\(\int \frac {(a+b x^4)^3}{x^3} \, dx\) [641]

Optimal result

Integrand size = 13, antiderivative size = 43 \[ \int \frac {\left (a+b x^4\right )^3}{x^3} \, dx=-\frac {a^3}{2 x^2}+\frac {3}{2} a^2 b x^2+\frac {1}{2} a b^2 x^6+\frac {b^3 x^{10}}{10} \]

[Out]

-1/2*a^3/x^2+3/2*a^2*b*x^2+1/2*a*b^2*x^6+1/10*b^3*x^10

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, 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} \[ \int \frac {\left (a+b x^4\right )^3}{x^3} \, dx=-\frac {a^3}{2 x^2}+\frac {3}{2} a^2 b x^2+\frac {1}{2} a b^2 x^6+\frac {b^3 x^{10}}{10} \]

[In]

Int[(a + b*x^4)^3/x^3,x]

[Out]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b x^4\right )^3}{x^3} \, dx=-\frac {a^3}{2 x^2}+\frac {3}{2} a^2 b x^2+\frac {1}{2} a b^2 x^6+\frac {b^3 x^{10}}{10} \]

[In]

Integrate[(a + b*x^4)^3/x^3,x]

[Out]

-1/2*a^3/x^2 + (3*a^2*b*x^2)/2 + (a*b^2*x^6)/2 + (b^3*x^10)/10

Maple [A] (verified)

Time = 3.86 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {a^{3}}{2 x^{2}}+\frac {3 a^{2} b \,x^{2}}{2}+\frac {a \,b^{2} x^{6}}{2}+\frac {b^{3} x^{10}}{10}\)	\(36\)
risch	\(-\frac {a^{3}}{2 x^{2}}+\frac {3 a^{2} b \,x^{2}}{2}+\frac {a \,b^{2} x^{6}}{2}+\frac {b^{3} x^{10}}{10}\)	\(36\)
norman	\(\frac {\frac {1}{10} b^{3} x^{12}+\frac {1}{2} a \,b^{2} x^{8}+\frac {3}{2} a^{2} b \,x^{4}-\frac {1}{2} a^{3}}{x^{2}}\)	\(37\)
parallelrisch	\(\frac {b^{3} x^{12}+5 a \,b^{2} x^{8}+15 a^{2} b \,x^{4}-5 a^{3}}{10 x^{2}}\)	\(37\)
gosper	\(-\frac {-b^{3} x^{12}-5 a \,b^{2} x^{8}-15 a^{2} b \,x^{4}+5 a^{3}}{10 x^{2}}\)	\(38\)

[In]

int((b*x^4+a)^3/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*a^3/x^2+3/2*a^2*b*x^2+1/2*a*b^2*x^6+1/10*b^3*x^10

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^4\right )^3}{x^3} \, dx=\frac {b^{3} x^{12} + 5 \, a b^{2} x^{8} + 15 \, a^{2} b x^{4} - 5 \, a^{3}}{10 \, x^{2}} \]

[In]

integrate((b*x^4+a)^3/x^3,x, algorithm="fricas")

[Out]

1/10*(b^3*x^12 + 5*a*b^2*x^8 + 15*a^2*b*x^4 - 5*a^3)/x^2

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+b x^4\right )^3}{x^3} \, dx=- \frac {a^{3}}{2 x^{2}} + \frac {3 a^{2} b x^{2}}{2} + \frac {a b^{2} x^{6}}{2} + \frac {b^{3} x^{10}}{10} \]

[In]

integrate((b*x**4+a)**3/x**3,x)

[Out]

-a**3/(2*x**2) + 3*a**2*b*x**2/2 + a*b**2*x**6/2 + b**3*x**10/10

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x^4\right )^3}{x^3} \, dx=\frac {1}{10} \, b^{3} x^{10} + \frac {1}{2} \, a b^{2} x^{6} + \frac {3}{2} \, a^{2} b x^{2} - \frac {a^{3}}{2 \, x^{2}} \]

[In]

integrate((b*x^4+a)^3/x^3,x, algorithm="maxima")

[Out]

1/10*b^3*x^10 + 1/2*a*b^2*x^6 + 3/2*a^2*b*x^2 - 1/2*a^3/x^2

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x^4\right )^3}{x^3} \, dx=\frac {1}{10} \, b^{3} x^{10} + \frac {1}{2} \, a b^{2} x^{6} + \frac {3}{2} \, a^{2} b x^{2} - \frac {a^{3}}{2 \, x^{2}} \]

[In]

integrate((b*x^4+a)^3/x^3,x, algorithm="giac")

[Out]

1/10*b^3*x^10 + 1/2*a*b^2*x^6 + 3/2*a^2*b*x^2 - 1/2*a^3/x^2

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x^4\right )^3}{x^3} \, dx=\frac {b^3\,x^{10}}{10}-\frac {a^3}{2\,x^2}+\frac {3\,a^2\,b\,x^2}{2}+\frac {a\,b^2\,x^6}{2} \]

[In]

int((a + b*x^4)^3/x^3,x)

[Out]

(b^3*x^10)/10 - a^3/(2*x^2) + (3*a^2*b*x^2)/2 + (a*b^2*x^6)/2