3.3.45 \(\int \frac {(a+b x^3)^3}{x^{13}} \, dx\) [245]

Optimal. Leaf size=19 \[ -\frac {\left (a+b x^3\right )^4}{12 a x^{12}} \]

[Out]

-1/12*(b*x^3+a)^4/a/x^12

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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \begin {gather*} -\frac {\left (a+b x^3\right )^4}{12 a x^{12}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^3/x^13,x]

[Out]

-1/12*(a + b*x^3)^4/(a*x^12)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^3}{x^{13}} \, dx &=-\frac {\left (a+b x^3\right )^4}{12 a x^{12}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(19)=38\).
time = 0.00, size = 43, normalized size = 2.26 \begin {gather*} -\frac {a^3}{12 x^{12}}-\frac {a^2 b}{3 x^9}-\frac {a b^2}{2 x^6}-\frac {b^3}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^3/x^13,x]

[Out]

-1/12*a^3/x^12 - (a^2*b)/(3*x^9) - (a*b^2)/(2*x^6) - b^3/(3*x^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(35\) vs. \(2(17)=34\).
time = 0.13, size = 36, normalized size = 1.89

method	result	size
gosper	\(-\frac {4 b^{3} x^{9}+6 a \,b^{2} x^{6}+4 a^{2} b \,x^{3}+a^{3}}{12 x^{12}}\)	\(36\)
default	\(-\frac {a^{2} b}{3 x^{9}}-\frac {a^{3}}{12 x^{12}}-\frac {a \,b^{2}}{2 x^{6}}-\frac {b^{3}}{3 x^{3}}\)	\(36\)
norman	\(\frac {-\frac {1}{3} b^{3} x^{9}-\frac {1}{2} a \,b^{2} x^{6}-\frac {1}{3} a^{2} b \,x^{3}-\frac {1}{12} a^{3}}{x^{12}}\)	\(37\)
risch	\(\frac {-\frac {1}{3} b^{3} x^{9}-\frac {1}{2} a \,b^{2} x^{6}-\frac {1}{3} a^{2} b \,x^{3}-\frac {1}{12} a^{3}}{x^{12}}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^3/x^13,x,method=_RETURNVERBOSE)

[Out]

-1/3*a^2*b/x^9-1/12*a^3/x^12-1/2*a*b^2/x^6-1/3*b^3/x^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).
time = 0.30, size = 35, normalized size = 1.84 \begin {gather*} -\frac {4 \, b^{3} x^{9} + 6 \, a b^{2} x^{6} + 4 \, a^{2} b x^{3} + a^{3}}{12 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^13,x, algorithm="maxima")

[Out]

-1/12*(4*b^3*x^9 + 6*a*b^2*x^6 + 4*a^2*b*x^3 + a^3)/x^12

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).
time = 0.33, size = 35, normalized size = 1.84 \begin {gather*} -\frac {4 \, b^{3} x^{9} + 6 \, a b^{2} x^{6} + 4 \, a^{2} b x^{3} + a^{3}}{12 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^13,x, algorithm="fricas")

[Out]

-1/12*(4*b^3*x^9 + 6*a*b^2*x^6 + 4*a^2*b*x^3 + a^3)/x^12

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).
time = 0.14, size = 37, normalized size = 1.95 \begin {gather*} \frac {- a^{3} - 4 a^{2} b x^{3} - 6 a b^{2} x^{6} - 4 b^{3} x^{9}}{12 x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**3/x**13,x)

[Out]

(-a**3 - 4*a**2*b*x**3 - 6*a*b**2*x**6 - 4*b**3*x**9)/(12*x**12)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).
time = 1.41, size = 35, normalized size = 1.84 \begin {gather*} -\frac {4 \, b^{3} x^{9} + 6 \, a b^{2} x^{6} + 4 \, a^{2} b x^{3} + a^{3}}{12 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^13,x, algorithm="giac")

[Out]

-1/12*(4*b^3*x^9 + 6*a*b^2*x^6 + 4*a^2*b*x^3 + a^3)/x^12

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Mupad [B]
time = 0.03, size = 37, normalized size = 1.95 \begin {gather*} -\frac {\frac {a^3}{12}+\frac {a^2\,b\,x^3}{3}+\frac {a\,b^2\,x^6}{2}+\frac {b^3\,x^9}{3}}{x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^3/x^13,x)

[Out]

-(a^3/12 + (b^3*x^9)/3 + (a^2*b*x^3)/3 + (a*b^2*x^6)/2)/x^12

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