3.3.49 \(\int x^4 (a+b x^3)^3 \, dx\) [249]

Optimal. Leaf size=43 \[ \frac {a^3 x^5}{5}+\frac {3}{8} a^2 b x^8+\frac {3}{11} a b^2 x^{11}+\frac {b^3 x^{14}}{14} \]

[Out]

1/5*a^3*x^5+3/8*a^2*b*x^8+3/11*a*b^2*x^11+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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276} \begin {gather*} \frac {a^3 x^5}{5}+\frac {3}{8} a^2 b x^8+\frac {3}{11} a b^2 x^{11}+\frac {b^3 x^{14}}{14} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^3)^3,x]

[Out]

(a^3*x^5)/5 + (3*a^2*b*x^8)/8 + (3*a*b^2*x^11)/11 + (b^3*x^14)/14

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^4 \left (a+b x^3\right )^3 \, dx &=\int \left (a^3 x^4+3 a^2 b x^7+3 a b^2 x^{10}+b^3 x^{13}\right ) \, dx\\ &=\frac {a^3 x^5}{5}+\frac {3}{8} a^2 b x^8+\frac {3}{11} a b^2 x^{11}+\frac {b^3 x^{14}}{14}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} \frac {a^3 x^5}{5}+\frac {3}{8} a^2 b x^8+\frac {3}{11} a b^2 x^{11}+\frac {b^3 x^{14}}{14} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^3)^3,x]

[Out]

(a^3*x^5)/5 + (3*a^2*b*x^8)/8 + (3*a*b^2*x^11)/11 + (b^3*x^14)/14

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Maple [A]
time = 0.13, size = 36, normalized size = 0.84

method	result	size
gosper	\(\frac {1}{5} a^{3} x^{5}+\frac {3}{8} a^{2} b \,x^{8}+\frac {3}{11} a \,b^{2} x^{11}+\frac {1}{14} b^{3} x^{14}\)	\(36\)
default	\(\frac {1}{5} a^{3} x^{5}+\frac {3}{8} a^{2} b \,x^{8}+\frac {3}{11} a \,b^{2} x^{11}+\frac {1}{14} b^{3} x^{14}\)	\(36\)
norman	\(\frac {1}{5} a^{3} x^{5}+\frac {3}{8} a^{2} b \,x^{8}+\frac {3}{11} a \,b^{2} x^{11}+\frac {1}{14} b^{3} x^{14}\)	\(36\)
risch	\(\frac {1}{5} a^{3} x^{5}+\frac {3}{8} a^{2} b \,x^{8}+\frac {3}{11} a \,b^{2} x^{11}+\frac {1}{14} b^{3} x^{14}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a^3*x^5+3/8*a^2*b*x^8+3/11*a*b^2*x^11+1/14*b^3*x^14

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Maxima [A]
time = 0.29, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{14} \, b^{3} x^{14} + \frac {3}{11} \, a b^{2} x^{11} + \frac {3}{8} \, a^{2} b x^{8} + \frac {1}{5} \, a^{3} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*b^3*x^14 + 3/11*a*b^2*x^11 + 3/8*a^2*b*x^8 + 1/5*a^3*x^5

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Fricas [A]
time = 0.33, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{14} \, b^{3} x^{14} + \frac {3}{11} \, a b^{2} x^{11} + \frac {3}{8} \, a^{2} b x^{8} + \frac {1}{5} \, a^{3} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*b^3*x^14 + 3/11*a*b^2*x^11 + 3/8*a^2*b*x^8 + 1/5*a^3*x^5

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Sympy [A]
time = 0.01, size = 39, normalized size = 0.91 \begin {gather*} \frac {a^{3} x^{5}}{5} + \frac {3 a^{2} b x^{8}}{8} + \frac {3 a b^{2} x^{11}}{11} + \frac {b^{3} x^{14}}{14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**5/5 + 3*a**2*b*x**8/8 + 3*a*b**2*x**11/11 + b**3*x**14/14

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Giac [A]
time = 1.76, size = 35, normalized size = 0.81 \begin {gather*} \frac {1}{14} \, b^{3} x^{14} + \frac {3}{11} \, a b^{2} x^{11} + \frac {3}{8} \, a^{2} b x^{8} + \frac {1}{5} \, a^{3} x^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*b^3*x^14 + 3/11*a*b^2*x^11 + 3/8*a^2*b*x^8 + 1/5*a^3*x^5

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Mupad [B]
time = 0.04, size = 35, normalized size = 0.81 \begin {gather*} \frac {a^3\,x^5}{5}+\frac {3\,a^2\,b\,x^8}{8}+\frac {3\,a\,b^2\,x^{11}}{11}+\frac {b^3\,x^{14}}{14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^3)^3,x)

[Out]

(a^3*x^5)/5 + (b^3*x^14)/14 + (3*a^2*b*x^8)/8 + (3*a*b^2*x^11)/11

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