3.3.54 \(\int \frac {(a+b x^3)^3}{x^3} \, dx\) [254]

Optimal. Leaf size=39 \[ -\frac {a^3}{2 x^2}+3 a^2 b x+\frac {3}{4} a b^2 x^4+\frac {b^3 x^7}{7} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 39, 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}{2 x^2}+3 a^2 b x+\frac {3}{4} a b^2 x^4+\frac {b^3 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^3/x^3,x]

[Out]

-1/2*a^3/x^2 + 3*a^2*b*x + (3*a*b^2*x^4)/4 + (b^3*x^7)/7

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

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Mathematica [A]
time = 0.00, size = 39, normalized size = 1.00 \begin {gather*} -\frac {a^3}{2 x^2}+3 a^2 b x+\frac {3}{4} a b^2 x^4+\frac {b^3 x^7}{7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^3/x^3,x]

[Out]

-1/2*a^3/x^2 + 3*a^2*b*x + (3*a*b^2*x^4)/4 + (b^3*x^7)/7

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Maple [A]
time = 0.12, size = 34, normalized size = 0.87

method	result	size
default	\(-\frac {a^{3}}{2 x^{2}}+3 a^{2} b x +\frac {3 a \,b^{2} x^{4}}{4}+\frac {b^{3} x^{7}}{7}\)	\(34\)
risch	\(-\frac {a^{3}}{2 x^{2}}+3 a^{2} b x +\frac {3 a \,b^{2} x^{4}}{4}+\frac {b^{3} x^{7}}{7}\)	\(34\)
norman	\(\frac {\frac {1}{7} b^{3} x^{9}+\frac {3}{4} a \,b^{2} x^{6}+3 a^{2} b \,x^{3}-\frac {1}{2} a^{3}}{x^{2}}\)	\(37\)
gosper	\(-\frac {-4 b^{3} x^{9}-21 a \,b^{2} x^{6}-84 a^{2} b \,x^{3}+14 a^{3}}{28 x^{2}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 33, normalized size = 0.85 \begin {gather*} \frac {1}{7} \, b^{3} x^{7} + \frac {3}{4} \, a b^{2} x^{4} + 3 \, a^{2} b x - \frac {a^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.35, size = 37, normalized size = 0.95 \begin {gather*} \frac {4 \, b^{3} x^{9} + 21 \, a b^{2} x^{6} + 84 \, a^{2} b x^{3} - 14 \, a^{3}}{28 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*(4*b^3*x^9 + 21*a*b^2*x^6 + 84*a^2*b*x^3 - 14*a^3)/x^2

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Sympy [A]
time = 0.03, size = 36, normalized size = 0.92 \begin {gather*} - \frac {a^{3}}{2 x^{2}} + 3 a^{2} b x + \frac {3 a b^{2} x^{4}}{4} + \frac {b^{3} x^{7}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.34, size = 33, normalized size = 0.85 \begin {gather*} \frac {1}{7} \, b^{3} x^{7} + \frac {3}{4} \, a b^{2} x^{4} + 3 \, a^{2} b x - \frac {a^{3}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.04, size = 33, normalized size = 0.85 \begin {gather*} \frac {b^3\,x^7}{7}-\frac {a^3}{2\,x^2}+\frac {3\,a\,b^2\,x^4}{4}+3\,a^2\,b\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^3/x^3,x)

[Out]

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

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