3.268 \(\int \frac{(a+b x^3)^5}{x^{13}} \, dx\)

Optimal. Leaf size=66 \[ -\frac{5 a^3 b^2}{3 x^6}-\frac{10 a^2 b^3}{3 x^3}-\frac{5 a^4 b}{9 x^9}-\frac{a^5}{12 x^{12}}+5 a b^4 \log (x)+\frac{b^5 x^3}{3} \]

[Out]

-a^5/(12*x^12) - (5*a^4*b)/(9*x^9) - (5*a^3*b^2)/(3*x^6) - (10*a^2*b^3)/(3*x^3) + (b^5*x^3)/3 + 5*a*b^4*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0307271, 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} \[ -\frac{5 a^3 b^2}{3 x^6}-\frac{10 a^2 b^3}{3 x^3}-\frac{5 a^4 b}{9 x^9}-\frac{a^5}{12 x^{12}}+5 a b^4 \log (x)+\frac{b^5 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^5/x^13,x]

[Out]

-a^5/(12*x^12) - (5*a^4*b)/(9*x^9) - (5*a^3*b^2)/(3*x^6) - (10*a^2*b^3)/(3*x^3) + (b^5*x^3)/3 + 5*a*b^4*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^{13}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^5} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (b^5+\frac{a^5}{x^5}+\frac{5 a^4 b}{x^4}+\frac{10 a^3 b^2}{x^3}+\frac{10 a^2 b^3}{x^2}+\frac{5 a b^4}{x}\right ) \, dx,x,x^3\right )\\ &=-\frac{a^5}{12 x^{12}}-\frac{5 a^4 b}{9 x^9}-\frac{5 a^3 b^2}{3 x^6}-\frac{10 a^2 b^3}{3 x^3}+\frac{b^5 x^3}{3}+5 a b^4 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0045832, size = 66, normalized size = 1. \[ -\frac{5 a^3 b^2}{3 x^6}-\frac{10 a^2 b^3}{3 x^3}-\frac{5 a^4 b}{9 x^9}-\frac{a^5}{12 x^{12}}+5 a b^4 \log (x)+\frac{b^5 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^5/x^13,x]

[Out]

-a^5/(12*x^12) - (5*a^4*b)/(9*x^9) - (5*a^3*b^2)/(3*x^6) - (10*a^2*b^3)/(3*x^3) + (b^5*x^3)/3 + 5*a*b^4*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{12\,{x}^{12}}}-{\frac{5\,{a}^{4}b}{9\,{x}^{9}}}-{\frac{5\,{a}^{3}{b}^{2}}{3\,{x}^{6}}}-{\frac{10\,{a}^{2}{b}^{3}}{3\,{x}^{3}}}+{\frac{{b}^{5}{x}^{3}}{3}}+5\,a{b}^{4}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5/x^13,x)

[Out]

-1/12*a^5/x^12-5/9*a^4*b/x^9-5/3*a^3*b^2/x^6-10/3*a^2*b^3/x^3+1/3*b^5*x^3+5*a*b^4*ln(x)

________________________________________________________________________________________

Maxima [A]  time = 0.983119, size = 82, normalized size = 1.24 \begin{align*} \frac{1}{3} \, b^{5} x^{3} + \frac{5}{3} \, a b^{4} \log \left (x^{3}\right ) - \frac{120 \, a^{2} b^{3} x^{9} + 60 \, a^{3} b^{2} x^{6} + 20 \, a^{4} b x^{3} + 3 \, a^{5}}{36 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^13,x, algorithm="maxima")

[Out]

1/3*b^5*x^3 + 5/3*a*b^4*log(x^3) - 1/36*(120*a^2*b^3*x^9 + 60*a^3*b^2*x^6 + 20*a^4*b*x^3 + 3*a^5)/x^12

________________________________________________________________________________________

Fricas [A]  time = 1.63863, size = 144, normalized size = 2.18 \begin{align*} \frac{12 \, b^{5} x^{15} + 180 \, a b^{4} x^{12} \log \left (x\right ) - 120 \, a^{2} b^{3} x^{9} - 60 \, a^{3} b^{2} x^{6} - 20 \, a^{4} b x^{3} - 3 \, a^{5}}{36 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^13,x, algorithm="fricas")

[Out]

1/36*(12*b^5*x^15 + 180*a*b^4*x^12*log(x) - 120*a^2*b^3*x^9 - 60*a^3*b^2*x^6 - 20*a^4*b*x^3 - 3*a^5)/x^12

________________________________________________________________________________________

Sympy [A]  time = 0.745922, size = 61, normalized size = 0.92 \begin{align*} 5 a b^{4} \log{\left (x \right )} + \frac{b^{5} x^{3}}{3} - \frac{3 a^{5} + 20 a^{4} b x^{3} + 60 a^{3} b^{2} x^{6} + 120 a^{2} b^{3} x^{9}}{36 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5/x**13,x)

[Out]

5*a*b**4*log(x) + b**5*x**3/3 - (3*a**5 + 20*a**4*b*x**3 + 60*a**3*b**2*x**6 + 120*a**2*b**3*x**9)/(36*x**12)

________________________________________________________________________________________

Giac [A]  time = 1.11978, size = 93, normalized size = 1.41 \begin{align*} \frac{1}{3} \, b^{5} x^{3} + 5 \, a b^{4} \log \left ({\left | x \right |}\right ) - \frac{125 \, a b^{4} x^{12} + 120 \, a^{2} b^{3} x^{9} + 60 \, a^{3} b^{2} x^{6} + 20 \, a^{4} b x^{3} + 3 \, a^{5}}{36 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5/x^13,x, algorithm="giac")

[Out]

1/3*b^5*x^3 + 5*a*b^4*log(abs(x)) - 1/36*(125*a*b^4*x^12 + 120*a^2*b^3*x^9 + 60*a^3*b^2*x^6 + 20*a^4*b*x^3 + 3
*a^5)/x^12