3.295 \(\int \frac {(a+b x^3)^8}{x^{10}} \, dx\)

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

[Out]

-1/9*a^8/x^9-4/3*a^7*b/x^6-28/3*a^6*b^2/x^3+70/3*a^4*b^4*x^3+28/3*a^3*b^5*x^6+28/9*a^2*b^6*x^9+2/3*a*b^7*x^12+
1/15*b^8*x^15+56*a^5*b^3*ln(x)

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Rubi [A]  time = 0.06, 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 = {266, 43} \[ \frac {28}{9} a^2 b^6 x^9+\frac {28}{3} a^3 b^5 x^6+\frac {70}{3} a^4 b^4 x^3-\frac {28 a^6 b^2}{3 x^3}+56 a^5 b^3 \log (x)-\frac {4 a^7 b}{3 x^6}-\frac {a^8}{9 x^9}+\frac {2}{3} a b^7 x^{12}+\frac {b^8 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^10,x]

[Out]

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

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])

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]]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^8}{x^{10}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(a+b x)^8}{x^4} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (70 a^4 b^4+\frac {a^8}{x^4}+\frac {8 a^7 b}{x^3}+\frac {28 a^6 b^2}{x^2}+\frac {56 a^5 b^3}{x}+56 a^3 b^5 x+28 a^2 b^6 x^2+8 a b^7 x^3+b^8 x^4\right ) \, dx,x,x^3\right )\\ &=-\frac {a^8}{9 x^9}-\frac {4 a^7 b}{3 x^6}-\frac {28 a^6 b^2}{3 x^3}+\frac {70}{3} a^4 b^4 x^3+\frac {28}{3} a^3 b^5 x^6+\frac {28}{9} a^2 b^6 x^9+\frac {2}{3} a b^7 x^{12}+\frac {b^8 x^{15}}{15}+56 a^5 b^3 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 105, normalized size = 1.00 \[ -\frac {a^8}{9 x^9}-\frac {4 a^7 b}{3 x^6}-\frac {28 a^6 b^2}{3 x^3}+56 a^5 b^3 \log (x)+\frac {70}{3} a^4 b^4 x^3+\frac {28}{3} a^3 b^5 x^6+\frac {28}{9} a^2 b^6 x^9+\frac {2}{3} a b^7 x^{12}+\frac {b^8 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^10,x]

[Out]

-1/9*a^8/x^9 - (4*a^7*b)/(3*x^6) - (28*a^6*b^2)/(3*x^3) + (70*a^4*b^4*x^3)/3 + (28*a^3*b^5*x^6)/3 + (28*a^2*b^
6*x^9)/9 + (2*a*b^7*x^12)/3 + (b^8*x^15)/15 + 56*a^5*b^3*Log[x]

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fricas [A]  time = 0.77, size = 94, normalized size = 0.90 \[ \frac {3 \, b^{8} x^{24} + 30 \, a b^{7} x^{21} + 140 \, a^{2} b^{6} x^{18} + 420 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 2520 \, a^{5} b^{3} x^{9} \log \relax (x) - 420 \, a^{6} b^{2} x^{6} - 60 \, a^{7} b x^{3} - 5 \, a^{8}}{45 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^10,x, algorithm="fricas")

[Out]

1/45*(3*b^8*x^24 + 30*a*b^7*x^21 + 140*a^2*b^6*x^18 + 420*a^3*b^5*x^15 + 1050*a^4*b^4*x^12 + 2520*a^5*b^3*x^9*
log(x) - 420*a^6*b^2*x^6 - 60*a^7*b*x^3 - 5*a^8)/x^9

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giac [A]  time = 0.15, size = 102, normalized size = 0.97 \[ \frac {1}{15} \, b^{8} x^{15} + \frac {2}{3} \, a b^{7} x^{12} + \frac {28}{9} \, a^{2} b^{6} x^{9} + \frac {28}{3} \, a^{3} b^{5} x^{6} + \frac {70}{3} \, a^{4} b^{4} x^{3} + 56 \, a^{5} b^{3} \log \left ({\left | x \right |}\right ) - \frac {308 \, a^{5} b^{3} x^{9} + 84 \, a^{6} b^{2} x^{6} + 12 \, a^{7} b x^{3} + a^{8}}{9 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^10,x, algorithm="giac")

[Out]

1/15*b^8*x^15 + 2/3*a*b^7*x^12 + 28/9*a^2*b^6*x^9 + 28/3*a^3*b^5*x^6 + 70/3*a^4*b^4*x^3 + 56*a^5*b^3*log(abs(x
)) - 1/9*(308*a^5*b^3*x^9 + 84*a^6*b^2*x^6 + 12*a^7*b*x^3 + a^8)/x^9

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maple [A]  time = 0.01, size = 90, normalized size = 0.86 \[ \frac {b^{8} x^{15}}{15}+\frac {2 a \,b^{7} x^{12}}{3}+\frac {28 a^{2} b^{6} x^{9}}{9}+\frac {28 a^{3} b^{5} x^{6}}{3}+\frac {70 a^{4} b^{4} x^{3}}{3}+56 a^{5} b^{3} \ln \relax (x )-\frac {28 a^{6} b^{2}}{3 x^{3}}-\frac {4 a^{7} b}{3 x^{6}}-\frac {a^{8}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^10,x)

[Out]

-1/9*a^8/x^9-4/3*a^7*b/x^6-28/3*a^6*b^2/x^3+70/3*a^4*b^4*x^3+28/3*a^3*b^5*x^6+28/9*a^2*b^6*x^9+2/3*a*b^7*x^12+
1/15*b^8*x^15+56*a^5*b^3*ln(x)

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maxima [A]  time = 1.30, size = 92, normalized size = 0.88 \[ \frac {1}{15} \, b^{8} x^{15} + \frac {2}{3} \, a b^{7} x^{12} + \frac {28}{9} \, a^{2} b^{6} x^{9} + \frac {28}{3} \, a^{3} b^{5} x^{6} + \frac {70}{3} \, a^{4} b^{4} x^{3} + \frac {56}{3} \, a^{5} b^{3} \log \left (x^{3}\right ) - \frac {84 \, a^{6} b^{2} x^{6} + 12 \, a^{7} b x^{3} + a^{8}}{9 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^10,x, algorithm="maxima")

[Out]

1/15*b^8*x^15 + 2/3*a*b^7*x^12 + 28/9*a^2*b^6*x^9 + 28/3*a^3*b^5*x^6 + 70/3*a^4*b^4*x^3 + 56/3*a^5*b^3*log(x^3
) - 1/9*(84*a^6*b^2*x^6 + 12*a^7*b*x^3 + a^8)/x^9

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mupad [B]  time = 0.96, size = 92, normalized size = 0.88 \[ \frac {b^8\,x^{15}}{15}-\frac {\frac {a^8}{9}+\frac {4\,a^7\,b\,x^3}{3}+\frac {28\,a^6\,b^2\,x^6}{3}}{x^9}+\frac {2\,a\,b^7\,x^{12}}{3}+\frac {70\,a^4\,b^4\,x^3}{3}+\frac {28\,a^3\,b^5\,x^6}{3}+\frac {28\,a^2\,b^6\,x^9}{9}+56\,a^5\,b^3\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8/x^10,x)

[Out]

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

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sympy [A]  time = 0.47, size = 104, normalized size = 0.99 \[ 56 a^{5} b^{3} \log {\relax (x )} + \frac {70 a^{4} b^{4} x^{3}}{3} + \frac {28 a^{3} b^{5} x^{6}}{3} + \frac {28 a^{2} b^{6} x^{9}}{9} + \frac {2 a b^{7} x^{12}}{3} + \frac {b^{8} x^{15}}{15} + \frac {- a^{8} - 12 a^{7} b x^{3} - 84 a^{6} b^{2} x^{6}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**10,x)

[Out]

56*a**5*b**3*log(x) + 70*a**4*b**4*x**3/3 + 28*a**3*b**5*x**6/3 + 28*a**2*b**6*x**9/9 + 2*a*b**7*x**12/3 + b**
8*x**15/15 + (-a**8 - 12*a**7*b*x**3 - 84*a**6*b**2*x**6)/(9*x**9)

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