3.3.35 \(\int \frac {(a+b x^3)^2}{x^{13}} \, dx\) [235]

Optimal. Leaf size=30 \[ -\frac {a^2}{12 x^{12}}-\frac {2 a b}{9 x^9}-\frac {b^2}{6 x^6} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \begin {gather*} -\frac {a^2}{12 x^{12}}-\frac {2 a b}{9 x^9}-\frac {b^2}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^13,x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^2}{x^{13}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^2}{x^5} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^2}{x^5}+\frac {2 a b}{x^4}+\frac {b^2}{x^3}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^2}{12 x^{12}}-\frac {2 a b}{9 x^9}-\frac {b^2}{6 x^6}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} -\frac {a^2}{12 x^{12}}-\frac {2 a b}{9 x^9}-\frac {b^2}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^13,x]

[Out]

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

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Maple [A]
time = 0.11, size = 25, normalized size = 0.83

method	result	size
default	\(-\frac {a^{2}}{12 x^{12}}-\frac {2 a b}{9 x^{9}}-\frac {b^{2}}{6 x^{6}}\)	\(25\)
norman	\(\frac {-\frac {1}{6} b^{2} x^{6}-\frac {2}{9} a b \,x^{3}-\frac {1}{12} a^{2}}{x^{12}}\)	\(26\)
risch	\(\frac {-\frac {1}{6} b^{2} x^{6}-\frac {2}{9} a b \,x^{3}-\frac {1}{12} a^{2}}{x^{12}}\)	\(26\)
gosper	\(-\frac {6 b^{2} x^{6}+8 a b \,x^{3}+3 a^{2}}{36 x^{12}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2/x^13,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 26, normalized size = 0.87 \begin {gather*} -\frac {6 \, b^{2} x^{6} + 8 \, a b x^{3} + 3 \, a^{2}}{36 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^13,x, algorithm="maxima")

[Out]

-1/36*(6*b^2*x^6 + 8*a*b*x^3 + 3*a^2)/x^12

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Fricas [A]
time = 0.34, size = 26, normalized size = 0.87 \begin {gather*} -\frac {6 \, b^{2} x^{6} + 8 \, a b x^{3} + 3 \, a^{2}}{36 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^13,x, algorithm="fricas")

[Out]

-1/36*(6*b^2*x^6 + 8*a*b*x^3 + 3*a^2)/x^12

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Sympy [A]
time = 0.10, size = 27, normalized size = 0.90 \begin {gather*} \frac {- 3 a^{2} - 8 a b x^{3} - 6 b^{2} x^{6}}{36 x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2/x**13,x)

[Out]

(-3*a**2 - 8*a*b*x**3 - 6*b**2*x**6)/(36*x**12)

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Giac [A]
time = 1.33, size = 26, normalized size = 0.87 \begin {gather*} -\frac {6 \, b^{2} x^{6} + 8 \, a b x^{3} + 3 \, a^{2}}{36 \, x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^13,x, algorithm="giac")

[Out]

-1/36*(6*b^2*x^6 + 8*a*b*x^3 + 3*a^2)/x^12

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Mupad [B]
time = 0.04, size = 26, normalized size = 0.87 \begin {gather*} -\frac {\frac {a^2}{12}+\frac {2\,a\,b\,x^3}{9}+\frac {b^2\,x^6}{6}}{x^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^2/x^13,x)

[Out]

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

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