\(\int \frac {(a+b x) (a c-b c x)^3}{x^3} \, dx\) [7]

Optimal result

Integrand size = 20, antiderivative size = 18 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^3} \, dx=-\frac {c^3 (a-b x)^4}{2 x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {75} \[ \int \frac {(a+b x) (a c-b c x)^3}{x^3} \, dx=-\frac {c^3 (a-b x)^4}{2 x^2} \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x^3,x]

[Out]

-1/2*(c^3*(a - b*x)^4)/x^2

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {c^3 (a-b x)^4}{2 x^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(41\) vs. \(2(18)=36\).

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.28 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^3} \, dx=c^3 \left (-\frac {a^4}{2 x^2}+\frac {2 a^3 b}{x}+2 a b^3 x-\frac {b^4 x^2}{2}\right ) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^3,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(35\) vs. \(2(16)=32\).

Time = 0.37 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.00

method	result	size
gosper	\(-\frac {c^{3} \left (b^{4} x^{4}-4 a \,b^{3} x^{3}-4 a^{3} b x +a^{4}\right )}{2 x^{2}}\)	\(36\)
default	\(c^{3} \left (-\frac {b^{4} x^{2}}{2}+2 a \,b^{3} x +\frac {2 a^{3} b}{x}-\frac {a^{4}}{2 x^{2}}\right )\)	\(38\)
risch	\(-\frac {b^{4} c^{3} x^{2}}{2}+2 b^{3} c^{3} a x +\frac {2 a^{3} b \,c^{3} x -\frac {1}{2} a^{4} c^{3}}{x^{2}}\)	\(46\)
parallelrisch	\(-\frac {b^{4} c^{3} x^{4}-4 a \,b^{3} c^{3} x^{3}-4 a^{3} b \,c^{3} x +a^{4} c^{3}}{2 x^{2}}\)	\(46\)
norman	\(\frac {-\frac {1}{2} a^{4} c^{3}-\frac {1}{2} b^{4} c^{3} x^{4}+2 a \,b^{3} c^{3} x^{3}+2 a^{3} b \,c^{3} x}{x^{2}}\)	\(47\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (17) = 34\).

Time = 0.21 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.50 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^3} \, dx=-\frac {b^{4} c^{3} x^{4} - 4 \, a b^{3} c^{3} x^{3} - 4 \, a^{3} b c^{3} x + a^{4} c^{3}}{2 \, x^{2}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (15) = 30\).

Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.56 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^3} \, dx=2 a b^{3} c^{3} x - \frac {b^{4} c^{3} x^{2}}{2} - \frac {a^{4} c^{3} - 4 a^{3} b c^{3} x}{2 x^{2}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**3/x**3,x)

[Out]

2*a*b**3*c**3*x - b**4*c**3*x**2/2 - (a**4*c**3 - 4*a**3*b*c**3*x)/(2*x**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (17) = 34\).

Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.56 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^3} \, dx=-\frac {1}{2} \, b^{4} c^{3} x^{2} + 2 \, a b^{3} c^{3} x + \frac {4 \, a^{3} b c^{3} x - a^{4} c^{3}}{2 \, x^{2}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (17) = 34\).

Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.56 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^3} \, dx=-\frac {1}{2} \, b^{4} c^{3} x^{2} + 2 \, a b^{3} c^{3} x + \frac {4 \, a^{3} b c^{3} x - a^{4} c^{3}}{2 \, x^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^3} \, dx=-\frac {c^3\,\left (a^4-4\,a^3\,b\,x-4\,a\,b^3\,x^3+b^4\,x^4\right )}{2\,x^2} \]

[In]

int(((a*c - b*c*x)^3*(a + b*x))/x^3,x)

[Out]

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