\(\int \frac {(a+b x^3)^8}{x^{13}} \, dx\) [296]

Optimal result

Integrand size = 13, antiderivative size = 105 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=-\frac {a^8}{12 x^{12}}-\frac {8 a^7 b}{9 x^9}-\frac {14 a^6 b^2}{3 x^6}-\frac {56 a^5 b^3}{3 x^3}+\frac {56}{3} a^3 b^5 x^3+\frac {14}{3} a^2 b^6 x^6+\frac {8}{9} a b^7 x^9+\frac {b^8 x^{12}}{12}+70 a^4 b^4 \log (x) \]

[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)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=-\frac {a^8}{12 x^{12}}-\frac {8 a^7 b}{9 x^9}-\frac {14 a^6 b^2}{3 x^6}-\frac {56 a^5 b^3}{3 x^3}+70 a^4 b^4 \log (x)+\frac {56}{3} a^3 b^5 x^3+\frac {14}{3} a^2 b^6 x^6+\frac {8}{9} a b^7 x^9+\frac {b^8 x^{12}}{12} \]

[In]

Int[(a + b*x^3)^8/x^13,x]

[Out]

-1/12*a^8/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 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*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^8}{x^5} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {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] (verified)

Time = 0.00 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=-\frac {a^8}{12 x^{12}}-\frac {8 a^7 b}{9 x^9}-\frac {14 a^6 b^2}{3 x^6}-\frac {56 a^5 b^3}{3 x^3}+\frac {56}{3} a^3 b^5 x^3+\frac {14}{3} a^2 b^6 x^6+\frac {8}{9} a b^7 x^9+\frac {b^8 x^{12}}{12}+70 a^4 b^4 \log (x) \]

[In]

Integrate[(a + b*x^3)^8/x^13,x]

[Out]

-1/12*a^8/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]

Maple [A] (verified)

Time = 3.64 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {a^{8}}{12 x^{12}}-\frac {8 a^{7} b}{9 x^{9}}-\frac {14 a^{6} b^{2}}{3 x^{6}}-\frac {56 a^{5} b^{3}}{3 x^{3}}+\frac {56 a^{3} b^{5} x^{3}}{3}+\frac {14 a^{2} x^{6} b^{6}}{3}+\frac {8 a \,b^{7} x^{9}}{9}+\frac {b^{8} x^{12}}{12}+70 a^{4} b^{4} \ln \left (x \right )\)	\(90\)
norman	\(\frac {-\frac {1}{12} a^{8}+\frac {1}{12} b^{8} x^{24}+\frac {8}{9} a \,b^{7} x^{21}+\frac {14}{3} a^{2} b^{6} x^{18}+\frac {56}{3} a^{3} b^{5} x^{15}-\frac {14}{3} a^{6} b^{2} x^{6}-\frac {8}{9} x^{3} b \,a^{7}-\frac {56}{3} x^{9} b^{3} a^{5}}{x^{12}}+70 a^{4} b^{4} \ln \left (x \right )\)	\(92\)
risch	\(\frac {b^{8} x^{12}}{12}+\frac {8 a \,b^{7} x^{9}}{9}+\frac {14 a^{2} x^{6} b^{6}}{3}+\frac {56 a^{3} b^{5} x^{3}}{3}+\frac {-\frac {56}{3} x^{9} b^{3} a^{5}-\frac {14}{3} a^{6} b^{2} x^{6}-\frac {8}{9} x^{3} b \,a^{7}-\frac {1}{12} a^{8}}{x^{12}}+70 a^{4} b^{4} \ln \left (x \right )\)	\(92\)
parallelrisch	\(\frac {3 b^{8} x^{24}+32 a \,b^{7} x^{21}+168 a^{2} b^{6} x^{18}+672 a^{3} b^{5} x^{15}+2520 a^{4} b^{4} \ln \left (x \right ) x^{12}-672 x^{9} b^{3} a^{5}-168 a^{6} b^{2} x^{6}-32 x^{3} b \,a^{7}-3 a^{8}}{36 x^{12}}\)	\(95\)

[In]

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

[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*x^6*b^6+8/9*a*b^7*x^9
+1/12*b^8*x^12+70*a^4*b^4*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=\frac {3 \, b^{8} x^{24} + 32 \, a b^{7} x^{21} + 168 \, a^{2} b^{6} x^{18} + 672 \, a^{3} b^{5} x^{15} + 2520 \, a^{4} b^{4} x^{12} \log \left (x\right ) - 672 \, a^{5} b^{3} x^{9} - 168 \, a^{6} b^{2} x^{6} - 32 \, a^{7} b x^{3} - 3 \, a^{8}}{36 \, x^{12}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=70 a^{4} b^{4} \log {\left (x \right )} + \frac {56 a^{3} b^{5} x^{3}}{3} + \frac {14 a^{2} b^{6} x^{6}}{3} + \frac {8 a b^{7} x^{9}}{9} + \frac {b^{8} x^{12}}{12} + \frac {- 3 a^{8} - 32 a^{7} b x^{3} - 168 a^{6} b^{2} x^{6} - 672 a^{5} b^{3} x^{9}}{36 x^{12}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=\frac {1}{12} \, b^{8} x^{12} + \frac {8}{9} \, a b^{7} x^{9} + \frac {14}{3} \, a^{2} b^{6} x^{6} + \frac {56}{3} \, a^{3} b^{5} x^{3} + \frac {70}{3} \, a^{4} b^{4} \log \left (x^{3}\right ) - \frac {672 \, a^{5} b^{3} x^{9} + 168 \, a^{6} b^{2} x^{6} + 32 \, a^{7} b x^{3} + 3 \, a^{8}}{36 \, x^{12}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=\frac {1}{12} \, b^{8} x^{12} + \frac {8}{9} \, a b^{7} x^{9} + \frac {14}{3} \, a^{2} b^{6} x^{6} + \frac {56}{3} \, a^{3} b^{5} x^{3} + 70 \, a^{4} b^{4} \log \left ({\left | x \right |}\right ) - \frac {1750 \, a^{4} b^{4} x^{12} + 672 \, a^{5} b^{3} x^{9} + 168 \, a^{6} b^{2} x^{6} + 32 \, a^{7} b x^{3} + 3 \, a^{8}}{36 \, x^{12}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^3\right )^8}{x^{13}} \, dx=\frac {b^8\,x^{12}}{12}-\frac {\frac {a^8}{12}+\frac {8\,a^7\,b\,x^3}{9}+\frac {14\,a^6\,b^2\,x^6}{3}+\frac {56\,a^5\,b^3\,x^9}{3}}{x^{12}}+\frac {8\,a\,b^7\,x^9}{9}+\frac {56\,a^3\,b^5\,x^3}{3}+\frac {14\,a^2\,b^6\,x^6}{3}+70\,a^4\,b^4\,\ln \left (x\right ) \]

[In]

int((a + b*x^3)^8/x^13,x)

[Out]

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