\(\int (\frac {a}{x}+b x)^4 \, dx\) [346]

Optimal result

Integrand size = 11, antiderivative size = 50 \[ \int \left (\frac {a}{x}+b x\right )^4 \, dx=-\frac {a^4}{3 x^3}-\frac {4 a^3 b}{x}+6 a^2 b^2 x+\frac {4}{3} a b^3 x^3+\frac {b^4 x^5}{5} \]

[Out]

-1/3*a^4/x^3-4*a^3*b/x+6*a^2*b^2*x+4/3*a*b^3*x^3+1/5*b^4*x^5

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1607, 276} \[ \int \left (\frac {a}{x}+b x\right )^4 \, dx=-\frac {a^4}{3 x^3}-\frac {4 a^3 b}{x}+6 a^2 b^2 x+\frac {4}{3} a b^3 x^3+\frac {b^4 x^5}{5} \]

[In]

Int[(a/x + b*x)^4,x]

[Out]

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

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]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int \left (\frac {a}{x}+b x\right )^4 \, dx=-\frac {a^4}{3 x^3}-\frac {4 a^3 b}{x}+6 a^2 b^2 x+\frac {4}{3} a b^3 x^3+\frac {b^4 x^5}{5} \]

[In]

Integrate[(a/x + b*x)^4,x]

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.90

method	result	size
default	\(-\frac {a^{4}}{3 x^{3}}-\frac {4 a^{3} b}{x}+6 a^{2} b^{2} x +\frac {4 a \,b^{3} x^{3}}{3}+\frac {b^{4} x^{5}}{5}\)	\(45\)
risch	\(\frac {b^{4} x^{5}}{5}+\frac {4 a \,b^{3} x^{3}}{3}+6 a^{2} b^{2} x +\frac {-4 a^{3} b \,x^{2}-\frac {1}{3} a^{4}}{x^{3}}\)	\(47\)
norman	\(\frac {\frac {1}{5} x^{8} b^{4}+\frac {4}{3} a \,b^{3} x^{6}+6 a^{2} x^{4} b^{2}-4 a^{3} b \,x^{2}-\frac {1}{3} a^{4}}{x^{3}}\)	\(48\)
gosper	\(-\frac {-3 x^{8} b^{4}-20 a \,b^{3} x^{6}-90 a^{2} x^{4} b^{2}+60 a^{3} b \,x^{2}+5 a^{4}}{15 x^{3}}\)	\(49\)
parallelrisch	\(\frac {3 x^{8} b^{4}+20 a \,b^{3} x^{6}+90 a^{2} x^{4} b^{2}-60 a^{3} b \,x^{2}-5 a^{4}}{15 x^{3}}\)	\(49\)

[In]

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

[Out]

-1/3*a^4/x^3-4*a^3*b/x+6*a^2*b^2*x+4/3*a*b^3*x^3+1/5*b^4*x^5

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.96 \[ \int \left (\frac {a}{x}+b x\right )^4 \, dx=\frac {3 \, b^{4} x^{8} + 20 \, a b^{3} x^{6} + 90 \, a^{2} b^{2} x^{4} - 60 \, a^{3} b x^{2} - 5 \, a^{4}}{15 \, x^{3}} \]

[In]

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

[Out]

1/15*(3*b^4*x^8 + 20*a*b^3*x^6 + 90*a^2*b^2*x^4 - 60*a^3*b*x^2 - 5*a^4)/x^3

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.98 \[ \int \left (\frac {a}{x}+b x\right )^4 \, dx=6 a^{2} b^{2} x + \frac {4 a b^{3} x^{3}}{3} + \frac {b^{4} x^{5}}{5} + \frac {- a^{4} - 12 a^{3} b x^{2}}{3 x^{3}} \]

[In]

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

[Out]

6*a**2*b**2*x + 4*a*b**3*x**3/3 + b**4*x**5/5 + (-a**4 - 12*a**3*b*x**2)/(3*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88 \[ \int \left (\frac {a}{x}+b x\right )^4 \, dx=\frac {1}{5} \, b^{4} x^{5} + \frac {4}{3} \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x - \frac {4 \, a^{3} b}{x} - \frac {a^{4}}{3 \, x^{3}} \]

[In]

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

[Out]

1/5*b^4*x^5 + 4/3*a*b^3*x^3 + 6*a^2*b^2*x - 4*a^3*b/x - 1/3*a^4/x^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.90 \[ \int \left (\frac {a}{x}+b x\right )^4 \, dx=\frac {1}{5} \, b^{4} x^{5} + \frac {4}{3} \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x - \frac {12 \, a^{3} b x^{2} + a^{4}}{3 \, x^{3}} \]

[In]

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

[Out]

1/5*b^4*x^5 + 4/3*a*b^3*x^3 + 6*a^2*b^2*x - 1/3*(12*a^3*b*x^2 + a^4)/x^3

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.94 \[ \int \left (\frac {a}{x}+b x\right )^4 \, dx=\frac {b^4\,x^5}{5}-\frac {\frac {a^4}{3}+4\,b\,a^3\,x^2}{x^3}+6\,a^2\,b^2\,x+\frac {4\,a\,b^3\,x^3}{3} \]

[In]

int((b*x + a/x)^4,x)

[Out]

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