\(\int \frac {(a+b x^3)^5}{x^{13}} \, dx\) [268]

Optimal result

Integrand size = 13, antiderivative size = 66 \[ \int \frac {\left (a+b x^3\right )^5}{x^{13}} \, dx=-\frac {a^5}{12 x^{12}}-\frac {5 a^4 b}{9 x^9}-\frac {5 a^3 b^2}{3 x^6}-\frac {10 a^2 b^3}{3 x^3}+\frac {b^5 x^3}{3}+5 a b^4 \log (x) \]

[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)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {\left (a+b x^3\right )^5}{x^{13}} \, dx=-\frac {a^5}{12 x^{12}}-\frac {5 a^4 b}{9 x^9}-\frac {5 a^3 b^2}{3 x^6}-\frac {10 a^2 b^3}{3 x^3}+5 a b^4 \log (x)+\frac {b^5 x^3}{3} \]

[In]

Int[(a + b*x^3)^5/x^13,x]

[Out]

-1/12*a^5/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 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*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^5}{x^5} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {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] (verified)

Time = 0.00 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5}{x^{13}} \, dx=-\frac {a^5}{12 x^{12}}-\frac {5 a^4 b}{9 x^9}-\frac {5 a^3 b^2}{3 x^6}-\frac {10 a^2 b^3}{3 x^3}+\frac {b^5 x^3}{3}+5 a b^4 \log (x) \]

[In]

Integrate[(a + b*x^3)^5/x^13,x]

[Out]

-1/12*a^5/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] (verified)

Time = 3.58 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {a^{5}}{12 x^{12}}-\frac {5 a^{4} b}{9 x^{9}}-\frac {5 a^{3} b^{2}}{3 x^{6}}-\frac {10 a^{2} b^{3}}{3 x^{3}}+\frac {x^{3} b^{5}}{3}+5 a \,b^{4} \ln \left (x \right )\)	\(57\)
norman	\(\frac {-\frac {1}{12} a^{5}+\frac {1}{3} b^{5} x^{15}-\frac {10}{3} a^{2} b^{3} x^{9}-\frac {5}{3} a^{3} b^{2} x^{6}-\frac {5}{9} a^{4} b \,x^{3}}{x^{12}}+5 a \,b^{4} \ln \left (x \right )\)	\(59\)
risch	\(\frac {x^{3} b^{5}}{3}+\frac {-\frac {10}{3} a^{2} b^{3} x^{9}-\frac {5}{3} a^{3} b^{2} x^{6}-\frac {5}{9} a^{4} b \,x^{3}-\frac {1}{12} a^{5}}{x^{12}}+5 a \,b^{4} \ln \left (x \right )\)	\(59\)
parallelrisch	\(\frac {12 b^{5} x^{15}+180 a \,b^{4} \ln \left (x \right ) x^{12}-120 a^{2} b^{3} x^{9}-60 a^{3} b^{2} x^{6}-20 a^{4} b \,x^{3}-3 a^{5}}{36 x^{12}}\)	\(62\)

[In]

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

[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*x^3*b^5+5*a*b^4*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^3\right )^5}{x^{13}} \, dx=\frac {12 \, b^{5} x^{15} + 180 \, a b^{4} x^{12} \log \left (x\right ) - 120 \, a^{2} b^{3} x^{9} - 60 \, a^{3} b^{2} x^{6} - 20 \, a^{4} b x^{3} - 3 \, a^{5}}{36 \, x^{12}} \]

[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] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b x^3\right )^5}{x^{13}} \, dx=5 a b^{4} \log {\left (x \right )} + \frac {b^{5} x^{3}}{3} + \frac {- 3 a^{5} - 20 a^{4} b x^{3} - 60 a^{3} b^{2} x^{6} - 120 a^{2} b^{3} x^{9}}{36 x^{12}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^3\right )^5}{x^{13}} \, dx=\frac {1}{3} \, b^{5} x^{3} + \frac {5}{3} \, a b^{4} \log \left (x^{3}\right ) - \frac {120 \, a^{2} b^{3} x^{9} + 60 \, a^{3} b^{2} x^{6} + 20 \, a^{4} b x^{3} + 3 \, a^{5}}{36 \, x^{12}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a+b x^3\right )^5}{x^{13}} \, dx=\frac {1}{3} \, b^{5} x^{3} + 5 \, a b^{4} \log \left ({\left | x \right |}\right ) - \frac {125 \, a b^{4} x^{12} + 120 \, a^{2} b^{3} x^{9} + 60 \, a^{3} b^{2} x^{6} + 20 \, a^{4} b x^{3} + 3 \, a^{5}}{36 \, x^{12}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^3\right )^5}{x^{13}} \, dx=\frac {b^5\,x^3}{3}-\frac {\frac {a^5}{12}+\frac {5\,a^4\,b\,x^3}{9}+\frac {5\,a^3\,b^2\,x^6}{3}+\frac {10\,a^2\,b^3\,x^9}{3}}{x^{12}}+5\,a\,b^4\,\ln \left (x\right ) \]

[In]

int((a + b*x^3)^5/x^13,x)

[Out]

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