\(\int \frac {(a+b x^3)^8}{x^{19}} \, dx\) [298]

Optimal result

Integrand size = 13, antiderivative size = 105 \[ \int \frac {\left (a+b x^3\right )^8}{x^{19}} \, dx=-\frac {a^8}{18 x^{18}}-\frac {8 a^7 b}{15 x^{15}}-\frac {7 a^6 b^2}{3 x^{12}}-\frac {56 a^5 b^3}{9 x^9}-\frac {35 a^4 b^4}{3 x^6}-\frac {56 a^3 b^5}{3 x^3}+\frac {8}{3} a b^7 x^3+\frac {b^8 x^6}{6}+28 a^2 b^6 \log (x) \]

[Out]

-1/18*a^8/x^18-8/15*a^7*b/x^15-7/3*a^6*b^2/x^12-56/9*a^5*b^3/x^9-35/3*a^4*b^4/x^6-56/3*a^3*b^5/x^3+8/3*a*b^7*x
^3+1/6*b^8*x^6+28*a^2*b^6*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} \[ \int \frac {\left (a+b x^3\right )^8}{x^{19}} \, dx=-\frac {a^8}{18 x^{18}}-\frac {8 a^7 b}{15 x^{15}}-\frac {7 a^6 b^2}{3 x^{12}}-\frac {56 a^5 b^3}{9 x^9}-\frac {35 a^4 b^4}{3 x^6}-\frac {56 a^3 b^5}{3 x^3}+28 a^2 b^6 \log (x)+\frac {8}{3} a b^7 x^3+\frac {b^8 x^6}{6} \]

[In]

Int[(a + b*x^3)^8/x^19,x]

[Out]

-1/18*a^8/x^18 - (8*a^7*b)/(15*x^15) - (7*a^6*b^2)/(3*x^12) - (56*a^5*b^3)/(9*x^9) - (35*a^4*b^4)/(3*x^6) - (5
6*a^3*b^5)/(3*x^3) + (8*a*b^7*x^3)/3 + (b^8*x^6)/6 + 28*a^2*b^6*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^7} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (8 a b^7+\frac {a^8}{x^7}+\frac {8 a^7 b}{x^6}+\frac {28 a^6 b^2}{x^5}+\frac {56 a^5 b^3}{x^4}+\frac {70 a^4 b^4}{x^3}+\frac {56 a^3 b^5}{x^2}+\frac {28 a^2 b^6}{x}+b^8 x\right ) \, dx,x,x^3\right ) \\ & = -\frac {a^8}{18 x^{18}}-\frac {8 a^7 b}{15 x^{15}}-\frac {7 a^6 b^2}{3 x^{12}}-\frac {56 a^5 b^3}{9 x^9}-\frac {35 a^4 b^4}{3 x^6}-\frac {56 a^3 b^5}{3 x^3}+\frac {8}{3} a b^7 x^3+\frac {b^8 x^6}{6}+28 a^2 b^6 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^8}{x^{19}} \, dx=-\frac {a^8}{18 x^{18}}-\frac {8 a^7 b}{15 x^{15}}-\frac {7 a^6 b^2}{3 x^{12}}-\frac {56 a^5 b^3}{9 x^9}-\frac {35 a^4 b^4}{3 x^6}-\frac {56 a^3 b^5}{3 x^3}+\frac {8}{3} a b^7 x^3+\frac {b^8 x^6}{6}+28 a^2 b^6 \log (x) \]

[In]

Integrate[(a + b*x^3)^8/x^19,x]

[Out]

-1/18*a^8/x^18 - (8*a^7*b)/(15*x^15) - (7*a^6*b^2)/(3*x^12) - (56*a^5*b^3)/(9*x^9) - (35*a^4*b^4)/(3*x^6) - (5
6*a^3*b^5)/(3*x^3) + (8*a*b^7*x^3)/3 + (b^8*x^6)/6 + 28*a^2*b^6*Log[x]

Maple [A] (verified)

Time = 3.68 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {a^{8}}{18 x^{18}}-\frac {8 a^{7} b}{15 x^{15}}-\frac {7 a^{6} b^{2}}{3 x^{12}}-\frac {56 a^{5} b^{3}}{9 x^{9}}-\frac {35 a^{4} b^{4}}{3 x^{6}}-\frac {56 a^{3} b^{5}}{3 x^{3}}+\frac {8 a \,b^{7} x^{3}}{3}+\frac {b^{8} x^{6}}{6}+28 a^{2} b^{6} \ln \left (x \right )\)	\(90\)
norman	\(\frac {-\frac {1}{18} a^{8}+\frac {1}{6} b^{8} x^{24}+\frac {8}{3} a \,b^{7} x^{21}-\frac {56}{3} a^{3} b^{5} x^{15}-\frac {35}{3} a^{4} b^{4} x^{12}-\frac {7}{3} a^{6} b^{2} x^{6}-\frac {8}{15} x^{3} b \,a^{7}-\frac {56}{9} x^{9} b^{3} a^{5}}{x^{18}}+28 a^{2} b^{6} \ln \left (x \right )\)	\(92\)
parallelrisch	\(\frac {15 b^{8} x^{24}+240 a \,b^{7} x^{21}+2520 a^{2} b^{6} \ln \left (x \right ) x^{18}-1680 a^{3} b^{5} x^{15}-1050 a^{4} b^{4} x^{12}-560 x^{9} b^{3} a^{5}-210 a^{6} b^{2} x^{6}-48 x^{3} b \,a^{7}-5 a^{8}}{90 x^{18}}\)	\(95\)
risch	\(\frac {b^{8} x^{6}}{6}+\frac {8 a \,b^{7} x^{3}}{3}+\frac {32 a^{2} b^{6}}{3}+\frac {-\frac {1}{18} a^{8}-\frac {8}{15} x^{3} b \,a^{7}-\frac {7}{3} a^{6} b^{2} x^{6}-\frac {56}{9} x^{9} b^{3} a^{5}-\frac {35}{3} a^{4} b^{4} x^{12}-\frac {56}{3} a^{3} b^{5} x^{15}}{x^{18}}+28 a^{2} b^{6} \ln \left (x \right )\)	\(100\)

[In]

int((b*x^3+a)^8/x^19,x,method=_RETURNVERBOSE)

[Out]

-1/18*a^8/x^18-8/15*a^7*b/x^15-7/3*a^6*b^2/x^12-56/9*a^5*b^3/x^9-35/3*a^4*b^4/x^6-56/3*a^3*b^5/x^3+8/3*a*b^7*x
^3+1/6*b^8*x^6+28*a^2*b^6*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{19}} \, dx=\frac {15 \, b^{8} x^{24} + 240 \, a b^{7} x^{21} + 2520 \, a^{2} b^{6} x^{18} \log \left (x\right ) - 1680 \, a^{3} b^{5} x^{15} - 1050 \, a^{4} b^{4} x^{12} - 560 \, a^{5} b^{3} x^{9} - 210 \, a^{6} b^{2} x^{6} - 48 \, a^{7} b x^{3} - 5 \, a^{8}}{90 \, x^{18}} \]

[In]

integrate((b*x^3+a)^8/x^19,x, algorithm="fricas")

[Out]

1/90*(15*b^8*x^24 + 240*a*b^7*x^21 + 2520*a^2*b^6*x^18*log(x) - 1680*a^3*b^5*x^15 - 1050*a^4*b^4*x^12 - 560*a^
5*b^3*x^9 - 210*a^6*b^2*x^6 - 48*a^7*b*x^3 - 5*a^8)/x^18

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^3\right )^8}{x^{19}} \, dx=28 a^{2} b^{6} \log {\left (x \right )} + \frac {8 a b^{7} x^{3}}{3} + \frac {b^{8} x^{6}}{6} + \frac {- 5 a^{8} - 48 a^{7} b x^{3} - 210 a^{6} b^{2} x^{6} - 560 a^{5} b^{3} x^{9} - 1050 a^{4} b^{4} x^{12} - 1680 a^{3} b^{5} x^{15}}{90 x^{18}} \]

[In]

integrate((b*x**3+a)**8/x**19,x)

[Out]

28*a**2*b**6*log(x) + 8*a*b**7*x**3/3 + b**8*x**6/6 + (-5*a**8 - 48*a**7*b*x**3 - 210*a**6*b**2*x**6 - 560*a**
5*b**3*x**9 - 1050*a**4*b**4*x**12 - 1680*a**3*b**5*x**15)/(90*x**18)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^3\right )^8}{x^{19}} \, dx=\frac {1}{6} \, b^{8} x^{6} + \frac {8}{3} \, a b^{7} x^{3} + \frac {28}{3} \, a^{2} b^{6} \log \left (x^{3}\right ) - \frac {1680 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 560 \, a^{5} b^{3} x^{9} + 210 \, a^{6} b^{2} x^{6} + 48 \, a^{7} b x^{3} + 5 \, a^{8}}{90 \, x^{18}} \]

[In]

integrate((b*x^3+a)^8/x^19,x, algorithm="maxima")

[Out]

1/6*b^8*x^6 + 8/3*a*b^7*x^3 + 28/3*a^2*b^6*log(x^3) - 1/90*(1680*a^3*b^5*x^15 + 1050*a^4*b^4*x^12 + 560*a^5*b^
3*x^9 + 210*a^6*b^2*x^6 + 48*a^7*b*x^3 + 5*a^8)/x^18

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b x^3\right )^8}{x^{19}} \, dx=\frac {1}{6} \, b^{8} x^{6} + \frac {8}{3} \, a b^{7} x^{3} + 28 \, a^{2} b^{6} \log \left ({\left | x \right |}\right ) - \frac {2058 \, a^{2} b^{6} x^{18} + 1680 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 560 \, a^{5} b^{3} x^{9} + 210 \, a^{6} b^{2} x^{6} + 48 \, a^{7} b x^{3} + 5 \, a^{8}}{90 \, x^{18}} \]

[In]

integrate((b*x^3+a)^8/x^19,x, algorithm="giac")

[Out]

1/6*b^8*x^6 + 8/3*a*b^7*x^3 + 28*a^2*b^6*log(abs(x)) - 1/90*(2058*a^2*b^6*x^18 + 1680*a^3*b^5*x^15 + 1050*a^4*
b^4*x^12 + 560*a^5*b^3*x^9 + 210*a^6*b^2*x^6 + 48*a^7*b*x^3 + 5*a^8)/x^18

Mupad [B] (verification not implemented)

Time = 5.45 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^3\right )^8}{x^{19}} \, dx=\frac {b^8\,x^6}{6}-\frac {\frac {a^8}{18}+\frac {8\,a^7\,b\,x^3}{15}+\frac {7\,a^6\,b^2\,x^6}{3}+\frac {56\,a^5\,b^3\,x^9}{9}+\frac {35\,a^4\,b^4\,x^{12}}{3}+\frac {56\,a^3\,b^5\,x^{15}}{3}}{x^{18}}+\frac {8\,a\,b^7\,x^3}{3}+28\,a^2\,b^6\,\ln \left (x\right ) \]

[In]

int((a + b*x^3)^8/x^19,x)

[Out]

(b^8*x^6)/6 - (a^8/18 + (8*a^7*b*x^3)/15 + (7*a^6*b^2*x^6)/3 + (56*a^5*b^3*x^9)/9 + (35*a^4*b^4*x^12)/3 + (56*
a^3*b^5*x^15)/3)/x^18 + (8*a*b^7*x^3)/3 + 28*a^2*b^6*log(x)