3.296 \(\int \frac{(a+b x^3)^8}{x^{13}} \, dx\)

Optimal. Leaf size=105 \[ \frac{14}{3} a^2 b^6 x^6+\frac{56}{3} a^3 b^5 x^3-\frac{56 a^5 b^3}{3 x^3}-\frac{14 a^6 b^2}{3 x^6}+70 a^4 b^4 \log (x)-\frac{8 a^7 b}{9 x^9}-\frac{a^8}{12 x^{12}}+\frac{8}{9} a b^7 x^9+\frac{b^8 x^{12}}{12} \]

[Out]

-a^8/(12*x^12) - (8*a^7*b)/(9*x^9) - (14*a^6*b^2)/(3*x^6) - (56*a^5*b^3)/(3*x^3) + (56*a^3*b^5*x^3)/3 + (14*a^
2*b^6*x^6)/3 + (8*a*b^7*x^9)/9 + (b^8*x^12)/12 + 70*a^4*b^4*Log[x]

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Rubi [A]  time = 0.0560571, 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} \[ \frac{14}{3} a^2 b^6 x^6+\frac{56}{3} a^3 b^5 x^3-\frac{56 a^5 b^3}{3 x^3}-\frac{14 a^6 b^2}{3 x^6}+70 a^4 b^4 \log (x)-\frac{8 a^7 b}{9 x^9}-\frac{a^8}{12 x^{12}}+\frac{8}{9} a b^7 x^9+\frac{b^8 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^13,x]

[Out]

-a^8/(12*x^12) - (8*a^7*b)/(9*x^9) - (14*a^6*b^2)/(3*x^6) - (56*a^5*b^3)/(3*x^3) + (56*a^3*b^5*x^3)/3 + (14*a^
2*b^6*x^6)/3 + (8*a*b^7*x^9)/9 + (b^8*x^12)/12 + 70*a^4*b^4*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^{13}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^5} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (56 a^3 b^5+\frac{a^8}{x^5}+\frac{8 a^7 b}{x^4}+\frac{28 a^6 b^2}{x^3}+\frac{56 a^5 b^3}{x^2}+\frac{70 a^4 b^4}{x}+28 a^2 b^6 x+8 a b^7 x^2+b^8 x^3\right ) \, dx,x,x^3\right )\\ &=-\frac{a^8}{12 x^{12}}-\frac{8 a^7 b}{9 x^9}-\frac{14 a^6 b^2}{3 x^6}-\frac{56 a^5 b^3}{3 x^3}+\frac{56}{3} a^3 b^5 x^3+\frac{14}{3} a^2 b^6 x^6+\frac{8}{9} a b^7 x^9+\frac{b^8 x^{12}}{12}+70 a^4 b^4 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0052424, size = 105, normalized size = 1. \[ \frac{14}{3} a^2 b^6 x^6+\frac{56}{3} a^3 b^5 x^3-\frac{56 a^5 b^3}{3 x^3}-\frac{14 a^6 b^2}{3 x^6}+70 a^4 b^4 \log (x)-\frac{8 a^7 b}{9 x^9}-\frac{a^8}{12 x^{12}}+\frac{8}{9} a b^7 x^9+\frac{b^8 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^13,x]

[Out]

-a^8/(12*x^12) - (8*a^7*b)/(9*x^9) - (14*a^6*b^2)/(3*x^6) - (56*a^5*b^3)/(3*x^3) + (56*a^3*b^5*x^3)/3 + (14*a^
2*b^6*x^6)/3 + (8*a*b^7*x^9)/9 + (b^8*x^12)/12 + 70*a^4*b^4*Log[x]

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Maple [A]  time = 0.007, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{12\,{x}^{12}}}-{\frac{8\,{a}^{7}b}{9\,{x}^{9}}}-{\frac{14\,{a}^{6}{b}^{2}}{3\,{x}^{6}}}-{\frac{56\,{a}^{5}{b}^{3}}{3\,{x}^{3}}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{3}}{3}}+{\frac{14\,{a}^{2}{b}^{6}{x}^{6}}{3}}+{\frac{8\,a{b}^{7}{x}^{9}}{9}}+{\frac{{b}^{8}{x}^{12}}{12}}+70\,{a}^{4}{b}^{4}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^13,x)

[Out]

-1/12*a^8/x^12-8/9*a^7*b/x^9-14/3*a^6*b^2/x^6-56/3*a^5*b^3/x^3+56/3*a^3*b^5*x^3+14/3*a^2*b^6*x^6+8/9*a*b^7*x^9
+1/12*b^8*x^12+70*a^4*b^4*ln(x)

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Maxima [A]  time = 0.981447, size = 127, normalized size = 1.21 \begin{align*} \frac{1}{12} \, b^{8} x^{12} + \frac{8}{9} \, a b^{7} x^{9} + \frac{14}{3} \, a^{2} b^{6} x^{6} + \frac{56}{3} \, a^{3} b^{5} x^{3} + \frac{70}{3} \, a^{4} b^{4} \log \left (x^{3}\right ) - \frac{672 \, a^{5} b^{3} x^{9} + 168 \, a^{6} b^{2} x^{6} + 32 \, a^{7} b x^{3} + 3 \, a^{8}}{36 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^13,x, algorithm="maxima")

[Out]

1/12*b^8*x^12 + 8/9*a*b^7*x^9 + 14/3*a^2*b^6*x^6 + 56/3*a^3*b^5*x^3 + 70/3*a^4*b^4*log(x^3) - 1/36*(672*a^5*b^
3*x^9 + 168*a^6*b^2*x^6 + 32*a^7*b*x^3 + 3*a^8)/x^12

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Fricas [A]  time = 1.58588, size = 221, normalized size = 2.1 \begin{align*} \frac{3 \, b^{8} x^{24} + 32 \, a b^{7} x^{21} + 168 \, a^{2} b^{6} x^{18} + 672 \, a^{3} b^{5} x^{15} + 2520 \, a^{4} b^{4} x^{12} \log \left (x\right ) - 672 \, a^{5} b^{3} x^{9} - 168 \, a^{6} b^{2} x^{6} - 32 \, a^{7} b x^{3} - 3 \, a^{8}}{36 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^13,x, algorithm="fricas")

[Out]

1/36*(3*b^8*x^24 + 32*a*b^7*x^21 + 168*a^2*b^6*x^18 + 672*a^3*b^5*x^15 + 2520*a^4*b^4*x^12*log(x) - 672*a^5*b^
3*x^9 - 168*a^6*b^2*x^6 - 32*a^7*b*x^3 - 3*a^8)/x^12

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Sympy [A]  time = 0.852344, size = 102, normalized size = 0.97 \begin{align*} 70 a^{4} b^{4} \log{\left (x \right )} + \frac{56 a^{3} b^{5} x^{3}}{3} + \frac{14 a^{2} b^{6} x^{6}}{3} + \frac{8 a b^{7} x^{9}}{9} + \frac{b^{8} x^{12}}{12} - \frac{3 a^{8} + 32 a^{7} b x^{3} + 168 a^{6} b^{2} x^{6} + 672 a^{5} b^{3} x^{9}}{36 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**13,x)

[Out]

70*a**4*b**4*log(x) + 56*a**3*b**5*x**3/3 + 14*a**2*b**6*x**6/3 + 8*a*b**7*x**9/9 + b**8*x**12/12 - (3*a**8 +
32*a**7*b*x**3 + 168*a**6*b**2*x**6 + 672*a**5*b**3*x**9)/(36*x**12)

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Giac [A]  time = 1.12443, size = 140, normalized size = 1.33 \begin{align*} \frac{1}{12} \, b^{8} x^{12} + \frac{8}{9} \, a b^{7} x^{9} + \frac{14}{3} \, a^{2} b^{6} x^{6} + \frac{56}{3} \, a^{3} b^{5} x^{3} + 70 \, a^{4} b^{4} \log \left ({\left | x \right |}\right ) - \frac{1750 \, a^{4} b^{4} x^{12} + 672 \, a^{5} b^{3} x^{9} + 168 \, a^{6} b^{2} x^{6} + 32 \, a^{7} b x^{3} + 3 \, a^{8}}{36 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^13,x, algorithm="giac")

[Out]

1/12*b^8*x^12 + 8/9*a*b^7*x^9 + 14/3*a^2*b^6*x^6 + 56/3*a^3*b^5*x^3 + 70*a^4*b^4*log(abs(x)) - 1/36*(1750*a^4*
b^4*x^12 + 672*a^5*b^3*x^9 + 168*a^6*b^2*x^6 + 32*a^7*b*x^3 + 3*a^8)/x^12