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

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

[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]

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Rubi [A]  time = 0.0556131, antiderivative size = 105, normalized size of antiderivative = 1., 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 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]]

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])

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*}

Mathematica [A]  time = 0.0100133, size = 105, normalized size = 1. \[ \frac{28}{9} a^2 b^6 x^9+\frac{28}{3} a^3 b^5 x^6+\frac{70}{3} a^4 b^4 x^3-\frac{28 a^6 b^2}{3 x^3}+56 a^5 b^3 \log (x)-\frac{4 a^7 b}{3 x^6}-\frac{a^8}{9 x^9}+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]

Integrate[(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]

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Maple [A]  time = 0.006, size = 90, normalized size = 0.9 \begin{align*} -{\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\,{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}}+56\,{a}^{5}{b}^{3}\ln \left ( x \right ) \end{align*}

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 = 0.955577, size = 124, normalized size = 1.18 \begin{align*} \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}} \end{align*}

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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Fricas [A]  time = 1.62914, size = 221, normalized size = 2.1 \begin{align*} \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 \left (x\right ) - 420 \, a^{6} b^{2} x^{6} - 60 \, a^{7} b x^{3} - 5 \, a^{8}}{45 \, x^{9}} \end{align*}

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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Sympy [A]  time = 0.686961, size = 102, normalized size = 0.97 \begin{align*} 56 a^{5} b^{3} \log{\left (x \right )} + \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}} \end{align*}

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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Giac [A]  time = 1.23617, size = 138, normalized size = 1.31 \begin{align*} \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}} \end{align*}

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