\(\int \frac {(a+b x^3)^5}{x^9} \, dx\) [284]

Optimal result

Integrand size = 13, antiderivative size = 61 \[ \int \frac {\left (a+b x^3\right )^5}{x^9} \, dx=-\frac {a^5}{8 x^8}-\frac {a^4 b}{x^5}-\frac {5 a^3 b^2}{x^2}+10 a^2 b^3 x+\frac {5}{4} a b^4 x^4+\frac {b^5 x^7}{7} \]

[Out]

-1/8*a^5/x^8-a^4*b/x^5-5*a^3*b^2/x^2+10*a^2*b^3*x+5/4*a*b^4*x^4+1/7*b^5*x^7

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276} \[ \int \frac {\left (a+b x^3\right )^5}{x^9} \, dx=-\frac {a^5}{8 x^8}-\frac {a^4 b}{x^5}-\frac {5 a^3 b^2}{x^2}+10 a^2 b^3 x+\frac {5}{4} a b^4 x^4+\frac {b^5 x^7}{7} \]

[In]

Int[(a + b*x^3)^5/x^9,x]

[Out]

-1/8*a^5/x^8 - (a^4*b)/x^5 - (5*a^3*b^2)/x^2 + 10*a^2*b^3*x + (5*a*b^4*x^4)/4 + (b^5*x^7)/7

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*} \text {integral}& = \int \left (10 a^2 b^3+\frac {a^5}{x^9}+\frac {5 a^4 b}{x^6}+\frac {10 a^3 b^2}{x^3}+5 a b^4 x^3+b^5 x^6\right ) \, dx \\ & = -\frac {a^5}{8 x^8}-\frac {a^4 b}{x^5}-\frac {5 a^3 b^2}{x^2}+10 a^2 b^3 x+\frac {5}{4} a b^4 x^4+\frac {b^5 x^7}{7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5}{x^9} \, dx=-\frac {a^5}{8 x^8}-\frac {a^4 b}{x^5}-\frac {5 a^3 b^2}{x^2}+10 a^2 b^3 x+\frac {5}{4} a b^4 x^4+\frac {b^5 x^7}{7} \]

[In]

Integrate[(a + b*x^3)^5/x^9,x]

[Out]

-1/8*a^5/x^8 - (a^4*b)/x^5 - (5*a^3*b^2)/x^2 + 10*a^2*b^3*x + (5*a*b^4*x^4)/4 + (b^5*x^7)/7

Maple [A] (verified)

Time = 3.58 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92

method	result	size
default	\(-\frac {a^{5}}{8 x^{8}}-\frac {a^{4} b}{x^{5}}-\frac {5 a^{3} b^{2}}{x^{2}}+10 a^{2} b^{3} x +\frac {5 a \,b^{4} x^{4}}{4}+\frac {b^{5} x^{7}}{7}\)	\(56\)
risch	\(\frac {b^{5} x^{7}}{7}+\frac {5 a \,b^{4} x^{4}}{4}+10 a^{2} b^{3} x +\frac {-5 a^{3} b^{2} x^{6}-a^{4} b \,x^{3}-\frac {1}{8} a^{5}}{x^{8}}\)	\(58\)
norman	\(\frac {-\frac {1}{8} a^{5}-a^{4} b \,x^{3}-5 a^{3} b^{2} x^{6}+10 a^{2} b^{3} x^{9}+\frac {5}{4} a \,b^{4} x^{12}+\frac {1}{7} b^{5} x^{15}}{x^{8}}\)	\(59\)
gosper	\(-\frac {-8 b^{5} x^{15}-70 a \,b^{4} x^{12}-560 a^{2} b^{3} x^{9}+280 a^{3} b^{2} x^{6}+56 a^{4} b \,x^{3}+7 a^{5}}{56 x^{8}}\)	\(60\)
parallelrisch	\(\frac {8 b^{5} x^{15}+70 a \,b^{4} x^{12}+560 a^{2} b^{3} x^{9}-280 a^{3} b^{2} x^{6}-56 a^{4} b \,x^{3}-7 a^{5}}{56 x^{8}}\)	\(60\)

[In]

int((b*x^3+a)^5/x^9,x,method=_RETURNVERBOSE)

[Out]

-1/8*a^5/x^8-a^4*b/x^5-5*a^3*b^2/x^2+10*a^2*b^3*x+5/4*a*b^4*x^4+1/7*b^5*x^7

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.97 \[ \int \frac {\left (a+b x^3\right )^5}{x^9} \, dx=\frac {8 \, b^{5} x^{15} + 70 \, a b^{4} x^{12} + 560 \, a^{2} b^{3} x^{9} - 280 \, a^{3} b^{2} x^{6} - 56 \, a^{4} b x^{3} - 7 \, a^{5}}{56 \, x^{8}} \]

[In]

integrate((b*x^3+a)^5/x^9,x, algorithm="fricas")

[Out]

1/56*(8*b^5*x^15 + 70*a*b^4*x^12 + 560*a^2*b^3*x^9 - 280*a^3*b^2*x^6 - 56*a^4*b*x^3 - 7*a^5)/x^8

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5}{x^9} \, dx=10 a^{2} b^{3} x + \frac {5 a b^{4} x^{4}}{4} + \frac {b^{5} x^{7}}{7} + \frac {- a^{5} - 8 a^{4} b x^{3} - 40 a^{3} b^{2} x^{6}}{8 x^{8}} \]

[In]

integrate((b*x**3+a)**5/x**9,x)

[Out]

10*a**2*b**3*x + 5*a*b**4*x**4/4 + b**5*x**7/7 + (-a**5 - 8*a**4*b*x**3 - 40*a**3*b**2*x**6)/(8*x**8)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^3\right )^5}{x^9} \, dx=\frac {1}{7} \, b^{5} x^{7} + \frac {5}{4} \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x - \frac {40 \, a^{3} b^{2} x^{6} + 8 \, a^{4} b x^{3} + a^{5}}{8 \, x^{8}} \]

[In]

integrate((b*x^3+a)^5/x^9,x, algorithm="maxima")

[Out]

1/7*b^5*x^7 + 5/4*a*b^4*x^4 + 10*a^2*b^3*x - 1/8*(40*a^3*b^2*x^6 + 8*a^4*b*x^3 + a^5)/x^8

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b x^3\right )^5}{x^9} \, dx=\frac {1}{7} \, b^{5} x^{7} + \frac {5}{4} \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x - \frac {40 \, a^{3} b^{2} x^{6} + 8 \, a^{4} b x^{3} + a^{5}}{8 \, x^{8}} \]

[In]

integrate((b*x^3+a)^5/x^9,x, algorithm="giac")

[Out]

1/7*b^5*x^7 + 5/4*a*b^4*x^4 + 10*a^2*b^3*x - 1/8*(40*a^3*b^2*x^6 + 8*a^4*b*x^3 + a^5)/x^8

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a+b x^3\right )^5}{x^9} \, dx=\frac {b^5\,x^7}{7}-\frac {\frac {a^5}{8}+a^4\,b\,x^3+5\,a^3\,b^2\,x^6}{x^8}+10\,a^2\,b^3\,x+\frac {5\,a\,b^4\,x^4}{4} \]

[In]

int((a + b*x^3)^5/x^9,x)

[Out]

(b^5*x^7)/7 - (a^5/8 + a^4*b*x^3 + 5*a^3*b^2*x^6)/x^8 + 10*a^2*b^3*x + (5*a*b^4*x^4)/4