\(\int \frac {(a+b x) (a c-b c x)^4}{x^7} \, dx\) [24]

Optimal result

Integrand size = 20, antiderivative size = 41 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^7} \, dx=-\frac {c^4 (a-b x)^5}{6 x^6}-\frac {7 b c^4 (a-b x)^5}{30 a x^5} \]

[Out]

-1/6*c^4*(-b*x+a)^5/x^6-7/30*b*c^4*(-b*x+a)^5/a/x^5

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {79, 37} \[ \int \frac {(a+b x) (a c-b c x)^4}{x^7} \, dx=-\frac {c^4 (a-b x)^5}{6 x^6}-\frac {7 b c^4 (a-b x)^5}{30 a x^5} \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^4)/x^7,x]

[Out]

-1/6*(c^4*(a - b*x)^5)/x^6 - (7*b*c^4*(a - b*x)^5)/(30*a*x^5)

Rule 37

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

Rule 79

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

Rubi steps \begin{align*} \text {integral}& = -\frac {c^4 (a-b x)^5}{6 x^6}+\frac {1}{6} (7 b) \int \frac {(a c-b c x)^4}{x^6} \, dx \\ & = -\frac {c^4 (a-b x)^5}{6 x^6}-\frac {7 b c^4 (a-b x)^5}{30 a x^5} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(41)=82\).

Time = 0.01 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.07 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^7} \, dx=-\frac {a^5 c^4}{6 x^6}+\frac {3 a^4 b c^4}{5 x^5}-\frac {a^3 b^2 c^4}{2 x^4}-\frac {2 a^2 b^3 c^4}{3 x^3}+\frac {3 a b^4 c^4}{2 x^2}-\frac {b^5 c^4}{x} \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^7,x]

[Out]

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

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.49

method	result	size
gosper	\(-\frac {c^{4} \left (30 b^{5} x^{5}-45 a \,b^{4} x^{4}+20 a^{2} b^{3} x^{3}+15 a^{3} b^{2} x^{2}-18 a^{4} b x +5 a^{5}\right )}{30 x^{6}}\)	\(61\)
default	\(c^{4} \left (-\frac {a^{5}}{6 x^{6}}-\frac {2 a^{2} b^{3}}{3 x^{3}}-\frac {b^{5}}{x}+\frac {3 a \,b^{4}}{2 x^{2}}-\frac {a^{3} b^{2}}{2 x^{4}}+\frac {3 a^{4} b}{5 x^{5}}\right )\)	\(62\)
norman	\(\frac {-\frac {1}{6} a^{5} c^{4}-b^{5} c^{4} x^{5}+\frac {3}{2} a \,b^{4} c^{4} x^{4}-\frac {2}{3} a^{2} b^{3} c^{4} x^{3}-\frac {1}{2} a^{3} b^{2} c^{4} x^{2}+\frac {3}{5} a^{4} b \,c^{4} x}{x^{6}}\)	\(75\)
risch	\(\frac {-\frac {1}{6} a^{5} c^{4}-b^{5} c^{4} x^{5}+\frac {3}{2} a \,b^{4} c^{4} x^{4}-\frac {2}{3} a^{2} b^{3} c^{4} x^{3}-\frac {1}{2} a^{3} b^{2} c^{4} x^{2}+\frac {3}{5} a^{4} b \,c^{4} x}{x^{6}}\)	\(75\)
parallelrisch	\(\frac {-30 b^{5} c^{4} x^{5}+45 a \,b^{4} c^{4} x^{4}-20 a^{2} b^{3} c^{4} x^{3}-15 a^{3} b^{2} c^{4} x^{2}+18 a^{4} b \,c^{4} x -5 a^{5} c^{4}}{30 x^{6}}\)	\(76\)

[In]

int((b*x+a)*(-b*c*x+a*c)^4/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/30*c^4*(30*b^5*x^5-45*a*b^4*x^4+20*a^2*b^3*x^3+15*a^3*b^2*x^2-18*a^4*b*x+5*a^5)/x^6

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^7} \, dx=-\frac {30 \, b^{5} c^{4} x^{5} - 45 \, a b^{4} c^{4} x^{4} + 20 \, a^{2} b^{3} c^{4} x^{3} + 15 \, a^{3} b^{2} c^{4} x^{2} - 18 \, a^{4} b c^{4} x + 5 \, a^{5} c^{4}}{30 \, x^{6}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^7,x, algorithm="fricas")

[Out]

-1/30*(30*b^5*c^4*x^5 - 45*a*b^4*c^4*x^4 + 20*a^2*b^3*c^4*x^3 + 15*a^3*b^2*c^4*x^2 - 18*a^4*b*c^4*x + 5*a^5*c^
4)/x^6

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (36) = 72\).

Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.95 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^7} \, dx=\frac {- 5 a^{5} c^{4} + 18 a^{4} b c^{4} x - 15 a^{3} b^{2} c^{4} x^{2} - 20 a^{2} b^{3} c^{4} x^{3} + 45 a b^{4} c^{4} x^{4} - 30 b^{5} c^{4} x^{5}}{30 x^{6}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**4/x**7,x)

[Out]

(-5*a**5*c**4 + 18*a**4*b*c**4*x - 15*a**3*b**2*c**4*x**2 - 20*a**2*b**3*c**4*x**3 + 45*a*b**4*c**4*x**4 - 30*
b**5*c**4*x**5)/(30*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^7} \, dx=-\frac {30 \, b^{5} c^{4} x^{5} - 45 \, a b^{4} c^{4} x^{4} + 20 \, a^{2} b^{3} c^{4} x^{3} + 15 \, a^{3} b^{2} c^{4} x^{2} - 18 \, a^{4} b c^{4} x + 5 \, a^{5} c^{4}}{30 \, x^{6}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^7,x, algorithm="maxima")

[Out]

-1/30*(30*b^5*c^4*x^5 - 45*a*b^4*c^4*x^4 + 20*a^2*b^3*c^4*x^3 + 15*a^3*b^2*c^4*x^2 - 18*a^4*b*c^4*x + 5*a^5*c^
4)/x^6

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.83 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^7} \, dx=-\frac {30 \, b^{5} c^{4} x^{5} - 45 \, a b^{4} c^{4} x^{4} + 20 \, a^{2} b^{3} c^{4} x^{3} + 15 \, a^{3} b^{2} c^{4} x^{2} - 18 \, a^{4} b c^{4} x + 5 \, a^{5} c^{4}}{30 \, x^{6}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^7,x, algorithm="giac")

[Out]

-1/30*(30*b^5*c^4*x^5 - 45*a*b^4*c^4*x^4 + 20*a^2*b^3*c^4*x^3 + 15*a^3*b^2*c^4*x^2 - 18*a^4*b*c^4*x + 5*a^5*c^
4)/x^6

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.80 \[ \int \frac {(a+b x) (a c-b c x)^4}{x^7} \, dx=-\frac {\frac {a^5\,c^4}{6}-\frac {3\,a^4\,b\,c^4\,x}{5}+\frac {a^3\,b^2\,c^4\,x^2}{2}+\frac {2\,a^2\,b^3\,c^4\,x^3}{3}-\frac {3\,a\,b^4\,c^4\,x^4}{2}+b^5\,c^4\,x^5}{x^6} \]

[In]

int(((a*c - b*c*x)^4*(a + b*x))/x^7,x)

[Out]

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