\(\int \frac {(a+b x^3)^8}{x} \, dx\) [292]

Optimal result

Integrand size = 13, antiderivative size = 104 \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=\frac {8}{3} a^7 b x^3+\frac {14}{3} a^6 b^2 x^6+\frac {56}{9} a^5 b^3 x^9+\frac {35}{6} a^4 b^4 x^{12}+\frac {56}{15} a^3 b^5 x^{15}+\frac {14}{9} a^2 b^6 x^{18}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{24}}{24}+a^8 \log (x) \]

[Out]

8/3*a^7*b*x^3+14/3*a^6*b^2*x^6+56/9*a^5*b^3*x^9+35/6*a^4*b^4*x^12+56/15*a^3*b^5*x^15+14/9*a^2*b^6*x^18+8/21*a*
b^7*x^21+1/24*b^8*x^24+a^8*ln(x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 104, 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} \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=a^8 \log (x)+\frac {8}{3} a^7 b x^3+\frac {14}{3} a^6 b^2 x^6+\frac {56}{9} a^5 b^3 x^9+\frac {35}{6} a^4 b^4 x^{12}+\frac {56}{15} a^3 b^5 x^{15}+\frac {14}{9} a^2 b^6 x^{18}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{24}}{24} \]

[In]

Int[(a + b*x^3)^8/x,x]

[Out]

(8*a^7*b*x^3)/3 + (14*a^6*b^2*x^6)/3 + (56*a^5*b^3*x^9)/9 + (35*a^4*b^4*x^12)/6 + (56*a^3*b^5*x^15)/15 + (14*a
^2*b^6*x^18)/9 + (8*a*b^7*x^21)/21 + (b^8*x^24)/24 + a^8*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} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (8 a^7 b+\frac {a^8}{x}+28 a^6 b^2 x+56 a^5 b^3 x^2+70 a^4 b^4 x^3+56 a^3 b^5 x^4+28 a^2 b^6 x^5+8 a b^7 x^6+b^8 x^7\right ) \, dx,x,x^3\right ) \\ & = \frac {8}{3} a^7 b x^3+\frac {14}{3} a^6 b^2 x^6+\frac {56}{9} a^5 b^3 x^9+\frac {35}{6} a^4 b^4 x^{12}+\frac {56}{15} a^3 b^5 x^{15}+\frac {14}{9} a^2 b^6 x^{18}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{24}}{24}+a^8 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=\frac {8}{3} a^7 b x^3+\frac {14}{3} a^6 b^2 x^6+\frac {56}{9} a^5 b^3 x^9+\frac {35}{6} a^4 b^4 x^{12}+\frac {56}{15} a^3 b^5 x^{15}+\frac {14}{9} a^2 b^6 x^{18}+\frac {8}{21} a b^7 x^{21}+\frac {b^8 x^{24}}{24}+a^8 \log (x) \]

[In]

Integrate[(a + b*x^3)^8/x,x]

[Out]

(8*a^7*b*x^3)/3 + (14*a^6*b^2*x^6)/3 + (56*a^5*b^3*x^9)/9 + (35*a^4*b^4*x^12)/6 + (56*a^3*b^5*x^15)/15 + (14*a
^2*b^6*x^18)/9 + (8*a*b^7*x^21)/21 + (b^8*x^24)/24 + a^8*Log[x]

Maple [A] (verified)

Time = 3.74 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {8 x^{3} b \,a^{7}}{3}+\frac {14 a^{6} b^{2} x^{6}}{3}+\frac {56 x^{9} b^{3} a^{5}}{9}+\frac {35 a^{4} b^{4} x^{12}}{6}+\frac {56 a^{3} b^{5} x^{15}}{15}+\frac {14 a^{2} b^{6} x^{18}}{9}+\frac {8 a \,b^{7} x^{21}}{21}+\frac {b^{8} x^{24}}{24}+a^{8} \ln \left (x \right )\)	\(89\)
norman	\(\frac {8 x^{3} b \,a^{7}}{3}+\frac {14 a^{6} b^{2} x^{6}}{3}+\frac {56 x^{9} b^{3} a^{5}}{9}+\frac {35 a^{4} b^{4} x^{12}}{6}+\frac {56 a^{3} b^{5} x^{15}}{15}+\frac {14 a^{2} b^{6} x^{18}}{9}+\frac {8 a \,b^{7} x^{21}}{21}+\frac {b^{8} x^{24}}{24}+a^{8} \ln \left (x \right )\)	\(89\)
parallelrisch	\(\frac {8 x^{3} b \,a^{7}}{3}+\frac {14 a^{6} b^{2} x^{6}}{3}+\frac {56 x^{9} b^{3} a^{5}}{9}+\frac {35 a^{4} b^{4} x^{12}}{6}+\frac {56 a^{3} b^{5} x^{15}}{15}+\frac {14 a^{2} b^{6} x^{18}}{9}+\frac {8 a \,b^{7} x^{21}}{21}+\frac {b^{8} x^{24}}{24}+a^{8} \ln \left (x \right )\)	\(89\)
risch	\(\frac {352 a^{8}}{315}+\frac {b^{8} x^{24}}{24}+\frac {8 x^{3} b \,a^{7}}{3}+\frac {8 a \,b^{7} x^{21}}{21}+\frac {56 a^{3} b^{5} x^{15}}{15}+\frac {14 a^{2} b^{6} x^{18}}{9}+\frac {35 a^{4} b^{4} x^{12}}{6}+a^{8} \ln \left (x \right )+\frac {56 x^{9} b^{3} a^{5}}{9}+\frac {14 a^{6} b^{2} x^{6}}{3}\)	\(94\)

[In]

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

[Out]

8/3*x^3*b*a^7+14/3*a^6*b^2*x^6+56/9*x^9*b^3*a^5+35/6*a^4*b^4*x^12+56/15*a^3*b^5*x^15+14/9*a^2*b^6*x^18+8/21*a*
b^7*x^21+1/24*b^8*x^24+a^8*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=\frac {1}{24} \, b^{8} x^{24} + \frac {8}{21} \, a b^{7} x^{21} + \frac {14}{9} \, a^{2} b^{6} x^{18} + \frac {56}{15} \, a^{3} b^{5} x^{15} + \frac {35}{6} \, a^{4} b^{4} x^{12} + \frac {56}{9} \, a^{5} b^{3} x^{9} + \frac {14}{3} \, a^{6} b^{2} x^{6} + \frac {8}{3} \, a^{7} b x^{3} + a^{8} \log \left (x\right ) \]

[In]

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

[Out]

1/24*b^8*x^24 + 8/21*a*b^7*x^21 + 14/9*a^2*b^6*x^18 + 56/15*a^3*b^5*x^15 + 35/6*a^4*b^4*x^12 + 56/9*a^5*b^3*x^
9 + 14/3*a^6*b^2*x^6 + 8/3*a^7*b*x^3 + a^8*log(x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=a^{8} \log {\left (x \right )} + \frac {8 a^{7} b x^{3}}{3} + \frac {14 a^{6} b^{2} x^{6}}{3} + \frac {56 a^{5} b^{3} x^{9}}{9} + \frac {35 a^{4} b^{4} x^{12}}{6} + \frac {56 a^{3} b^{5} x^{15}}{15} + \frac {14 a^{2} b^{6} x^{18}}{9} + \frac {8 a b^{7} x^{21}}{21} + \frac {b^{8} x^{24}}{24} \]

[In]

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

[Out]

a**8*log(x) + 8*a**7*b*x**3/3 + 14*a**6*b**2*x**6/3 + 56*a**5*b**3*x**9/9 + 35*a**4*b**4*x**12/6 + 56*a**3*b**
5*x**15/15 + 14*a**2*b**6*x**18/9 + 8*a*b**7*x**21/21 + b**8*x**24/24

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=\frac {1}{24} \, b^{8} x^{24} + \frac {8}{21} \, a b^{7} x^{21} + \frac {14}{9} \, a^{2} b^{6} x^{18} + \frac {56}{15} \, a^{3} b^{5} x^{15} + \frac {35}{6} \, a^{4} b^{4} x^{12} + \frac {56}{9} \, a^{5} b^{3} x^{9} + \frac {14}{3} \, a^{6} b^{2} x^{6} + \frac {8}{3} \, a^{7} b x^{3} + \frac {1}{3} \, a^{8} \log \left (x^{3}\right ) \]

[In]

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

[Out]

1/24*b^8*x^24 + 8/21*a*b^7*x^21 + 14/9*a^2*b^6*x^18 + 56/15*a^3*b^5*x^15 + 35/6*a^4*b^4*x^12 + 56/9*a^5*b^3*x^
9 + 14/3*a^6*b^2*x^6 + 8/3*a^7*b*x^3 + 1/3*a^8*log(x^3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=\frac {1}{24} \, b^{8} x^{24} + \frac {8}{21} \, a b^{7} x^{21} + \frac {14}{9} \, a^{2} b^{6} x^{18} + \frac {56}{15} \, a^{3} b^{5} x^{15} + \frac {35}{6} \, a^{4} b^{4} x^{12} + \frac {56}{9} \, a^{5} b^{3} x^{9} + \frac {14}{3} \, a^{6} b^{2} x^{6} + \frac {8}{3} \, a^{7} b x^{3} + a^{8} \log \left ({\left | x \right |}\right ) \]

[In]

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

[Out]

1/24*b^8*x^24 + 8/21*a*b^7*x^21 + 14/9*a^2*b^6*x^18 + 56/15*a^3*b^5*x^15 + 35/6*a^4*b^4*x^12 + 56/9*a^5*b^3*x^
9 + 14/3*a^6*b^2*x^6 + 8/3*a^7*b*x^3 + a^8*log(abs(x))

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^3\right )^8}{x} \, dx=a^8\,\ln \left (x\right )+\frac {b^8\,x^{24}}{24}+\frac {8\,a^7\,b\,x^3}{3}+\frac {8\,a\,b^7\,x^{21}}{21}+\frac {14\,a^6\,b^2\,x^6}{3}+\frac {56\,a^5\,b^3\,x^9}{9}+\frac {35\,a^4\,b^4\,x^{12}}{6}+\frac {56\,a^3\,b^5\,x^{15}}{15}+\frac {14\,a^2\,b^6\,x^{18}}{9} \]

[In]

int((a + b*x^3)^8/x,x)

[Out]

a^8*log(x) + (b^8*x^24)/24 + (8*a^7*b*x^3)/3 + (8*a*b^7*x^21)/21 + (14*a^6*b^2*x^6)/3 + (56*a^5*b^3*x^9)/9 + (
35*a^4*b^4*x^12)/6 + (56*a^3*b^5*x^15)/15 + (14*a^2*b^6*x^18)/9