

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ \frac {a x^4}{4}+\frac {b x^8}{8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^4),x]

[Out]

(a*x^4)/4 + (b*x^8)/8

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fricas [A]  time = 0.63, size = 13, normalized size = 0.76 \[ \frac {1}{8} x^{8} b + \frac {1}{4} x^{4} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^4+a),x, algorithm="fricas")

[Out]

1/8*x^8*b + 1/4*x^4*a

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giac [A]  time = 0.15, size = 13, normalized size = 0.76 \[ \frac {1}{8} \, b x^{8} + \frac {1}{4} \, a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^4+a),x, algorithm="giac")

[Out]

1/8*b*x^8 + 1/4*a*x^4

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \[ \frac {1}{8} b \,x^{8}+\frac {1}{4} a \,x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^4+a),x)

[Out]

1/4*a*x^4+1/8*b*x^8

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maxima [A]  time = 1.29, size = 14, normalized size = 0.82 \[ \frac {{\left (b x^{4} + a\right )}^{2}}{8 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^4+a),x, algorithm="maxima")

[Out]

1/8*(b*x^4 + a)^2/b

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mupad [B]  time = 0.02, size = 13, normalized size = 0.76 \[ \frac {b\,x^8}{8}+\frac {a\,x^4}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^4),x)

[Out]

(a*x^4)/4 + (b*x^8)/8

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sympy [A]  time = 0.18, size = 12, normalized size = 0.71 \[ \frac {a x^{4}}{4} + \frac {b x^{8}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**4+a),x)

[Out]

a*x**4/4 + b*x**8/8

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