3.297 \(\int \frac{(a+b x^3)^8}{x^{16}} \, dx\)

Optimal. Leaf size=105 \[ \frac{28}{3} a^2 b^6 x^3-\frac{70 a^4 b^4}{3 x^3}-\frac{28 a^5 b^3}{3 x^6}-\frac{28 a^6 b^2}{9 x^9}+56 a^3 b^5 \log (x)-\frac{2 a^7 b}{3 x^{12}}-\frac{a^8}{15 x^{15}}+\frac{4}{3} a b^7 x^6+\frac{b^8 x^9}{9} \]

[Out]

-a^8/(15*x^15) - (2*a^7*b)/(3*x^12) - (28*a^6*b^2)/(9*x^9) - (28*a^5*b^3)/(3*x^6) - (70*a^4*b^4)/(3*x^3) + (28
*a^2*b^6*x^3)/3 + (4*a*b^7*x^6)/3 + (b^8*x^9)/9 + 56*a^3*b^5*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0598809, antiderivative size = 105, normalized size of antiderivative = 1., 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} \[ \frac{28}{3} a^2 b^6 x^3-\frac{70 a^4 b^4}{3 x^3}-\frac{28 a^5 b^3}{3 x^6}-\frac{28 a^6 b^2}{9 x^9}+56 a^3 b^5 \log (x)-\frac{2 a^7 b}{3 x^{12}}-\frac{a^8}{15 x^{15}}+\frac{4}{3} a b^7 x^6+\frac{b^8 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^16,x]

[Out]

-a^8/(15*x^15) - (2*a^7*b)/(3*x^12) - (28*a^6*b^2)/(9*x^9) - (28*a^5*b^3)/(3*x^6) - (70*a^4*b^4)/(3*x^3) + (28
*a^2*b^6*x^3)/3 + (4*a*b^7*x^6)/3 + (b^8*x^9)/9 + 56*a^3*b^5*Log[x]

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]]

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])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^8}{x^{16}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^6} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (28 a^2 b^6+\frac{a^8}{x^6}+\frac{8 a^7 b}{x^5}+\frac{28 a^6 b^2}{x^4}+\frac{56 a^5 b^3}{x^3}+\frac{70 a^4 b^4}{x^2}+\frac{56 a^3 b^5}{x}+8 a b^7 x+b^8 x^2\right ) \, dx,x,x^3\right )\\ &=-\frac{a^8}{15 x^{15}}-\frac{2 a^7 b}{3 x^{12}}-\frac{28 a^6 b^2}{9 x^9}-\frac{28 a^5 b^3}{3 x^6}-\frac{70 a^4 b^4}{3 x^3}+\frac{28}{3} a^2 b^6 x^3+\frac{4}{3} a b^7 x^6+\frac{b^8 x^9}{9}+56 a^3 b^5 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0068886, size = 105, normalized size = 1. \[ \frac{28}{3} a^2 b^6 x^3-\frac{70 a^4 b^4}{3 x^3}-\frac{28 a^5 b^3}{3 x^6}-\frac{28 a^6 b^2}{9 x^9}+56 a^3 b^5 \log (x)-\frac{2 a^7 b}{3 x^{12}}-\frac{a^8}{15 x^{15}}+\frac{4}{3} a b^7 x^6+\frac{b^8 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^16,x]

[Out]

-a^8/(15*x^15) - (2*a^7*b)/(3*x^12) - (28*a^6*b^2)/(9*x^9) - (28*a^5*b^3)/(3*x^6) - (70*a^4*b^4)/(3*x^3) + (28
*a^2*b^6*x^3)/3 + (4*a*b^7*x^6)/3 + (b^8*x^9)/9 + 56*a^3*b^5*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{15\,{x}^{15}}}-{\frac{2\,{a}^{7}b}{3\,{x}^{12}}}-{\frac{28\,{a}^{6}{b}^{2}}{9\,{x}^{9}}}-{\frac{28\,{a}^{5}{b}^{3}}{3\,{x}^{6}}}-{\frac{70\,{a}^{4}{b}^{4}}{3\,{x}^{3}}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{3}}{3}}+{\frac{4\,a{b}^{7}{x}^{6}}{3}}+{\frac{{b}^{8}{x}^{9}}{9}}+56\,{a}^{3}{b}^{5}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^16,x)

[Out]

-1/15*a^8/x^15-2/3*a^7*b/x^12-28/9*a^6*b^2/x^9-28/3*a^5*b^3/x^6-70/3*a^4*b^4/x^3+28/3*a^2*b^6*x^3+4/3*a*b^7*x^
6+1/9*b^8*x^9+56*a^3*b^5*ln(x)

________________________________________________________________________________________

Maxima [A]  time = 0.965338, size = 127, normalized size = 1.21 \begin{align*} \frac{1}{9} \, b^{8} x^{9} + \frac{4}{3} \, a b^{7} x^{6} + \frac{28}{3} \, a^{2} b^{6} x^{3} + \frac{56}{3} \, a^{3} b^{5} \log \left (x^{3}\right ) - \frac{1050 \, a^{4} b^{4} x^{12} + 420 \, a^{5} b^{3} x^{9} + 140 \, a^{6} b^{2} x^{6} + 30 \, a^{7} b x^{3} + 3 \, a^{8}}{45 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^16,x, algorithm="maxima")

[Out]

1/9*b^8*x^9 + 4/3*a*b^7*x^6 + 28/3*a^2*b^6*x^3 + 56/3*a^3*b^5*log(x^3) - 1/45*(1050*a^4*b^4*x^12 + 420*a^5*b^3
*x^9 + 140*a^6*b^2*x^6 + 30*a^7*b*x^3 + 3*a^8)/x^15

________________________________________________________________________________________

Fricas [A]  time = 1.6638, size = 223, normalized size = 2.12 \begin{align*} \frac{5 \, b^{8} x^{24} + 60 \, a b^{7} x^{21} + 420 \, a^{2} b^{6} x^{18} + 2520 \, a^{3} b^{5} x^{15} \log \left (x\right ) - 1050 \, a^{4} b^{4} x^{12} - 420 \, a^{5} b^{3} x^{9} - 140 \, a^{6} b^{2} x^{6} - 30 \, a^{7} b x^{3} - 3 \, a^{8}}{45 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^16,x, algorithm="fricas")

[Out]

1/45*(5*b^8*x^24 + 60*a*b^7*x^21 + 420*a^2*b^6*x^18 + 2520*a^3*b^5*x^15*log(x) - 1050*a^4*b^4*x^12 - 420*a^5*b
^3*x^9 - 140*a^6*b^2*x^6 - 30*a^7*b*x^3 - 3*a^8)/x^15

________________________________________________________________________________________

Sympy [A]  time = 1.05132, size = 100, normalized size = 0.95 \begin{align*} 56 a^{3} b^{5} \log{\left (x \right )} + \frac{28 a^{2} b^{6} x^{3}}{3} + \frac{4 a b^{7} x^{6}}{3} + \frac{b^{8} x^{9}}{9} - \frac{3 a^{8} + 30 a^{7} b x^{3} + 140 a^{6} b^{2} x^{6} + 420 a^{5} b^{3} x^{9} + 1050 a^{4} b^{4} x^{12}}{45 x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**16,x)

[Out]

56*a**3*b**5*log(x) + 28*a**2*b**6*x**3/3 + 4*a*b**7*x**6/3 + b**8*x**9/9 - (3*a**8 + 30*a**7*b*x**3 + 140*a**
6*b**2*x**6 + 420*a**5*b**3*x**9 + 1050*a**4*b**4*x**12)/(45*x**15)

________________________________________________________________________________________

Giac [A]  time = 1.1516, size = 140, normalized size = 1.33 \begin{align*} \frac{1}{9} \, b^{8} x^{9} + \frac{4}{3} \, a b^{7} x^{6} + \frac{28}{3} \, a^{2} b^{6} x^{3} + 56 \, a^{3} b^{5} \log \left ({\left | x \right |}\right ) - \frac{1918 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 420 \, a^{5} b^{3} x^{9} + 140 \, a^{6} b^{2} x^{6} + 30 \, a^{7} b x^{3} + 3 \, a^{8}}{45 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^16,x, algorithm="giac")

[Out]

1/9*b^8*x^9 + 4/3*a*b^7*x^6 + 28/3*a^2*b^6*x^3 + 56*a^3*b^5*log(abs(x)) - 1/45*(1918*a^3*b^5*x^15 + 1050*a^4*b
^4*x^12 + 420*a^5*b^3*x^9 + 140*a^6*b^2*x^6 + 30*a^7*b*x^3 + 3*a^8)/x^15