\(\int \frac {(a+b x^3)^8}{x^{28}} \, dx\) [2727]

Optimal result

Integrand size = 13, antiderivative size = 19 \[ \int \frac {\left (a+b x^3\right )^8}{x^{28}} \, dx=-\frac {\left (a+b x^3\right )^9}{27 a x^{27}} \]

[Out]

-1/27*(b*x^3+a)^9/a/x^27

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} \[ \int \frac {\left (a+b x^3\right )^8}{x^{28}} \, dx=-\frac {\left (a+b x^3\right )^9}{27 a x^{27}} \]

[In]

Int[(a + b*x^3)^8/x^28,x]

[Out]

-1/27*(a + b*x^3)^9/(a*x^27)

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 )^9}{27 a x^{27}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(108\) vs. \(2(19)=38\).

Time = 0.01 (sec) , antiderivative size = 108, normalized size of antiderivative = 5.68 \[ \int \frac {\left (a+b x^3\right )^8}{x^{28}} \, dx=-\frac {a^8}{27 x^{27}}-\frac {a^7 b}{3 x^{24}}-\frac {4 a^6 b^2}{3 x^{21}}-\frac {28 a^5 b^3}{9 x^{18}}-\frac {14 a^4 b^4}{3 x^{15}}-\frac {14 a^3 b^5}{3 x^{12}}-\frac {28 a^2 b^6}{9 x^9}-\frac {4 a b^7}{3 x^6}-\frac {b^8}{3 x^3} \]

[In]

Integrate[(a + b*x^3)^8/x^28,x]

[Out]

-1/27*a^8/x^27 - (a^7*b)/(3*x^24) - (4*a^6*b^2)/(3*x^21) - (28*a^5*b^3)/(9*x^18) - (14*a^4*b^4)/(3*x^15) - (14
*a^3*b^5)/(3*x^12) - (28*a^2*b^6)/(9*x^9) - (4*a*b^7)/(3*x^6) - b^8/(3*x^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(17)=34\).

Time = 3.73 (sec) , antiderivative size = 91, normalized size of antiderivative = 4.79

method	result	size
gosper	\(-\frac {9 b^{8} x^{24}+36 a \,b^{7} x^{21}+84 a^{2} b^{6} x^{18}+126 a^{3} b^{5} x^{15}+126 a^{4} b^{4} x^{12}+84 x^{9} b^{3} a^{5}+36 a^{6} b^{2} x^{6}+9 x^{3} b \,a^{7}+a^{8}}{27 x^{27}}\)	\(91\)
default	\(-\frac {4 a \,b^{7}}{3 x^{6}}-\frac {14 a^{3} b^{5}}{3 x^{12}}-\frac {4 a^{6} b^{2}}{3 x^{21}}-\frac {14 a^{4} b^{4}}{3 x^{15}}-\frac {28 a^{5} b^{3}}{9 x^{18}}-\frac {b^{8}}{3 x^{3}}-\frac {28 a^{2} b^{6}}{9 x^{9}}-\frac {a^{7} b}{3 x^{24}}-\frac {a^{8}}{27 x^{27}}\)	\(91\)
norman	\(\frac {-\frac {14}{3} a^{3} b^{5} x^{15}-\frac {28}{9} a^{2} b^{6} x^{18}-\frac {28}{9} x^{9} b^{3} a^{5}-\frac {14}{3} a^{4} b^{4} x^{12}-\frac {1}{27} a^{8}-\frac {4}{3} a \,b^{7} x^{21}-\frac {1}{3} x^{3} b \,a^{7}-\frac {4}{3} a^{6} b^{2} x^{6}-\frac {1}{3} b^{8} x^{24}}{x^{27}}\)	\(92\)
risch	\(\frac {-\frac {14}{3} a^{3} b^{5} x^{15}-\frac {28}{9} a^{2} b^{6} x^{18}-\frac {28}{9} x^{9} b^{3} a^{5}-\frac {14}{3} a^{4} b^{4} x^{12}-\frac {1}{27} a^{8}-\frac {4}{3} a \,b^{7} x^{21}-\frac {1}{3} x^{3} b \,a^{7}-\frac {4}{3} a^{6} b^{2} x^{6}-\frac {1}{3} b^{8} x^{24}}{x^{27}}\)	\(92\)
parallelrisch	\(\frac {-9 b^{8} x^{24}-36 a \,b^{7} x^{21}-84 a^{2} b^{6} x^{18}-126 a^{3} b^{5} x^{15}-126 a^{4} b^{4} x^{12}-84 x^{9} b^{3} a^{5}-36 a^{6} b^{2} x^{6}-9 x^{3} b \,a^{7}-a^{8}}{27 x^{27}}\)	\(93\)

[In]

int((b*x^3+a)^8/x^28,x,method=_RETURNVERBOSE)

[Out]

-1/27*(9*b^8*x^24+36*a*b^7*x^21+84*a^2*b^6*x^18+126*a^3*b^5*x^15+126*a^4*b^4*x^12+84*a^5*b^3*x^9+36*a^6*b^2*x^
6+9*a^7*b*x^3+a^8)/x^27

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (17) = 34\).

Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.74 \[ \int \frac {\left (a+b x^3\right )^8}{x^{28}} \, dx=-\frac {9 \, b^{8} x^{24} + 36 \, a b^{7} x^{21} + 84 \, a^{2} b^{6} x^{18} + 126 \, a^{3} b^{5} x^{15} + 126 \, a^{4} b^{4} x^{12} + 84 \, a^{5} b^{3} x^{9} + 36 \, a^{6} b^{2} x^{6} + 9 \, a^{7} b x^{3} + a^{8}}{27 \, x^{27}} \]

[In]

integrate((b*x^3+a)^8/x^28,x, algorithm="fricas")

[Out]

-1/27*(9*b^8*x^24 + 36*a*b^7*x^21 + 84*a^2*b^6*x^18 + 126*a^3*b^5*x^15 + 126*a^4*b^4*x^12 + 84*a^5*b^3*x^9 + 3
6*a^6*b^2*x^6 + 9*a^7*b*x^3 + a^8)/x^27

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (15) = 30\).

Time = 0.56 (sec) , antiderivative size = 97, normalized size of antiderivative = 5.11 \[ \int \frac {\left (a+b x^3\right )^8}{x^{28}} \, dx=\frac {- a^{8} - 9 a^{7} b x^{3} - 36 a^{6} b^{2} x^{6} - 84 a^{5} b^{3} x^{9} - 126 a^{4} b^{4} x^{12} - 126 a^{3} b^{5} x^{15} - 84 a^{2} b^{6} x^{18} - 36 a b^{7} x^{21} - 9 b^{8} x^{24}}{27 x^{27}} \]

[In]

integrate((b*x**3+a)**8/x**28,x)

[Out]

(-a**8 - 9*a**7*b*x**3 - 36*a**6*b**2*x**6 - 84*a**5*b**3*x**9 - 126*a**4*b**4*x**12 - 126*a**3*b**5*x**15 - 8
4*a**2*b**6*x**18 - 36*a*b**7*x**21 - 9*b**8*x**24)/(27*x**27)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (17) = 34\).

Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.74 \[ \int \frac {\left (a+b x^3\right )^8}{x^{28}} \, dx=-\frac {9 \, b^{8} x^{24} + 36 \, a b^{7} x^{21} + 84 \, a^{2} b^{6} x^{18} + 126 \, a^{3} b^{5} x^{15} + 126 \, a^{4} b^{4} x^{12} + 84 \, a^{5} b^{3} x^{9} + 36 \, a^{6} b^{2} x^{6} + 9 \, a^{7} b x^{3} + a^{8}}{27 \, x^{27}} \]

[In]

integrate((b*x^3+a)^8/x^28,x, algorithm="maxima")

[Out]

-1/27*(9*b^8*x^24 + 36*a*b^7*x^21 + 84*a^2*b^6*x^18 + 126*a^3*b^5*x^15 + 126*a^4*b^4*x^12 + 84*a^5*b^3*x^9 + 3
6*a^6*b^2*x^6 + 9*a^7*b*x^3 + a^8)/x^27

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (17) = 34\).

Time = 0.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.74 \[ \int \frac {\left (a+b x^3\right )^8}{x^{28}} \, dx=-\frac {9 \, b^{8} x^{24} + 36 \, a b^{7} x^{21} + 84 \, a^{2} b^{6} x^{18} + 126 \, a^{3} b^{5} x^{15} + 126 \, a^{4} b^{4} x^{12} + 84 \, a^{5} b^{3} x^{9} + 36 \, a^{6} b^{2} x^{6} + 9 \, a^{7} b x^{3} + a^{8}}{27 \, x^{27}} \]

[In]

integrate((b*x^3+a)^8/x^28,x, algorithm="giac")

[Out]

-1/27*(9*b^8*x^24 + 36*a*b^7*x^21 + 84*a^2*b^6*x^18 + 126*a^3*b^5*x^15 + 126*a^4*b^4*x^12 + 84*a^5*b^3*x^9 + 3
6*a^6*b^2*x^6 + 9*a^7*b*x^3 + a^8)/x^27

Mupad [B] (verification not implemented)

Time = 5.81 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.84 \[ \int \frac {\left (a+b x^3\right )^8}{x^{28}} \, dx=-\frac {\frac {a^8}{27}+\frac {a^7\,b\,x^3}{3}+\frac {4\,a^6\,b^2\,x^6}{3}+\frac {28\,a^5\,b^3\,x^9}{9}+\frac {14\,a^4\,b^4\,x^{12}}{3}+\frac {14\,a^3\,b^5\,x^{15}}{3}+\frac {28\,a^2\,b^6\,x^{18}}{9}+\frac {4\,a\,b^7\,x^{21}}{3}+\frac {b^8\,x^{24}}{3}}{x^{27}} \]

[In]

int((a + b*x^3)^8/x^28,x)

[Out]

-(a^8/27 + (b^8*x^24)/3 + (a^7*b*x^3)/3 + (4*a*b^7*x^21)/3 + (4*a^6*b^2*x^6)/3 + (28*a^5*b^3*x^9)/9 + (14*a^4*
b^4*x^12)/3 + (14*a^3*b^5*x^15)/3 + (28*a^2*b^6*x^18)/9)/x^27