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

Optimal. Leaf size=43 \[ \frac {a^3 x^2}{2}+\frac {1}{2} a^2 b x^6+\frac {3}{10} a b^2 x^{10}+\frac {b^3 x^{14}}{14} \]

[Out]

1/2*a^3*x^2+1/2*a^2*b*x^6+3/10*a*b^2*x^10+1/14*b^3*x^14

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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 = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {270} \[ \frac {1}{2} a^2 b x^6+\frac {a^3 x^2}{2}+\frac {3}{10} a b^2 x^{10}+\frac {b^3 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^4)^3,x]

[Out]

(a^3*x^2)/2 + (a^2*b*x^6)/2 + (3*a*b^2*x^10)/10 + (b^3*x^14)/14

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 \left (a+b x^4\right )^3 \, dx &=\int \left (a^3 x+3 a^2 b x^5+3 a b^2 x^9+b^3 x^{13}\right ) \, dx\\ &=\frac {a^3 x^2}{2}+\frac {1}{2} a^2 b x^6+\frac {3}{10} a b^2 x^{10}+\frac {b^3 x^{14}}{14}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 43, normalized size = 1.00 \[ \frac {a^3 x^2}{2}+\frac {1}{2} a^2 b x^6+\frac {3}{10} a b^2 x^{10}+\frac {b^3 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^4)^3,x]

[Out]

(a^3*x^2)/2 + (a^2*b*x^6)/2 + (3*a*b^2*x^10)/10 + (b^3*x^14)/14

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fricas [A]  time = 0.45, size = 35, normalized size = 0.81 \[ \frac {1}{14} x^{14} b^{3} + \frac {3}{10} x^{10} b^{2} a + \frac {1}{2} x^{6} b a^{2} + \frac {1}{2} x^{2} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*x^14*b^3 + 3/10*x^10*b^2*a + 1/2*x^6*b*a^2 + 1/2*x^2*a^3

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giac [A]  time = 0.16, size = 35, normalized size = 0.81 \[ \frac {1}{14} \, b^{3} x^{14} + \frac {3}{10} \, a b^{2} x^{10} + \frac {1}{2} \, a^{2} b x^{6} + \frac {1}{2} \, a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*b^3*x^14 + 3/10*a*b^2*x^10 + 1/2*a^2*b*x^6 + 1/2*a^3*x^2

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maple [A]  time = 0.00, size = 36, normalized size = 0.84 \[ \frac {1}{14} b^{3} x^{14}+\frac {3}{10} a \,b^{2} x^{10}+\frac {1}{2} a^{2} b \,x^{6}+\frac {1}{2} a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^3*x^2+1/2*a^2*b*x^6+3/10*a*b^2*x^10+1/14*b^3*x^14

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maxima [A]  time = 1.34, size = 35, normalized size = 0.81 \[ \frac {1}{14} \, b^{3} x^{14} + \frac {3}{10} \, a b^{2} x^{10} + \frac {1}{2} \, a^{2} b x^{6} + \frac {1}{2} \, a^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*b^3*x^14 + 3/10*a*b^2*x^10 + 1/2*a^2*b*x^6 + 1/2*a^3*x^2

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mupad [B]  time = 0.04, size = 35, normalized size = 0.81 \[ \frac {a^3\,x^2}{2}+\frac {a^2\,b\,x^6}{2}+\frac {3\,a\,b^2\,x^{10}}{10}+\frac {b^3\,x^{14}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^4)^3,x)

[Out]

(a^3*x^2)/2 + (b^3*x^14)/14 + (a^2*b*x^6)/2 + (3*a*b^2*x^10)/10

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sympy [A]  time = 0.09, size = 37, normalized size = 0.86 \[ \frac {a^{3} x^{2}}{2} + \frac {a^{2} b x^{6}}{2} + \frac {3 a b^{2} x^{10}}{10} + \frac {b^{3} x^{14}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**2/2 + a**2*b*x**6/2 + 3*a*b**2*x**10/10 + b**3*x**14/14

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