\(\int \frac {(a+b x^3)^5}{x^{19}} \, dx\) [270]

Optimal result

Integrand size = 13, antiderivative size = 19 \[ \int \frac {\left (a+b x^3\right )^5}{x^{19}} \, dx=-\frac {\left (a+b x^3\right )^6}{18 a x^{18}} \]

[Out]

-1/18*(b*x^3+a)^6/a/x^18

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \int \frac {\left (a+b x^3\right )^5}{x^{19}} \, dx=-\frac {\left (a+b x^3\right )^6}{18 a x^{18}} \]

[In]

Int[(a + b*x^3)^5/x^19,x]

[Out]

-1/18*(a + b*x^3)^6/(a*x^18)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x^3\right )^6}{18 a x^{18}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(19)=38\).

Time = 0.00 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.63 \[ \int \frac {\left (a+b x^3\right )^5}{x^{19}} \, dx=-\frac {a^5}{18 x^{18}}-\frac {a^4 b}{3 x^{15}}-\frac {5 a^3 b^2}{6 x^{12}}-\frac {10 a^2 b^3}{9 x^9}-\frac {5 a b^4}{6 x^6}-\frac {b^5}{3 x^3} \]

[In]

Integrate[(a + b*x^3)^5/x^19,x]

[Out]

-1/18*a^5/x^18 - (a^4*b)/(3*x^15) - (5*a^3*b^2)/(6*x^12) - (10*a^2*b^3)/(9*x^9) - (5*a*b^4)/(6*x^6) - b^5/(3*x
^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(17)=34\).

Time = 3.66 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.05

method	result	size
gosper	\(-\frac {6 b^{5} x^{15}+15 a \,b^{4} x^{12}+20 a^{2} b^{3} x^{9}+15 a^{3} b^{2} x^{6}+6 a^{4} b \,x^{3}+a^{5}}{18 x^{18}}\)	\(58\)
default	\(-\frac {5 a \,b^{4}}{6 x^{6}}-\frac {5 a^{3} b^{2}}{6 x^{12}}-\frac {a^{4} b}{3 x^{15}}-\frac {a^{5}}{18 x^{18}}-\frac {b^{5}}{3 x^{3}}-\frac {10 a^{2} b^{3}}{9 x^{9}}\)	\(58\)
norman	\(\frac {-\frac {1}{18} a^{5}-\frac {1}{3} a^{4} b \,x^{3}-\frac {5}{6} a^{3} b^{2} x^{6}-\frac {10}{9} a^{2} b^{3} x^{9}-\frac {5}{6} a \,b^{4} x^{12}-\frac {1}{3} b^{5} x^{15}}{x^{18}}\)	\(59\)
risch	\(\frac {-\frac {1}{18} a^{5}-\frac {1}{3} a^{4} b \,x^{3}-\frac {5}{6} a^{3} b^{2} x^{6}-\frac {10}{9} a^{2} b^{3} x^{9}-\frac {5}{6} a \,b^{4} x^{12}-\frac {1}{3} b^{5} x^{15}}{x^{18}}\)	\(59\)
parallelrisch	\(\frac {-6 b^{5} x^{15}-15 a \,b^{4} x^{12}-20 a^{2} b^{3} x^{9}-15 a^{3} b^{2} x^{6}-6 a^{4} b \,x^{3}-a^{5}}{18 x^{18}}\)	\(60\)

[In]

int((b*x^3+a)^5/x^19,x,method=_RETURNVERBOSE)

[Out]

-1/18*(6*b^5*x^15+15*a*b^4*x^12+20*a^2*b^3*x^9+15*a^3*b^2*x^6+6*a^4*b*x^3+a^5)/x^18

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (17) = 34\).

Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.00 \[ \int \frac {\left (a+b x^3\right )^5}{x^{19}} \, dx=-\frac {6 \, b^{5} x^{15} + 15 \, a b^{4} x^{12} + 20 \, a^{2} b^{3} x^{9} + 15 \, a^{3} b^{2} x^{6} + 6 \, a^{4} b x^{3} + a^{5}}{18 \, x^{18}} \]

[In]

integrate((b*x^3+a)^5/x^19,x, algorithm="fricas")

[Out]

-1/18*(6*b^5*x^15 + 15*a*b^4*x^12 + 20*a^2*b^3*x^9 + 15*a^3*b^2*x^6 + 6*a^4*b*x^3 + a^5)/x^18

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (15) = 30\).

Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 3.21 \[ \int \frac {\left (a+b x^3\right )^5}{x^{19}} \, dx=\frac {- a^{5} - 6 a^{4} b x^{3} - 15 a^{3} b^{2} x^{6} - 20 a^{2} b^{3} x^{9} - 15 a b^{4} x^{12} - 6 b^{5} x^{15}}{18 x^{18}} \]

[In]

integrate((b*x**3+a)**5/x**19,x)

[Out]

(-a**5 - 6*a**4*b*x**3 - 15*a**3*b**2*x**6 - 20*a**2*b**3*x**9 - 15*a*b**4*x**12 - 6*b**5*x**15)/(18*x**18)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (17) = 34\).

Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.00 \[ \int \frac {\left (a+b x^3\right )^5}{x^{19}} \, dx=-\frac {6 \, b^{5} x^{15} + 15 \, a b^{4} x^{12} + 20 \, a^{2} b^{3} x^{9} + 15 \, a^{3} b^{2} x^{6} + 6 \, a^{4} b x^{3} + a^{5}}{18 \, x^{18}} \]

[In]

integrate((b*x^3+a)^5/x^19,x, algorithm="maxima")

[Out]

-1/18*(6*b^5*x^15 + 15*a*b^4*x^12 + 20*a^2*b^3*x^9 + 15*a^3*b^2*x^6 + 6*a^4*b*x^3 + a^5)/x^18

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (17) = 34\).

Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.00 \[ \int \frac {\left (a+b x^3\right )^5}{x^{19}} \, dx=-\frac {6 \, b^{5} x^{15} + 15 \, a b^{4} x^{12} + 20 \, a^{2} b^{3} x^{9} + 15 \, a^{3} b^{2} x^{6} + 6 \, a^{4} b x^{3} + a^{5}}{18 \, x^{18}} \]

[In]

integrate((b*x^3+a)^5/x^19,x, algorithm="giac")

[Out]

-1/18*(6*b^5*x^15 + 15*a*b^4*x^12 + 20*a^2*b^3*x^9 + 15*a^3*b^2*x^6 + 6*a^4*b*x^3 + a^5)/x^18

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 3.11 \[ \int \frac {\left (a+b x^3\right )^5}{x^{19}} \, dx=-\frac {\frac {a^5}{18}+\frac {a^4\,b\,x^3}{3}+\frac {5\,a^3\,b^2\,x^6}{6}+\frac {10\,a^2\,b^3\,x^9}{9}+\frac {5\,a\,b^4\,x^{12}}{6}+\frac {b^5\,x^{15}}{3}}{x^{18}} \]

[In]

int((a + b*x^3)^5/x^19,x)

[Out]

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