3.3.97 \(\int \frac {(a+b x^3)^8}{x^{16}} \, dx\) [297]

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

[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)

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 105, normalized size of antiderivative = 1.00, 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 = {272, 45} \begin {gather*} -\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}+56 a^3 b^5 \log (x)+\frac {28}{3} a^2 b^6 x^3+\frac {4}{3} a b^7 x^6+\frac {b^8 x^9}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^16,x]

[Out]

-1/15*a^8/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 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*} \int \frac {\left (a+b x^3\right )^8}{x^{16}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^8}{x^6} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {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 = 0.01, size = 105, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^16,x]

[Out]

-1/15*a^8/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 = 0.13, size = 90, normalized size = 0.86

method	result	size
default	\(-\frac {a^{8}}{15 x^{15}}-\frac {2 a^{7} b}{3 x^{12}}-\frac {28 a^{6} b^{2}}{9 x^{9}}-\frac {28 a^{5} b^{3}}{3 x^{6}}-\frac {70 a^{4} b^{4}}{3 x^{3}}+\frac {28 a^{2} b^{6} x^{3}}{3}+\frac {4 a \,b^{7} x^{6}}{3}+\frac {b^{8} x^{9}}{9}+56 a^{3} b^{5} \ln \left (x \right )\)	\(90\)
norman	\(\frac {-\frac {1}{15} a^{8}+\frac {1}{9} b^{8} x^{24}+\frac {4}{3} a \,b^{7} x^{21}+\frac {28}{3} a^{2} b^{6} x^{18}-\frac {70}{3} a^{4} b^{4} x^{12}-\frac {28}{3} a^{5} b^{3} x^{9}-\frac {28}{9} a^{6} b^{2} x^{6}-\frac {2}{3} a^{7} b \,x^{3}}{x^{15}}+56 a^{3} b^{5} \ln \left (x \right )\)	\(92\)
risch	\(\frac {b^{8} x^{9}}{9}+\frac {4 a \,b^{7} x^{6}}{3}+\frac {28 a^{2} b^{6} x^{3}}{3}+\frac {-\frac {1}{15} a^{8}-\frac {2}{3} a^{7} b \,x^{3}-\frac {28}{9} a^{6} b^{2} x^{6}-\frac {28}{3} a^{5} b^{3} x^{9}-\frac {70}{3} a^{4} b^{4} x^{12}}{x^{15}}+56 a^{3} b^{5} \ln \left (x \right )\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.29, size = 94, normalized size = 0.90 \begin {gather*} \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 {gather*}

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 = 0.36, size = 94, normalized size = 0.90 \begin {gather*} \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 {gather*}

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 = 0.26, size = 102, normalized size = 0.97 \begin {gather*} 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 {gather*}

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 = 1.44, size = 104, normalized size = 0.99 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 0.05, size = 92, normalized size = 0.88 \begin {gather*} \frac {b^8\,x^9}{9}-\frac {\frac {a^8}{15}+\frac {2\,a^7\,b\,x^3}{3}+\frac {28\,a^6\,b^2\,x^6}{9}+\frac {28\,a^5\,b^3\,x^9}{3}+\frac {70\,a^4\,b^4\,x^{12}}{3}}{x^{15}}+\frac {4\,a\,b^7\,x^6}{3}+\frac {28\,a^2\,b^6\,x^3}{3}+56\,a^3\,b^5\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8/x^16,x)

[Out]

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

________________________________________________________________________________________