3.269 \(\int \frac {(a+b x^3)^5}{x^{16}} \, dx\)

Optimal. Leaf size=65 \[ -\frac {a^5}{15 x^{15}}-\frac {5 a^4 b}{12 x^{12}}-\frac {10 a^3 b^2}{9 x^9}-\frac {5 a^2 b^3}{3 x^6}-\frac {5 a b^4}{3 x^3}+b^5 \log (x) \]

[Out]

-1/15*a^5/x^15-5/12*a^4*b/x^12-10/9*a^3*b^2/x^9-5/3*a^2*b^3/x^6-5/3*a*b^4/x^3+b^5*ln(x)

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Rubi [A]  time = 0.03, antiderivative size = 65, 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 = {266, 43} \[ -\frac {10 a^3 b^2}{9 x^9}-\frac {5 a^2 b^3}{3 x^6}-\frac {5 a^4 b}{12 x^{12}}-\frac {a^5}{15 x^{15}}-\frac {5 a b^4}{3 x^3}+b^5 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^16,x]

[Out]

-a^5/(15*x^15) - (5*a^4*b)/(12*x^12) - (10*a^3*b^2)/(9*x^9) - (5*a^2*b^3)/(3*x^6) - (5*a*b^4)/(3*x^3) + b^5*Lo
g[x]

Rule 43

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 266

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 )^5}{x^{16}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(a+b x)^5}{x^6} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {a^5}{x^6}+\frac {5 a^4 b}{x^5}+\frac {10 a^3 b^2}{x^4}+\frac {10 a^2 b^3}{x^3}+\frac {5 a b^4}{x^2}+\frac {b^5}{x}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^5}{15 x^{15}}-\frac {5 a^4 b}{12 x^{12}}-\frac {10 a^3 b^2}{9 x^9}-\frac {5 a^2 b^3}{3 x^6}-\frac {5 a b^4}{3 x^3}+b^5 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 65, normalized size = 1.00 \[ -\frac {a^5}{15 x^{15}}-\frac {5 a^4 b}{12 x^{12}}-\frac {10 a^3 b^2}{9 x^9}-\frac {5 a^2 b^3}{3 x^6}-\frac {5 a b^4}{3 x^3}+b^5 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^16,x]

[Out]

-1/15*a^5/x^15 - (5*a^4*b)/(12*x^12) - (10*a^3*b^2)/(9*x^9) - (5*a^2*b^3)/(3*x^6) - (5*a*b^4)/(3*x^3) + b^5*Lo
g[x]

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fricas [A]  time = 0.70, size = 61, normalized size = 0.94 \[ \frac {180 \, b^{5} x^{15} \log \relax (x) - 300 \, a b^{4} x^{12} - 300 \, a^{2} b^{3} x^{9} - 200 \, a^{3} b^{2} x^{6} - 75 \, a^{4} b x^{3} - 12 \, a^{5}}{180 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^16,x, algorithm="fricas")

[Out]

1/180*(180*b^5*x^15*log(x) - 300*a*b^4*x^12 - 300*a^2*b^3*x^9 - 200*a^3*b^2*x^6 - 75*a^4*b*x^3 - 12*a^5)/x^15

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giac [A]  time = 0.16, size = 67, normalized size = 1.03 \[ b^{5} \log \left ({\left | x \right |}\right ) - \frac {137 \, b^{5} x^{15} + 300 \, a b^{4} x^{12} + 300 \, a^{2} b^{3} x^{9} + 200 \, a^{3} b^{2} x^{6} + 75 \, a^{4} b x^{3} + 12 \, a^{5}}{180 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^16,x, algorithm="giac")

[Out]

b^5*log(abs(x)) - 1/180*(137*b^5*x^15 + 300*a*b^4*x^12 + 300*a^2*b^3*x^9 + 200*a^3*b^2*x^6 + 75*a^4*b*x^3 + 12
*a^5)/x^15

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maple [A]  time = 0.01, size = 56, normalized size = 0.86 \[ b^{5} \ln \relax (x )-\frac {5 a \,b^{4}}{3 x^{3}}-\frac {5 a^{2} b^{3}}{3 x^{6}}-\frac {10 a^{3} b^{2}}{9 x^{9}}-\frac {5 a^{4} b}{12 x^{12}}-\frac {a^{5}}{15 x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^16,x)

[Out]

-1/15*a^5/x^15-5/12*a^4*b/x^12-10/9*a^3*b^2/x^9-5/3*a^2*b^3/x^6-5/3*a*b^4/x^3+b^5*ln(x)

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maxima [A]  time = 1.35, size = 61, normalized size = 0.94 \[ \frac {1}{3} \, b^{5} \log \left (x^{3}\right ) - \frac {300 \, a b^{4} x^{12} + 300 \, a^{2} b^{3} x^{9} + 200 \, a^{3} b^{2} x^{6} + 75 \, a^{4} b x^{3} + 12 \, a^{5}}{180 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^16,x, algorithm="maxima")

[Out]

1/3*b^5*log(x^3) - 1/180*(300*a*b^4*x^12 + 300*a^2*b^3*x^9 + 200*a^3*b^2*x^6 + 75*a^4*b*x^3 + 12*a^5)/x^15

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mupad [B]  time = 1.01, size = 58, normalized size = 0.89 \[ b^5\,\ln \relax (x)-\frac {\frac {a^5}{15}+\frac {5\,a^4\,b\,x^3}{12}+\frac {10\,a^3\,b^2\,x^6}{9}+\frac {5\,a^2\,b^3\,x^9}{3}+\frac {5\,a\,b^4\,x^{12}}{3}}{x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^5/x^16,x)

[Out]

b^5*log(x) - (a^5/15 + (5*a^4*b*x^3)/12 + (5*a*b^4*x^12)/3 + (10*a^3*b^2*x^6)/9 + (5*a^2*b^3*x^9)/3)/x^15

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sympy [A]  time = 0.68, size = 61, normalized size = 0.94 \[ b^{5} \log {\relax (x )} + \frac {- 12 a^{5} - 75 a^{4} b x^{3} - 200 a^{3} b^{2} x^{6} - 300 a^{2} b^{3} x^{9} - 300 a b^{4} x^{12}}{180 x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**16,x)

[Out]

b**5*log(x) + (-12*a**5 - 75*a**4*b*x**3 - 200*a**3*b**2*x**6 - 300*a**2*b**3*x**9 - 300*a*b**4*x**12)/(180*x*
*15)

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