3.7.33 \(\int x^5 (a+b x^4)^3 \, dx\) [633]

Optimal. Leaf size=43 \[ \frac {a^3 x^6}{6}+\frac {3}{10} a^2 b x^{10}+\frac {3}{14} a b^2 x^{14}+\frac {b^3 x^{18}}{18} \]

[Out]

1/6*a^3*x^6+3/10*a^2*b*x^10+3/14*a*b^2*x^14+1/18*b^3*x^18

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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 = {276} \begin {gather*} \frac {a^3 x^6}{6}+\frac {3}{10} a^2 b x^{10}+\frac {3}{14} a b^2 x^{14}+\frac {b^3 x^{18}}{18} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^5*(a + b*x^4)^3,x]

[Out]

(a^3*x^6)/6 + (3*a^2*b*x^10)/10 + (3*a*b^2*x^14)/14 + (b^3*x^18)/18

Rule 276

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^5 \left (a+b x^4\right )^3 \, dx &=\int \left (a^3 x^5+3 a^2 b x^9+3 a b^2 x^{13}+b^3 x^{17}\right ) \, dx\\ &=\frac {a^3 x^6}{6}+\frac {3}{10} a^2 b x^{10}+\frac {3}{14} a b^2 x^{14}+\frac {b^3 x^{18}}{18}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} \frac {a^3 x^6}{6}+\frac {3}{10} a^2 b x^{10}+\frac {3}{14} a b^2 x^{14}+\frac {b^3 x^{18}}{18} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(a + b*x^4)^3,x]

[Out]

(a^3*x^6)/6 + (3*a^2*b*x^10)/10 + (3*a*b^2*x^14)/14 + (b^3*x^18)/18

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Maple [A]
time = 0.14, size = 36, normalized size = 0.84

method	result	size
gosper	\(\frac {1}{6} a^{3} x^{6}+\frac {3}{10} a^{2} b \,x^{10}+\frac {3}{14} a \,b^{2} x^{14}+\frac {1}{18} b^{3} x^{18}\)	\(36\)
default	\(\frac {1}{6} a^{3} x^{6}+\frac {3}{10} a^{2} b \,x^{10}+\frac {3}{14} a \,b^{2} x^{14}+\frac {1}{18} b^{3} x^{18}\)	\(36\)
norman	\(\frac {1}{6} a^{3} x^{6}+\frac {3}{10} a^{2} b \,x^{10}+\frac {3}{14} a \,b^{2} x^{14}+\frac {1}{18} b^{3} x^{18}\)	\(36\)
risch	\(\frac {1}{6} a^{3} x^{6}+\frac {3}{10} a^{2} b \,x^{10}+\frac {3}{14} a \,b^{2} x^{14}+\frac {1}{18} b^{3} x^{18}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(b*x^4+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/6*a^3*x^6+3/10*a^2*b*x^10+3/14*a*b^2*x^14+1/18*b^3*x^18

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Maxima [A]
time = 0.30, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{18} \, b^{3} x^{18} + \frac {3}{14} \, a b^{2} x^{14} + \frac {3}{10} \, a^{2} b x^{10} + \frac {1}{6} \, a^{3} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a)^3,x, algorithm="maxima")

[Out]

1/18*b^3*x^18 + 3/14*a*b^2*x^14 + 3/10*a^2*b*x^10 + 1/6*a^3*x^6

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Fricas [A]
time = 0.36, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{18} \, b^{3} x^{18} + \frac {3}{14} \, a b^{2} x^{14} + \frac {3}{10} \, a^{2} b x^{10} + \frac {1}{6} \, a^{3} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a)^3,x, algorithm="fricas")

[Out]

1/18*b^3*x^18 + 3/14*a*b^2*x^14 + 3/10*a^2*b*x^10 + 1/6*a^3*x^6

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Sympy [A]
time = 0.01, size = 39, normalized size = 0.91 \begin {gather*} \frac {a^{3} x^{6}}{6} + \frac {3 a^{2} b x^{10}}{10} + \frac {3 a b^{2} x^{14}}{14} + \frac {b^{3} x^{18}}{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(b*x**4+a)**3,x)

[Out]

a**3*x**6/6 + 3*a**2*b*x**10/10 + 3*a*b**2*x**14/14 + b**3*x**18/18

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Giac [A]
time = 0.59, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{18} \, b^{3} x^{18} + \frac {3}{14} \, a b^{2} x^{14} + \frac {3}{10} \, a^{2} b x^{10} + \frac {1}{6} \, a^{3} x^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x^4+a)^3,x, algorithm="giac")

[Out]

1/18*b^3*x^18 + 3/14*a*b^2*x^14 + 3/10*a^2*b*x^10 + 1/6*a^3*x^6

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Mupad [B]
time = 0.04, size = 35, normalized size = 0.81 \begin {gather*} \frac {a^3\,x^6}{6}+\frac {3\,a^2\,b\,x^{10}}{10}+\frac {3\,a\,b^2\,x^{14}}{14}+\frac {b^3\,x^{18}}{18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b*x^4)^3,x)

[Out]

(a^3*x^6)/6 + (b^3*x^18)/18 + (3*a^2*b*x^10)/10 + (3*a*b^2*x^14)/14

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