\(\int \frac {(a+b x^3)^5}{x^6} \, dx\) [282]

Optimal result

Integrand size = 13, antiderivative size = 65 \[ \int \frac {\left (a+b x^3\right )^5}{x^6} \, dx=-\frac {a^5}{5 x^5}-\frac {5 a^4 b}{2 x^2}+10 a^3 b^2 x+\frac {5}{2} a^2 b^3 x^4+\frac {5}{7} a b^4 x^7+\frac {b^5 x^{10}}{10} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 65, 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} \[ \int \frac {\left (a+b x^3\right )^5}{x^6} \, dx=-\frac {a^5}{5 x^5}-\frac {5 a^4 b}{2 x^2}+10 a^3 b^2 x+\frac {5}{2} a^2 b^3 x^4+\frac {5}{7} a b^4 x^7+\frac {b^5 x^{10}}{10} \]

[In]

Int[(a + b*x^3)^5/x^6,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5}{x^6} \, dx=-\frac {a^5}{5 x^5}-\frac {5 a^4 b}{2 x^2}+10 a^3 b^2 x+\frac {5}{2} a^2 b^3 x^4+\frac {5}{7} a b^4 x^7+\frac {b^5 x^{10}}{10} \]

[In]

Integrate[(a + b*x^3)^5/x^6,x]

[Out]

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

Maple [A] (verified)

Time = 3.61 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {a^{5}}{5 x^{5}}-\frac {5 a^{4} b}{2 x^{2}}+10 a^{3} b^{2} x +\frac {5 a^{2} b^{3} x^{4}}{2}+\frac {5 a \,b^{4} x^{7}}{7}+\frac {b^{5} x^{10}}{10}\)	\(56\)
risch	\(\frac {b^{5} x^{10}}{10}+\frac {5 a \,b^{4} x^{7}}{7}+\frac {5 a^{2} b^{3} x^{4}}{2}+10 a^{3} b^{2} x +\frac {-\frac {5}{2} a^{4} b \,x^{3}-\frac {1}{5} a^{5}}{x^{5}}\)	\(58\)
norman	\(\frac {-\frac {1}{5} a^{5}-\frac {5}{2} a^{4} b \,x^{3}+10 a^{3} b^{2} x^{6}+\frac {5}{2} a^{2} b^{3} x^{9}+\frac {5}{7} a \,b^{4} x^{12}+\frac {1}{10} b^{5} x^{15}}{x^{5}}\)	\(59\)
gosper	\(-\frac {-7 b^{5} x^{15}-50 a \,b^{4} x^{12}-175 a^{2} b^{3} x^{9}-700 a^{3} b^{2} x^{6}+175 a^{4} b \,x^{3}+14 a^{5}}{70 x^{5}}\)	\(60\)
parallelrisch	\(\frac {7 b^{5} x^{15}+50 a \,b^{4} x^{12}+175 a^{2} b^{3} x^{9}+700 a^{3} b^{2} x^{6}-175 a^{4} b \,x^{3}-14 a^{5}}{70 x^{5}}\)	\(60\)

[In]

int((b*x^3+a)^5/x^6,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^3\right )^5}{x^6} \, dx=\frac {7 \, b^{5} x^{15} + 50 \, a b^{4} x^{12} + 175 \, a^{2} b^{3} x^{9} + 700 \, a^{3} b^{2} x^{6} - 175 \, a^{4} b x^{3} - 14 \, a^{5}}{70 \, x^{5}} \]

[In]

integrate((b*x^3+a)^5/x^6,x, algorithm="fricas")

[Out]

1/70*(7*b^5*x^15 + 50*a*b^4*x^12 + 175*a^2*b^3*x^9 + 700*a^3*b^2*x^6 - 175*a^4*b*x^3 - 14*a^5)/x^5

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5}{x^6} \, dx=10 a^{3} b^{2} x + \frac {5 a^{2} b^{3} x^{4}}{2} + \frac {5 a b^{4} x^{7}}{7} + \frac {b^{5} x^{10}}{10} + \frac {- 2 a^{5} - 25 a^{4} b x^{3}}{10 x^{5}} \]

[In]

integrate((b*x**3+a)**5/x**6,x)

[Out]

10*a**3*b**2*x + 5*a**2*b**3*x**4/2 + 5*a*b**4*x**7/7 + b**5*x**10/10 + (-2*a**5 - 25*a**4*b*x**3)/(10*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^3\right )^5}{x^6} \, dx=\frac {1}{10} \, b^{5} x^{10} + \frac {5}{7} \, a b^{4} x^{7} + \frac {5}{2} \, a^{2} b^{3} x^{4} + 10 \, a^{3} b^{2} x - \frac {25 \, a^{4} b x^{3} + 2 \, a^{5}}{10 \, x^{5}} \]

[In]

integrate((b*x^3+a)^5/x^6,x, algorithm="maxima")

[Out]

1/10*b^5*x^10 + 5/7*a*b^4*x^7 + 5/2*a^2*b^3*x^4 + 10*a^3*b^2*x - 1/10*(25*a^4*b*x^3 + 2*a^5)/x^5

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^3\right )^5}{x^6} \, dx=\frac {1}{10} \, b^{5} x^{10} + \frac {5}{7} \, a b^{4} x^{7} + \frac {5}{2} \, a^{2} b^{3} x^{4} + 10 \, a^{3} b^{2} x - \frac {25 \, a^{4} b x^{3} + 2 \, a^{5}}{10 \, x^{5}} \]

[In]

integrate((b*x^3+a)^5/x^6,x, algorithm="giac")

[Out]

1/10*b^5*x^10 + 5/7*a*b^4*x^7 + 5/2*a^2*b^3*x^4 + 10*a^3*b^2*x - 1/10*(25*a^4*b*x^3 + 2*a^5)/x^5

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {\left (a+b x^3\right )^5}{x^6} \, dx=\frac {b^5\,x^{10}}{10}-\frac {\frac {a^5}{5}+\frac {5\,b\,a^4\,x^3}{2}}{x^5}+10\,a^3\,b^2\,x+\frac {5\,a\,b^4\,x^7}{7}+\frac {5\,a^2\,b^3\,x^4}{2} \]

[In]

int((a + b*x^3)^5/x^6,x)

[Out]

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