3.3.52 \(\int (a+b x^3)^3 \, dx\) [252]

Optimal. Leaf size=38 \[ a^3 x+\frac {3}{4} a^2 b x^4+\frac {3}{7} a b^2 x^7+\frac {b^3 x^{10}}{10} \]

[Out]

a^3*x+3/4*a^2*b*x^4+3/7*a*b^2*x^7+1/10*b^3*x^10

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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {200} \begin {gather*} a^3 x+\frac {3}{4} a^2 b x^4+\frac {3}{7} a b^2 x^7+\frac {b^3 x^{10}}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^3,x]

[Out]

a^3*x + (3*a^2*b*x^4)/4 + (3*a*b^2*x^7)/7 + (b^3*x^10)/10

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \left (a+b x^3\right )^3 \, dx &=\int \left (a^3+3 a^2 b x^3+3 a b^2 x^6+b^3 x^9\right ) \, dx\\ &=a^3 x+\frac {3}{4} a^2 b x^4+\frac {3}{7} a b^2 x^7+\frac {b^3 x^{10}}{10}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 38, normalized size = 1.00 \begin {gather*} a^3 x+\frac {3}{4} a^2 b x^4+\frac {3}{7} a b^2 x^7+\frac {b^3 x^{10}}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^3,x]

[Out]

a^3*x + (3*a^2*b*x^4)/4 + (3*a*b^2*x^7)/7 + (b^3*x^10)/10

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Maple [A]
time = 0.11, size = 33, normalized size = 0.87

method	result	size
gosper	\(a^{3} x +\frac {3}{4} a^{2} b \,x^{4}+\frac {3}{7} a \,b^{2} x^{7}+\frac {1}{10} b^{3} x^{10}\)	\(33\)
default	\(a^{3} x +\frac {3}{4} a^{2} b \,x^{4}+\frac {3}{7} a \,b^{2} x^{7}+\frac {1}{10} b^{3} x^{10}\)	\(33\)
norman	\(a^{3} x +\frac {3}{4} a^{2} b \,x^{4}+\frac {3}{7} a \,b^{2} x^{7}+\frac {1}{10} b^{3} x^{10}\)	\(33\)
risch	\(a^{3} x +\frac {3}{4} a^{2} b \,x^{4}+\frac {3}{7} a \,b^{2} x^{7}+\frac {1}{10} b^{3} x^{10}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^3*x+3/4*a^2*b*x^4+3/7*a*b^2*x^7+1/10*b^3*x^10

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Maxima [A]
time = 0.30, size = 32, normalized size = 0.84 \begin {gather*} \frac {1}{10} \, b^{3} x^{10} + \frac {3}{7} \, a b^{2} x^{7} + \frac {3}{4} \, a^{2} b x^{4} + a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^3*x^10 + 3/7*a*b^2*x^7 + 3/4*a^2*b*x^4 + a^3*x

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Fricas [A]
time = 0.34, size = 32, normalized size = 0.84 \begin {gather*} \frac {1}{10} \, b^{3} x^{10} + \frac {3}{7} \, a b^{2} x^{7} + \frac {3}{4} \, a^{2} b x^{4} + a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^3*x^10 + 3/7*a*b^2*x^7 + 3/4*a^2*b*x^4 + a^3*x

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Sympy [A]
time = 0.01, size = 36, normalized size = 0.95 \begin {gather*} a^{3} x + \frac {3 a^{2} b x^{4}}{4} + \frac {3 a b^{2} x^{7}}{7} + \frac {b^{3} x^{10}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x + 3*a**2*b*x**4/4 + 3*a*b**2*x**7/7 + b**3*x**10/10

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Giac [A]
time = 1.11, size = 32, normalized size = 0.84 \begin {gather*} \frac {1}{10} \, b^{3} x^{10} + \frac {3}{7} \, a b^{2} x^{7} + \frac {3}{4} \, a^{2} b x^{4} + a^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^3*x^10 + 3/7*a*b^2*x^7 + 3/4*a^2*b*x^4 + a^3*x

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Mupad [B]
time = 0.04, size = 32, normalized size = 0.84 \begin {gather*} a^3\,x+\frac {3\,a^2\,b\,x^4}{4}+\frac {3\,a\,b^2\,x^7}{7}+\frac {b^3\,x^{10}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^3,x)

[Out]

a^3*x + (b^3*x^10)/10 + (3*a^2*b*x^4)/4 + (3*a*b^2*x^7)/7

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