\(\int \frac {(a+b x^3)^5}{x^3} \, dx\) [280]

Optimal result

Integrand size = 13, antiderivative size = 65 \[ \int \frac {\left (a+b x^3\right )^5}{x^3} \, dx=-\frac {a^5}{2 x^2}+5 a^4 b x+\frac {5}{2} a^3 b^2 x^4+\frac {10}{7} a^2 b^3 x^7+\frac {1}{2} a b^4 x^{10}+\frac {b^5 x^{13}}{13} \]

[Out]

-1/2*a^5/x^2+5*a^4*b*x+5/2*a^3*b^2*x^4+10/7*a^2*b^3*x^7+1/2*a*b^4*x^10+1/13*b^5*x^13

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} \[ \int \frac {\left (a+b x^3\right )^5}{x^3} \, dx=-\frac {a^5}{2 x^2}+5 a^4 b x+\frac {5}{2} a^3 b^2 x^4+\frac {10}{7} a^2 b^3 x^7+\frac {1}{2} a b^4 x^{10}+\frac {b^5 x^{13}}{13} \]

[In]

Int[(a + b*x^3)^5/x^3,x]

[Out]

-1/2*a^5/x^2 + 5*a^4*b*x + (5*a^3*b^2*x^4)/2 + (10*a^2*b^3*x^7)/7 + (a*b^4*x^10)/2 + (b^5*x^13)/13

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 (5 a^4 b+\frac {a^5}{x^3}+10 a^3 b^2 x^3+10 a^2 b^3 x^6+5 a b^4 x^9+b^5 x^{12}\right ) \, dx \\ & = -\frac {a^5}{2 x^2}+5 a^4 b x+\frac {5}{2} a^3 b^2 x^4+\frac {10}{7} a^2 b^3 x^7+\frac {1}{2} a b^4 x^{10}+\frac {b^5 x^{13}}{13} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^3\right )^5}{x^3} \, dx=-\frac {a^5}{2 x^2}+5 a^4 b x+\frac {5}{2} a^3 b^2 x^4+\frac {10}{7} a^2 b^3 x^7+\frac {1}{2} a b^4 x^{10}+\frac {b^5 x^{13}}{13} \]

[In]

Integrate[(a + b*x^3)^5/x^3,x]

[Out]

-1/2*a^5/x^2 + 5*a^4*b*x + (5*a^3*b^2*x^4)/2 + (10*a^2*b^3*x^7)/7 + (a*b^4*x^10)/2 + (b^5*x^13)/13

Maple [A] (verified)

Time = 3.71 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {a^{5}}{2 x^{2}}+5 a^{4} b x +\frac {5 a^{3} b^{2} x^{4}}{2}+\frac {10 a^{2} b^{3} x^{7}}{7}+\frac {a \,b^{4} x^{10}}{2}+\frac {b^{5} x^{13}}{13}\)	\(56\)
risch	\(-\frac {a^{5}}{2 x^{2}}+5 a^{4} b x +\frac {5 a^{3} b^{2} x^{4}}{2}+\frac {10 a^{2} b^{3} x^{7}}{7}+\frac {a \,b^{4} x^{10}}{2}+\frac {b^{5} x^{13}}{13}\)	\(56\)
norman	\(\frac {-\frac {1}{2} a^{5}+5 a^{4} b \,x^{3}+\frac {5}{2} a^{3} b^{2} x^{6}+\frac {10}{7} a^{2} b^{3} x^{9}+\frac {1}{2} a \,b^{4} x^{12}+\frac {1}{13} b^{5} x^{15}}{x^{2}}\)	\(59\)
gosper	\(-\frac {-14 b^{5} x^{15}-91 a \,b^{4} x^{12}-260 a^{2} b^{3} x^{9}-455 a^{3} b^{2} x^{6}-910 a^{4} b \,x^{3}+91 a^{5}}{182 x^{2}}\)	\(60\)
parallelrisch	\(\frac {14 b^{5} x^{15}+91 a \,b^{4} x^{12}+260 a^{2} b^{3} x^{9}+455 a^{3} b^{2} x^{6}+910 a^{4} b \,x^{3}-91 a^{5}}{182 x^{2}}\)	\(60\)

[In]

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

[Out]

-1/2*a^5/x^2+5*a^4*b*x+5/2*a^3*b^2*x^4+10/7*a^2*b^3*x^7+1/2*a*b^4*x^10+1/13*b^5*x^13

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b x^3\right )^5}{x^3} \, dx=\frac {14 \, b^{5} x^{15} + 91 \, a b^{4} x^{12} + 260 \, a^{2} b^{3} x^{9} + 455 \, a^{3} b^{2} x^{6} + 910 \, a^{4} b x^{3} - 91 \, a^{5}}{182 \, x^{2}} \]

[In]

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

[Out]

1/182*(14*b^5*x^15 + 91*a*b^4*x^12 + 260*a^2*b^3*x^9 + 455*a^3*b^2*x^6 + 910*a^4*b*x^3 - 91*a^5)/x^2

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^3\right )^5}{x^3} \, dx=- \frac {a^{5}}{2 x^{2}} + 5 a^{4} b x + \frac {5 a^{3} b^{2} x^{4}}{2} + \frac {10 a^{2} b^{3} x^{7}}{7} + \frac {a b^{4} x^{10}}{2} + \frac {b^{5} x^{13}}{13} \]

[In]

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

[Out]

-a**5/(2*x**2) + 5*a**4*b*x + 5*a**3*b**2*x**4/2 + 10*a**2*b**3*x**7/7 + a*b**4*x**10/2 + b**5*x**13/13

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^3\right )^5}{x^3} \, dx=\frac {1}{13} \, b^{5} x^{13} + \frac {1}{2} \, a b^{4} x^{10} + \frac {10}{7} \, a^{2} b^{3} x^{7} + \frac {5}{2} \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x - \frac {a^{5}}{2 \, x^{2}} \]

[In]

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

[Out]

1/13*b^5*x^13 + 1/2*a*b^4*x^10 + 10/7*a^2*b^3*x^7 + 5/2*a^3*b^2*x^4 + 5*a^4*b*x - 1/2*a^5/x^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^3\right )^5}{x^3} \, dx=\frac {1}{13} \, b^{5} x^{13} + \frac {1}{2} \, a b^{4} x^{10} + \frac {10}{7} \, a^{2} b^{3} x^{7} + \frac {5}{2} \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x - \frac {a^{5}}{2 \, x^{2}} \]

[In]

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

[Out]

1/13*b^5*x^13 + 1/2*a*b^4*x^10 + 10/7*a^2*b^3*x^7 + 5/2*a^3*b^2*x^4 + 5*a^4*b*x - 1/2*a^5/x^2

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a+b x^3\right )^5}{x^3} \, dx=\frac {b^5\,x^{13}}{13}-\frac {a^5}{2\,x^2}+\frac {a\,b^4\,x^{10}}{2}+\frac {5\,a^3\,b^2\,x^4}{2}+\frac {10\,a^2\,b^3\,x^7}{7}+5\,a^4\,b\,x \]

[In]

int((a + b*x^3)^5/x^3,x)

[Out]

(b^5*x^13)/13 - a^5/(2*x^2) + (a*b^4*x^10)/2 + (5*a^3*b^2*x^4)/2 + (10*a^2*b^3*x^7)/7 + 5*a^4*b*x