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

Optimal result

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

[Out]

-2*a^3*b*c^3*x+2/3*a*b^3*c^3*x^3-1/4*b^4*c^3*x^4+a^4*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} \[ \int \frac {(a+b x) (a c-b c x)^3}{x} \, dx=a^4 c^3 \log (x)-2 a^3 b c^3 x+\frac {2}{3} a b^3 c^3 x^3-\frac {1}{4} b^4 c^3 x^4 \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^3)/x,x]

[Out]

-2*a^3*b*c^3*x + (2*a*b^3*c^3*x^3)/3 - (b^4*c^3*x^4)/4 + a^4*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 (-2 a^3 b c^3+\frac {a^4 c^3}{x}+2 a b^3 c^3 x^2-b^4 c^3 x^3\right ) \, dx \\ & = -2 a^3 b c^3 x+\frac {2}{3} a b^3 c^3 x^3-\frac {1}{4} b^4 c^3 x^4+a^4 c^3 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x) (a c-b c x)^3}{x} \, dx=c^3 \left (\frac {1}{12} \left (19 a^4-24 a^3 b x+8 a b^3 x^3-3 b^4 x^4\right )+a^4 \log (-b x)\right ) \]

[In]

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

[Out]

c^3*((19*a^4 - 24*a^3*b*x + 8*a*b^3*x^3 - 3*b^4*x^4)/12 + a^4*Log[-(b*x)])

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77

method	result	size
default	\(c^{3} \left (-\frac {b^{4} x^{4}}{4}+\frac {2 a \,b^{3} x^{3}}{3}-2 a^{3} b x +a^{4} \ln \left (x \right )\right )\)	\(36\)
norman	\(-2 a^{3} b \,c^{3} x +\frac {2 a \,b^{3} c^{3} x^{3}}{3}-\frac {b^{4} c^{3} x^{4}}{4}+a^{4} c^{3} \ln \left (x \right )\)	\(44\)
risch	\(-2 a^{3} b \,c^{3} x +\frac {2 a \,b^{3} c^{3} x^{3}}{3}-\frac {b^{4} c^{3} x^{4}}{4}+a^{4} c^{3} \ln \left (x \right )\)	\(44\)
parallelrisch	\(-2 a^{3} b \,c^{3} x +\frac {2 a \,b^{3} c^{3} x^{3}}{3}-\frac {b^{4} c^{3} x^{4}}{4}+a^{4} c^{3} \ln \left (x \right )\)	\(44\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x) (a c-b c x)^3}{x} \, dx=-\frac {1}{4} \, b^{4} c^{3} x^{4} + \frac {2}{3} \, a b^{3} c^{3} x^{3} - 2 \, a^{3} b c^{3} x + a^{4} c^{3} \log \left (x\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x) (a c-b c x)^3}{x} \, dx=a^{4} c^{3} \log {\left (x \right )} - 2 a^{3} b c^{3} x + \frac {2 a b^{3} c^{3} x^{3}}{3} - \frac {b^{4} c^{3} x^{4}}{4} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x) (a c-b c x)^3}{x} \, dx=-\frac {1}{4} \, b^{4} c^{3} x^{4} + \frac {2}{3} \, a b^{3} c^{3} x^{3} - 2 \, a^{3} b c^{3} x + a^{4} c^{3} \log \left (x\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b x) (a c-b c x)^3}{x} \, dx=-\frac {1}{4} \, b^{4} c^{3} x^{4} + \frac {2}{3} \, a b^{3} c^{3} x^{3} - 2 \, a^{3} b c^{3} x + a^{4} c^{3} \log \left ({\left | x \right |}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x) (a c-b c x)^3}{x} \, dx=a^4\,c^3\,\ln \left (x\right )-\frac {b^4\,c^3\,x^4}{4}+\frac {2\,a\,b^3\,c^3\,x^3}{3}-2\,a^3\,b\,c^3\,x \]

[In]

int(((a*c - b*c*x)^3*(a + b*x))/x,x)

[Out]

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