3.3.47 \(\int \frac {(a+b x^3)^3}{x^{19}} \, dx\) [247]

Optimal. Leaf size=43 \[ -\frac {a^3}{18 x^{18}}-\frac {a^2 b}{5 x^{15}}-\frac {a b^2}{4 x^{12}}-\frac {b^3}{9 x^9} \]

[Out]

-1/18*a^3/x^18-1/5*a^2*b/x^15-1/4*a*b^2/x^12-1/9*b^3/x^9

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Rubi [A]
time = 0.01, antiderivative size = 43, 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^3}{18 x^{18}}-\frac {a^2 b}{5 x^{15}}-\frac {a b^2}{4 x^{12}}-\frac {b^3}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^3/x^19,x]

[Out]

-1/18*a^3/x^18 - (a^2*b)/(5*x^15) - (a*b^2)/(4*x^12) - b^3/(9*x^9)

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 )^3}{x^{19}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^3}{x^7} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a^3}{x^7}+\frac {3 a^2 b}{x^6}+\frac {3 a b^2}{x^5}+\frac {b^3}{x^4}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^3}{18 x^{18}}-\frac {a^2 b}{5 x^{15}}-\frac {a b^2}{4 x^{12}}-\frac {b^3}{9 x^9}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} -\frac {a^3}{18 x^{18}}-\frac {a^2 b}{5 x^{15}}-\frac {a b^2}{4 x^{12}}-\frac {b^3}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^3/x^19,x]

[Out]

-1/18*a^3/x^18 - (a^2*b)/(5*x^15) - (a*b^2)/(4*x^12) - b^3/(9*x^9)

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Maple [A]
time = 0.12, size = 36, normalized size = 0.84

method	result	size
default	\(-\frac {a^{3}}{18 x^{18}}-\frac {a^{2} b}{5 x^{15}}-\frac {a \,b^{2}}{4 x^{12}}-\frac {b^{3}}{9 x^{9}}\)	\(36\)
norman	\(\frac {-\frac {1}{18} a^{3}-\frac {1}{5} a^{2} b \,x^{3}-\frac {1}{4} a \,b^{2} x^{6}-\frac {1}{9} b^{3} x^{9}}{x^{18}}\)	\(37\)
risch	\(\frac {-\frac {1}{18} a^{3}-\frac {1}{5} a^{2} b \,x^{3}-\frac {1}{4} a \,b^{2} x^{6}-\frac {1}{9} b^{3} x^{9}}{x^{18}}\)	\(37\)
gosper	\(-\frac {20 b^{3} x^{9}+45 a \,b^{2} x^{6}+36 a^{2} b \,x^{3}+10 a^{3}}{180 x^{18}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^3/x^19,x,method=_RETURNVERBOSE)

[Out]

-1/18*a^3/x^18-1/5*a^2*b/x^15-1/4*a*b^2/x^12-1/9*b^3/x^9

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Maxima [A]
time = 0.30, size = 37, normalized size = 0.86 \begin {gather*} -\frac {20 \, b^{3} x^{9} + 45 \, a b^{2} x^{6} + 36 \, a^{2} b x^{3} + 10 \, a^{3}}{180 \, x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^19,x, algorithm="maxima")

[Out]

-1/180*(20*b^3*x^9 + 45*a*b^2*x^6 + 36*a^2*b*x^3 + 10*a^3)/x^18

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Fricas [A]
time = 0.34, size = 37, normalized size = 0.86 \begin {gather*} -\frac {20 \, b^{3} x^{9} + 45 \, a b^{2} x^{6} + 36 \, a^{2} b x^{3} + 10 \, a^{3}}{180 \, x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^19,x, algorithm="fricas")

[Out]

-1/180*(20*b^3*x^9 + 45*a*b^2*x^6 + 36*a^2*b*x^3 + 10*a^3)/x^18

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Sympy [A]
time = 0.17, size = 39, normalized size = 0.91 \begin {gather*} \frac {- 10 a^{3} - 36 a^{2} b x^{3} - 45 a b^{2} x^{6} - 20 b^{3} x^{9}}{180 x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**3/x**19,x)

[Out]

(-10*a**3 - 36*a**2*b*x**3 - 45*a*b**2*x**6 - 20*b**3*x**9)/(180*x**18)

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Giac [A]
time = 1.19, size = 37, normalized size = 0.86 \begin {gather*} -\frac {20 \, b^{3} x^{9} + 45 \, a b^{2} x^{6} + 36 \, a^{2} b x^{3} + 10 \, a^{3}}{180 \, x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^3/x^19,x, algorithm="giac")

[Out]

-1/180*(20*b^3*x^9 + 45*a*b^2*x^6 + 36*a^2*b*x^3 + 10*a^3)/x^18

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Mupad [B]
time = 0.03, size = 37, normalized size = 0.86 \begin {gather*} -\frac {\frac {a^3}{18}+\frac {a^2\,b\,x^3}{5}+\frac {a\,b^2\,x^6}{4}+\frac {b^3\,x^9}{9}}{x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^3/x^19,x)

[Out]

-(a^3/18 + (b^3*x^9)/9 + (a^2*b*x^3)/5 + (a*b^2*x^6)/4)/x^18

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