3.267 \(\int \frac{(a+b x^3)^5}{x^{10}} \, dx\)

Optimal. Leaf size=66 \[ -\frac{10 a^3 b^2}{3 x^3}+10 a^2 b^3 \log (x)-\frac{5 a^4 b}{6 x^6}-\frac{a^5}{9 x^9}+\frac{5}{3} a b^4 x^3+\frac{b^5 x^6}{6} \]

[Out]

-a^5/(9*x^9) - (5*a^4*b)/(6*x^6) - (10*a^3*b^2)/(3*x^3) + (5*a*b^4*x^3)/3 + (b^5*x^6)/6 + 10*a^2*b^3*Log[x]

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Rubi [A]  time = 0.0331047, antiderivative size = 66, 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} \[ -\frac{10 a^3 b^2}{3 x^3}+10 a^2 b^3 \log (x)-\frac{5 a^4 b}{6 x^6}-\frac{a^5}{9 x^9}+\frac{5}{3} a b^4 x^3+\frac{b^5 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^10,x]

[Out]

-a^5/(9*x^9) - (5*a^4*b)/(6*x^6) - (10*a^3*b^2)/(3*x^3) + (5*a*b^4*x^3)/3 + (b^5*x^6)/6 + 10*a^2*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 )^5}{x^{10}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^4} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (5 a b^4+\frac{a^5}{x^4}+\frac{5 a^4 b}{x^3}+\frac{10 a^3 b^2}{x^2}+\frac{10 a^2 b^3}{x}+b^5 x\right ) \, dx,x,x^3\right )\\ &=-\frac{a^5}{9 x^9}-\frac{5 a^4 b}{6 x^6}-\frac{10 a^3 b^2}{3 x^3}+\frac{5}{3} a b^4 x^3+\frac{b^5 x^6}{6}+10 a^2 b^3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.006773, size = 66, normalized size = 1. \[ -\frac{10 a^3 b^2}{3 x^3}+10 a^2 b^3 \log (x)-\frac{5 a^4 b}{6 x^6}-\frac{a^5}{9 x^9}+\frac{5}{3} a b^4 x^3+\frac{b^5 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^10,x]

[Out]

-a^5/(9*x^9) - (5*a^4*b)/(6*x^6) - (10*a^3*b^2)/(3*x^3) + (5*a*b^4*x^3)/3 + (b^5*x^6)/6 + 10*a^2*b^3*Log[x]

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Maple [A]  time = 0.005, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{9\,{x}^{9}}}-{\frac{5\,{a}^{4}b}{6\,{x}^{6}}}-{\frac{10\,{a}^{3}{b}^{2}}{3\,{x}^{3}}}+{\frac{5\,a{b}^{4}{x}^{3}}{3}}+{\frac{{b}^{5}{x}^{6}}{6}}+10\,{a}^{2}{b}^{3}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^10,x)

[Out]

-1/9*a^5/x^9-5/6*a^4*b/x^6-10/3*a^3*b^2/x^3+5/3*a*b^4*x^3+1/6*b^5*x^6+10*a^2*b^3*ln(x)

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Maxima [A]  time = 0.983347, size = 82, normalized size = 1.24 \begin{align*} \frac{1}{6} \, b^{5} x^{6} + \frac{5}{3} \, a b^{4} x^{3} + \frac{10}{3} \, a^{2} b^{3} \log \left (x^{3}\right ) - \frac{60 \, a^{3} b^{2} x^{6} + 15 \, a^{4} b x^{3} + 2 \, a^{5}}{18 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^10,x, algorithm="maxima")

[Out]

1/6*b^5*x^6 + 5/3*a*b^4*x^3 + 10/3*a^2*b^3*log(x^3) - 1/18*(60*a^3*b^2*x^6 + 15*a^4*b*x^3 + 2*a^5)/x^9

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Fricas [A]  time = 1.67038, size = 140, normalized size = 2.12 \begin{align*} \frac{3 \, b^{5} x^{15} + 30 \, a b^{4} x^{12} + 180 \, a^{2} b^{3} x^{9} \log \left (x\right ) - 60 \, a^{3} b^{2} x^{6} - 15 \, a^{4} b x^{3} - 2 \, a^{5}}{18 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^10,x, algorithm="fricas")

[Out]

1/18*(3*b^5*x^15 + 30*a*b^4*x^12 + 180*a^2*b^3*x^9*log(x) - 60*a^3*b^2*x^6 - 15*a^4*b*x^3 - 2*a^5)/x^9

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Sympy [A]  time = 0.542234, size = 63, normalized size = 0.95 \begin{align*} 10 a^{2} b^{3} \log{\left (x \right )} + \frac{5 a b^{4} x^{3}}{3} + \frac{b^{5} x^{6}}{6} - \frac{2 a^{5} + 15 a^{4} b x^{3} + 60 a^{3} b^{2} x^{6}}{18 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**10,x)

[Out]

10*a**2*b**3*log(x) + 5*a*b**4*x**3/3 + b**5*x**6/6 - (2*a**5 + 15*a**4*b*x**3 + 60*a**3*b**2*x**6)/(18*x**9)

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Giac [A]  time = 1.12134, size = 96, normalized size = 1.45 \begin{align*} \frac{1}{6} \, b^{5} x^{6} + \frac{5}{3} \, a b^{4} x^{3} + 10 \, a^{2} b^{3} \log \left ({\left | x \right |}\right ) - \frac{110 \, a^{2} b^{3} x^{9} + 60 \, a^{3} b^{2} x^{6} + 15 \, a^{4} b x^{3} + 2 \, a^{5}}{18 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^10,x, algorithm="giac")

[Out]

1/6*b^5*x^6 + 5/3*a*b^4*x^3 + 10*a^2*b^3*log(abs(x)) - 1/18*(110*a^2*b^3*x^9 + 60*a^3*b^2*x^6 + 15*a^4*b*x^3 +
2*a^5)/x^9