\(\int \frac {(a+b x) (a c-b c x)^3}{x^2} \, dx\) [6]

Optimal result

Integrand size = 20, antiderivative size = 47 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^2} \, dx=-\frac {a^4 c^3}{x}+a b^3 c^3 x^2-\frac {1}{3} b^4 c^3 x^3-2 a^3 b c^3 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {76} \[ \int \frac {(a+b x) (a c-b c x)^3}{x^2} \, dx=-\frac {a^4 c^3}{x}-2 a^3 b c^3 \log (x)+a b^3 c^3 x^2-\frac {1}{3} b^4 c^3 x^3 \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x^2,x]

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^4 c^3}{x^2}-\frac {2 a^3 b c^3}{x}+2 a b^3 c^3 x-b^4 c^3 x^2\right ) \, dx \\ & = -\frac {a^4 c^3}{x}+a b^3 c^3 x^2-\frac {1}{3} b^4 c^3 x^3-2 a^3 b c^3 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^2} \, dx=c^3 \left (-\frac {a^4}{x}+a b^3 x^2-\frac {b^4 x^3}{3}-2 a^3 b \log (x)\right ) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^3)/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81

method	result	size
default	\(c^{3} \left (-\frac {b^{4} x^{3}}{3}+a \,b^{3} x^{2}-2 a^{3} b \ln \left (x \right )-\frac {a^{4}}{x}\right )\)	\(38\)
risch	\(-\frac {a^{4} c^{3}}{x}+a \,b^{3} c^{3} x^{2}-\frac {b^{4} c^{3} x^{3}}{3}-2 a^{3} b \,c^{3} \ln \left (x \right )\)	\(46\)
norman	\(\frac {a \,b^{3} c^{3} x^{3}-a^{4} c^{3}-\frac {1}{3} b^{4} c^{3} x^{4}}{x}-2 a^{3} b \,c^{3} \ln \left (x \right )\)	\(48\)
parallelrisch	\(-\frac {b^{4} c^{3} x^{4}-3 a \,b^{3} c^{3} x^{3}+6 a^{3} c^{3} b \ln \left (x \right ) x +3 a^{4} c^{3}}{3 x}\)	\(49\)

[In]

int((b*x+a)*(-b*c*x+a*c)^3/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^2} \, dx=-\frac {b^{4} c^{3} x^{4} - 3 \, a b^{3} c^{3} x^{3} + 6 \, a^{3} b c^{3} x \log \left (x\right ) + 3 \, a^{4} c^{3}}{3 \, x} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^2,x, algorithm="fricas")

[Out]

-1/3*(b^4*c^3*x^4 - 3*a*b^3*c^3*x^3 + 6*a^3*b*c^3*x*log(x) + 3*a^4*c^3)/x

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^2} \, dx=- \frac {a^{4} c^{3}}{x} - 2 a^{3} b c^{3} \log {\left (x \right )} + a b^{3} c^{3} x^{2} - \frac {b^{4} c^{3} x^{3}}{3} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**3/x**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^2} \, dx=-\frac {1}{3} \, b^{4} c^{3} x^{3} + a b^{3} c^{3} x^{2} - 2 \, a^{3} b c^{3} \log \left (x\right ) - \frac {a^{4} c^{3}}{x} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.98 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^2} \, dx=-\frac {1}{3} \, b^{4} c^{3} x^{3} + a b^{3} c^{3} x^{2} - 2 \, a^{3} b c^{3} \log \left ({\left | x \right |}\right ) - \frac {a^{4} c^{3}}{x} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^3/x^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x) (a c-b c x)^3}{x^2} \, dx=a\,b^3\,c^3\,x^2-\frac {b^4\,c^3\,x^3}{3}-\frac {a^4\,c^3}{x}-2\,a^3\,b\,c^3\,\ln \left (x\right ) \]

[In]

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

[Out]

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