\(\int \frac {(a+b x^3)^5}{x^{31}} \, dx\) [274]

Optimal result

Integrand size = 13, antiderivative size = 69 \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=-\frac {a^5}{30 x^{30}}-\frac {5 a^4 b}{27 x^{27}}-\frac {5 a^3 b^2}{12 x^{24}}-\frac {10 a^2 b^3}{21 x^{21}}-\frac {5 a b^4}{18 x^{18}}-\frac {b^5}{15 x^{15}} \]

[Out]

-1/30*a^5/x^30-5/27*a^4*b/x^27-5/12*a^3*b^2/x^24-10/21*a^2*b^3/x^21-5/18*a*b^4/x^18-1/15*b^5/x^15

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, 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} \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=-\frac {a^5}{30 x^{30}}-\frac {5 a^4 b}{27 x^{27}}-\frac {5 a^3 b^2}{12 x^{24}}-\frac {10 a^2 b^3}{21 x^{21}}-\frac {5 a b^4}{18 x^{18}}-\frac {b^5}{15 x^{15}} \]

[In]

Int[(a + b*x^3)^5/x^31,x]

[Out]

-1/30*a^5/x^30 - (5*a^4*b)/(27*x^27) - (5*a^3*b^2)/(12*x^24) - (10*a^2*b^3)/(21*x^21) - (5*a*b^4)/(18*x^18) -
b^5/(15*x^15)

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^{11}} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {a^5}{x^{11}}+\frac {5 a^4 b}{x^{10}}+\frac {10 a^3 b^2}{x^9}+\frac {10 a^2 b^3}{x^8}+\frac {5 a b^4}{x^7}+\frac {b^5}{x^6}\right ) \, dx,x,x^3\right ) \\ & = -\frac {a^5}{30 x^{30}}-\frac {5 a^4 b}{27 x^{27}}-\frac {5 a^3 b^2}{12 x^{24}}-\frac {10 a^2 b^3}{21 x^{21}}-\frac {5 a b^4}{18 x^{18}}-\frac {b^5}{15 x^{15}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=-\frac {a^5}{30 x^{30}}-\frac {5 a^4 b}{27 x^{27}}-\frac {5 a^3 b^2}{12 x^{24}}-\frac {10 a^2 b^3}{21 x^{21}}-\frac {5 a b^4}{18 x^{18}}-\frac {b^5}{15 x^{15}} \]

[In]

Integrate[(a + b*x^3)^5/x^31,x]

[Out]

-1/30*a^5/x^30 - (5*a^4*b)/(27*x^27) - (5*a^3*b^2)/(12*x^24) - (10*a^2*b^3)/(21*x^21) - (5*a*b^4)/(18*x^18) -
b^5/(15*x^15)

Maple [A] (verified)

Time = 3.59 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {a^{5}}{30 x^{30}}-\frac {5 a^{4} b}{27 x^{27}}-\frac {5 a^{3} b^{2}}{12 x^{24}}-\frac {10 a^{2} b^{3}}{21 x^{21}}-\frac {5 a \,b^{4}}{18 x^{18}}-\frac {b^{5}}{15 x^{15}}\)	\(58\)
norman	\(\frac {-\frac {1}{15} b^{5} x^{15}-\frac {1}{30} a^{5}-\frac {5}{27} a^{4} b \,x^{3}-\frac {5}{12} a^{3} b^{2} x^{6}-\frac {10}{21} a^{2} b^{3} x^{9}-\frac {5}{18} a \,b^{4} x^{12}}{x^{30}}\)	\(59\)
risch	\(\frac {-\frac {1}{15} b^{5} x^{15}-\frac {1}{30} a^{5}-\frac {5}{27} a^{4} b \,x^{3}-\frac {5}{12} a^{3} b^{2} x^{6}-\frac {10}{21} a^{2} b^{3} x^{9}-\frac {5}{18} a \,b^{4} x^{12}}{x^{30}}\)	\(59\)
gosper	\(-\frac {252 b^{5} x^{15}+1050 a \,b^{4} x^{12}+1800 a^{2} b^{3} x^{9}+1575 a^{3} b^{2} x^{6}+700 a^{4} b \,x^{3}+126 a^{5}}{3780 x^{30}}\)	\(60\)
parallelrisch	\(\frac {-252 b^{5} x^{15}-1050 a \,b^{4} x^{12}-1800 a^{2} b^{3} x^{9}-1575 a^{3} b^{2} x^{6}-700 a^{4} b \,x^{3}-126 a^{5}}{3780 x^{30}}\)	\(60\)

[In]

int((b*x^3+a)^5/x^31,x,method=_RETURNVERBOSE)

[Out]

-1/30*a^5/x^30-5/27*a^4*b/x^27-5/12*a^3*b^2/x^24-10/21*a^2*b^3/x^21-5/18*a*b^4/x^18-1/15*b^5/x^15

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=-\frac {252 \, b^{5} x^{15} + 1050 \, a b^{4} x^{12} + 1800 \, a^{2} b^{3} x^{9} + 1575 \, a^{3} b^{2} x^{6} + 700 \, a^{4} b x^{3} + 126 \, a^{5}}{3780 \, x^{30}} \]

[In]

integrate((b*x^3+a)^5/x^31,x, algorithm="fricas")

[Out]

-1/3780*(252*b^5*x^15 + 1050*a*b^4*x^12 + 1800*a^2*b^3*x^9 + 1575*a^3*b^2*x^6 + 700*a^4*b*x^3 + 126*a^5)/x^30

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=\frac {- 126 a^{5} - 700 a^{4} b x^{3} - 1575 a^{3} b^{2} x^{6} - 1800 a^{2} b^{3} x^{9} - 1050 a b^{4} x^{12} - 252 b^{5} x^{15}}{3780 x^{30}} \]

[In]

integrate((b*x**3+a)**5/x**31,x)

[Out]

(-126*a**5 - 700*a**4*b*x**3 - 1575*a**3*b**2*x**6 - 1800*a**2*b**3*x**9 - 1050*a*b**4*x**12 - 252*b**5*x**15)
/(3780*x**30)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=-\frac {252 \, b^{5} x^{15} + 1050 \, a b^{4} x^{12} + 1800 \, a^{2} b^{3} x^{9} + 1575 \, a^{3} b^{2} x^{6} + 700 \, a^{4} b x^{3} + 126 \, a^{5}}{3780 \, x^{30}} \]

[In]

integrate((b*x^3+a)^5/x^31,x, algorithm="maxima")

[Out]

-1/3780*(252*b^5*x^15 + 1050*a*b^4*x^12 + 1800*a^2*b^3*x^9 + 1575*a^3*b^2*x^6 + 700*a^4*b*x^3 + 126*a^5)/x^30

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=-\frac {252 \, b^{5} x^{15} + 1050 \, a b^{4} x^{12} + 1800 \, a^{2} b^{3} x^{9} + 1575 \, a^{3} b^{2} x^{6} + 700 \, a^{4} b x^{3} + 126 \, a^{5}}{3780 \, x^{30}} \]

[In]

integrate((b*x^3+a)^5/x^31,x, algorithm="giac")

[Out]

-1/3780*(252*b^5*x^15 + 1050*a*b^4*x^12 + 1800*a^2*b^3*x^9 + 1575*a^3*b^2*x^6 + 700*a^4*b*x^3 + 126*a^5)/x^30

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^3\right )^5}{x^{31}} \, dx=-\frac {\frac {a^5}{30}+\frac {5\,a^4\,b\,x^3}{27}+\frac {5\,a^3\,b^2\,x^6}{12}+\frac {10\,a^2\,b^3\,x^9}{21}+\frac {5\,a\,b^4\,x^{12}}{18}+\frac {b^5\,x^{15}}{15}}{x^{30}} \]

[In]

int((a + b*x^3)^5/x^31,x)

[Out]

-(a^5/30 + (b^5*x^15)/15 + (5*a^4*b*x^3)/27 + (5*a*b^4*x^12)/18 + (5*a^3*b^2*x^6)/12 + (10*a^2*b^3*x^9)/21)/x^
30