3.4.17 \(\int \frac {(a+b x^3)^8}{x^9} \, dx\) [317]

Optimal. Leaf size=100 \[ -\frac {a^8}{8 x^8}-\frac {8 a^7 b}{5 x^5}-\frac {14 a^6 b^2}{x^2}+56 a^5 b^3 x+\frac {35}{2} a^4 b^4 x^4+8 a^3 b^5 x^7+\frac {14}{5} a^2 b^6 x^{10}+\frac {8}{13} a b^7 x^{13}+\frac {b^8 x^{16}}{16} \]

[Out]

-1/8*a^8/x^8-8/5*a^7*b/x^5-14*a^6*b^2/x^2+56*a^5*b^3*x+35/2*a^4*b^4*x^4+8*a^3*b^5*x^7+14/5*a^2*b^6*x^10+8/13*a
*b^7*x^13+1/16*b^8*x^16

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Rubi [A]
time = 0.02, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276} \begin {gather*} -\frac {a^8}{8 x^8}-\frac {8 a^7 b}{5 x^5}-\frac {14 a^6 b^2}{x^2}+56 a^5 b^3 x+\frac {35}{2} a^4 b^4 x^4+8 a^3 b^5 x^7+\frac {14}{5} a^2 b^6 x^{10}+\frac {8}{13} a b^7 x^{13}+\frac {b^8 x^{16}}{16} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^8/x^9,x]

[Out]

-1/8*a^8/x^8 - (8*a^7*b)/(5*x^5) - (14*a^6*b^2)/x^2 + 56*a^5*b^3*x + (35*a^4*b^4*x^4)/2 + 8*a^3*b^5*x^7 + (14*
a^2*b^6*x^10)/5 + (8*a*b^7*x^13)/13 + (b^8*x^16)/16

Rule 276

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

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^8}{x^9} \, dx &=\int \left (56 a^5 b^3+\frac {a^8}{x^9}+\frac {8 a^7 b}{x^6}+\frac {28 a^6 b^2}{x^3}+70 a^4 b^4 x^3+56 a^3 b^5 x^6+28 a^2 b^6 x^9+8 a b^7 x^{12}+b^8 x^{15}\right ) \, dx\\ &=-\frac {a^8}{8 x^8}-\frac {8 a^7 b}{5 x^5}-\frac {14 a^6 b^2}{x^2}+56 a^5 b^3 x+\frac {35}{2} a^4 b^4 x^4+8 a^3 b^5 x^7+\frac {14}{5} a^2 b^6 x^{10}+\frac {8}{13} a b^7 x^{13}+\frac {b^8 x^{16}}{16}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 100, normalized size = 1.00 \begin {gather*} -\frac {a^8}{8 x^8}-\frac {8 a^7 b}{5 x^5}-\frac {14 a^6 b^2}{x^2}+56 a^5 b^3 x+\frac {35}{2} a^4 b^4 x^4+8 a^3 b^5 x^7+\frac {14}{5} a^2 b^6 x^{10}+\frac {8}{13} a b^7 x^{13}+\frac {b^8 x^{16}}{16} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^8/x^9,x]

[Out]

-1/8*a^8/x^8 - (8*a^7*b)/(5*x^5) - (14*a^6*b^2)/x^2 + 56*a^5*b^3*x + (35*a^4*b^4*x^4)/2 + 8*a^3*b^5*x^7 + (14*
a^2*b^6*x^10)/5 + (8*a*b^7*x^13)/13 + (b^8*x^16)/16

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Maple [A]
time = 0.12, size = 89, normalized size = 0.89

method	result	size
default	\(-\frac {a^{8}}{8 x^{8}}-\frac {8 a^{7} b}{5 x^{5}}-\frac {14 a^{6} b^{2}}{x^{2}}+56 a^{5} b^{3} x +\frac {35 a^{4} b^{4} x^{4}}{2}+8 a^{3} b^{5} x^{7}+\frac {14 a^{2} b^{6} x^{10}}{5}+\frac {8 a \,b^{7} x^{13}}{13}+\frac {b^{8} x^{16}}{16}\)	\(89\)
risch	\(\frac {b^{8} x^{16}}{16}+\frac {8 a \,b^{7} x^{13}}{13}+\frac {14 a^{2} b^{6} x^{10}}{5}+8 a^{3} b^{5} x^{7}+\frac {35 a^{4} b^{4} x^{4}}{2}+56 a^{5} b^{3} x +\frac {-14 a^{6} b^{2} x^{6}-\frac {8}{5} a^{7} b \,x^{3}-\frac {1}{8} a^{8}}{x^{8}}\)	\(91\)
norman	\(\frac {\frac {8}{13} a \,b^{7} x^{21}+\frac {1}{16} b^{8} x^{24}+\frac {14}{5} a^{2} b^{6} x^{18}+56 a^{5} b^{3} x^{9}+\frac {35}{2} a^{4} b^{4} x^{12}+8 a^{3} b^{5} x^{15}-14 a^{6} b^{2} x^{6}-\frac {8}{5} a^{7} b \,x^{3}-\frac {1}{8} a^{8}}{x^{8}}\)	\(92\)
gosper	\(-\frac {-65 b^{8} x^{24}-640 a \,b^{7} x^{21}-2912 a^{2} b^{6} x^{18}-8320 a^{3} b^{5} x^{15}-18200 a^{4} b^{4} x^{12}-58240 a^{5} b^{3} x^{9}+14560 a^{6} b^{2} x^{6}+1664 a^{7} b \,x^{3}+130 a^{8}}{1040 x^{8}}\)	\(93\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^8/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/8*a^8/x^8-8/5*a^7*b/x^5-14*a^6*b^2/x^2+56*a^5*b^3*x+35/2*a^4*b^4*x^4+8*a^3*b^5*x^7+14/5*a^2*b^6*x^10+8/13*a
*b^7*x^13+1/16*b^8*x^16

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Maxima [A]
time = 0.29, size = 91, normalized size = 0.91 \begin {gather*} \frac {1}{16} \, b^{8} x^{16} + \frac {8}{13} \, a b^{7} x^{13} + \frac {14}{5} \, a^{2} b^{6} x^{10} + 8 \, a^{3} b^{5} x^{7} + \frac {35}{2} \, a^{4} b^{4} x^{4} + 56 \, a^{5} b^{3} x - \frac {560 \, a^{6} b^{2} x^{6} + 64 \, a^{7} b x^{3} + 5 \, a^{8}}{40 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^9,x, algorithm="maxima")

[Out]

1/16*b^8*x^16 + 8/13*a*b^7*x^13 + 14/5*a^2*b^6*x^10 + 8*a^3*b^5*x^7 + 35/2*a^4*b^4*x^4 + 56*a^5*b^3*x - 1/40*(
560*a^6*b^2*x^6 + 64*a^7*b*x^3 + 5*a^8)/x^8

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Fricas [A]
time = 0.35, size = 92, normalized size = 0.92 \begin {gather*} \frac {65 \, b^{8} x^{24} + 640 \, a b^{7} x^{21} + 2912 \, a^{2} b^{6} x^{18} + 8320 \, a^{3} b^{5} x^{15} + 18200 \, a^{4} b^{4} x^{12} + 58240 \, a^{5} b^{3} x^{9} - 14560 \, a^{6} b^{2} x^{6} - 1664 \, a^{7} b x^{3} - 130 \, a^{8}}{1040 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^9,x, algorithm="fricas")

[Out]

1/1040*(65*b^8*x^24 + 640*a*b^7*x^21 + 2912*a^2*b^6*x^18 + 8320*a^3*b^5*x^15 + 18200*a^4*b^4*x^12 + 58240*a^5*
b^3*x^9 - 14560*a^6*b^2*x^6 - 1664*a^7*b*x^3 - 130*a^8)/x^8

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Sympy [A]
time = 0.11, size = 102, normalized size = 1.02 \begin {gather*} 56 a^{5} b^{3} x + \frac {35 a^{4} b^{4} x^{4}}{2} + 8 a^{3} b^{5} x^{7} + \frac {14 a^{2} b^{6} x^{10}}{5} + \frac {8 a b^{7} x^{13}}{13} + \frac {b^{8} x^{16}}{16} + \frac {- 5 a^{8} - 64 a^{7} b x^{3} - 560 a^{6} b^{2} x^{6}}{40 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**8/x**9,x)

[Out]

56*a**5*b**3*x + 35*a**4*b**4*x**4/2 + 8*a**3*b**5*x**7 + 14*a**2*b**6*x**10/5 + 8*a*b**7*x**13/13 + b**8*x**1
6/16 + (-5*a**8 - 64*a**7*b*x**3 - 560*a**6*b**2*x**6)/(40*x**8)

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Giac [A]
time = 1.55, size = 91, normalized size = 0.91 \begin {gather*} \frac {1}{16} \, b^{8} x^{16} + \frac {8}{13} \, a b^{7} x^{13} + \frac {14}{5} \, a^{2} b^{6} x^{10} + 8 \, a^{3} b^{5} x^{7} + \frac {35}{2} \, a^{4} b^{4} x^{4} + 56 \, a^{5} b^{3} x - \frac {560 \, a^{6} b^{2} x^{6} + 64 \, a^{7} b x^{3} + 5 \, a^{8}}{40 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^8/x^9,x, algorithm="giac")

[Out]

1/16*b^8*x^16 + 8/13*a*b^7*x^13 + 14/5*a^2*b^6*x^10 + 8*a^3*b^5*x^7 + 35/2*a^4*b^4*x^4 + 56*a^5*b^3*x - 1/40*(
560*a^6*b^2*x^6 + 64*a^7*b*x^3 + 5*a^8)/x^8

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Mupad [B]
time = 0.05, size = 91, normalized size = 0.91 \begin {gather*} \frac {b^8\,x^{16}}{16}-\frac {\frac {a^8}{8}+\frac {8\,a^7\,b\,x^3}{5}+14\,a^6\,b^2\,x^6}{x^8}+56\,a^5\,b^3\,x+\frac {8\,a\,b^7\,x^{13}}{13}+\frac {35\,a^4\,b^4\,x^4}{2}+8\,a^3\,b^5\,x^7+\frac {14\,a^2\,b^6\,x^{10}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^8/x^9,x)

[Out]

(b^8*x^16)/16 - (a^8/8 + (8*a^7*b*x^3)/5 + 14*a^6*b^2*x^6)/x^8 + 56*a^5*b^3*x + (8*a*b^7*x^13)/13 + (35*a^4*b^
4*x^4)/2 + 8*a^3*b^5*x^7 + (14*a^2*b^6*x^10)/5

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