\(\int \frac {(a+b x) (a c-b c x)^5}{x^4} \, dx\) [35]

Optimal result

Integrand size = 20, antiderivative size = 18 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^4} \, dx=-\frac {c^5 (a-b x)^6}{3 x^3} \]

[Out]

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

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} \[ \int \frac {(a+b x) (a c-b c x)^5}{x^4} \, dx=-\frac {c^5 (a-b x)^6}{3 x^3} \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^4,x]

[Out]

-1/3*(c^5*(a - b*x)^6)/x^3

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^5 (a-b x)^6}{3 x^3} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(63\) vs. \(2(18)=36\).

Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.50 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^4} \, dx=c^5 \left (-\frac {a^6}{3 x^3}+\frac {2 a^5 b}{x^2}-\frac {5 a^4 b^2}{x}-5 a^2 b^4 x+2 a b^5 x^2-\frac {b^6 x^3}{3}\right ) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^4,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(16)=32\).

Time = 0.85 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.22

method	result	size
gosper	\(-\frac {c^{5} \left (b^{6} x^{6}-6 a \,x^{5} b^{5}+15 a^{2} x^{4} b^{4}+15 a^{4} x^{2} b^{2}-6 a^{5} x b +a^{6}\right )}{3 x^{3}}\)	\(58\)
default	\(c^{5} \left (-\frac {b^{6} x^{3}}{3}+2 a \,b^{5} x^{2}-5 a^{2} b^{4} x -\frac {a^{6}}{3 x^{3}}-\frac {5 a^{4} b^{2}}{x}+\frac {2 a^{5} b}{x^{2}}\right )\)	\(60\)
risch	\(-\frac {b^{6} c^{5} x^{3}}{3}+2 b^{5} c^{5} a \,x^{2}-5 b^{4} c^{5} a^{2} x +\frac {-5 a^{4} b^{2} c^{5} x^{2}+2 a^{5} b \,c^{5} x -\frac {1}{3} a^{6} c^{5}}{x^{3}}\)	\(74\)
parallelrisch	\(-\frac {b^{6} c^{5} x^{6}-6 a \,b^{5} c^{5} x^{5}+15 a^{2} b^{4} c^{5} x^{4}+15 a^{4} b^{2} c^{5} x^{2}-6 a^{5} b \,c^{5} x +a^{6} c^{5}}{3 x^{3}}\)	\(74\)
norman	\(\frac {-\frac {1}{3} a^{6} c^{5}-\frac {1}{3} b^{6} c^{5} x^{6}+2 a \,b^{5} c^{5} x^{5}-5 a^{2} b^{4} c^{5} x^{4}-5 a^{4} b^{2} c^{5} x^{2}+2 a^{5} b \,c^{5} x}{x^{3}}\)	\(75\)

[In]

int((b*x+a)*(-b*c*x+a*c)^5/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*c^5*(b^6*x^6-6*a*b^5*x^5+15*a^2*b^4*x^4+15*a^4*b^2*x^2-6*a^5*b*x+a^6)/x^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (17) = 34\).

Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 4.06 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^4} \, dx=-\frac {b^{6} c^{5} x^{6} - 6 \, a b^{5} c^{5} x^{5} + 15 \, a^{2} b^{4} c^{5} x^{4} + 15 \, a^{4} b^{2} c^{5} x^{2} - 6 \, a^{5} b c^{5} x + a^{6} c^{5}}{3 \, x^{3}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^4,x, algorithm="fricas")

[Out]

-1/3*(b^6*c^5*x^6 - 6*a*b^5*c^5*x^5 + 15*a^2*b^4*c^5*x^4 + 15*a^4*b^2*c^5*x^2 - 6*a^5*b*c^5*x + a^6*c^5)/x^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (15) = 30\).

Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 4.22 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^4} \, dx=- 5 a^{2} b^{4} c^{5} x + 2 a b^{5} c^{5} x^{2} - \frac {b^{6} c^{5} x^{3}}{3} - \frac {a^{6} c^{5} - 6 a^{5} b c^{5} x + 15 a^{4} b^{2} c^{5} x^{2}}{3 x^{3}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**4,x)

[Out]

-5*a**2*b**4*c**5*x + 2*a*b**5*c**5*x**2 - b**6*c**5*x**3/3 - (a**6*c**5 - 6*a**5*b*c**5*x + 15*a**4*b**2*c**5
*x**2)/(3*x**3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (17) = 34\).

Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 4.06 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^4} \, dx=-\frac {1}{3} \, b^{6} c^{5} x^{3} + 2 \, a b^{5} c^{5} x^{2} - 5 \, a^{2} b^{4} c^{5} x - \frac {15 \, a^{4} b^{2} c^{5} x^{2} - 6 \, a^{5} b c^{5} x + a^{6} c^{5}}{3 \, x^{3}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^4,x, algorithm="maxima")

[Out]

-1/3*b^6*c^5*x^3 + 2*a*b^5*c^5*x^2 - 5*a^2*b^4*c^5*x - 1/3*(15*a^4*b^2*c^5*x^2 - 6*a^5*b*c^5*x + a^6*c^5)/x^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (17) = 34\).

Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 4.06 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^4} \, dx=-\frac {1}{3} \, b^{6} c^{5} x^{3} + 2 \, a b^{5} c^{5} x^{2} - 5 \, a^{2} b^{4} c^{5} x - \frac {15 \, a^{4} b^{2} c^{5} x^{2} - 6 \, a^{5} b c^{5} x + a^{6} c^{5}}{3 \, x^{3}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^4,x, algorithm="giac")

[Out]

-1/3*b^6*c^5*x^3 + 2*a*b^5*c^5*x^2 - 5*a^2*b^4*c^5*x - 1/3*(15*a^4*b^2*c^5*x^2 - 6*a^5*b*c^5*x + a^6*c^5)/x^3

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 4.11 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^4} \, dx=2\,a\,b^5\,c^5\,x^2-\frac {b^6\,c^5\,x^3}{3}-5\,a^2\,b^4\,c^5\,x-\frac {\frac {a^6\,c^5}{3}-2\,a^5\,b\,c^5\,x+5\,a^4\,b^2\,c^5\,x^2}{x^3} \]

[In]

int(((a*c - b*c*x)^5*(a + b*x))/x^4,x)

[Out]

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