3.1.25 \(\int \frac {(a+b x) (a c-b c x)^4}{x^8} \, dx\)

Optimal. Leaf size=84 \[ -\frac {a^5 c^4}{7 x^7}+\frac {a^4 b c^4}{2 x^6}-\frac {2 a^3 b^2 c^4}{5 x^5}-\frac {a^2 b^3 c^4}{2 x^4}+\frac {a b^4 c^4}{x^3}-\frac {b^5 c^4}{2 x^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {75} \begin {gather*} -\frac {2 a^3 b^2 c^4}{5 x^5}-\frac {a^2 b^3 c^4}{2 x^4}+\frac {a^4 b c^4}{2 x^6}-\frac {a^5 c^4}{7 x^7}+\frac {a b^4 c^4}{x^3}-\frac {b^5 c^4}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^4)/x^8,x]

[Out]

-(a^5*c^4)/(7*x^7) + (a^4*b*c^4)/(2*x^6) - (2*a^3*b^2*c^4)/(5*x^5) - (a^2*b^3*c^4)/(2*x^4) + (a*b^4*c^4)/x^3 -
(b^5*c^4)/(2*x^2)

Rule 75

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*} \int \frac {(a+b x) (a c-b c x)^4}{x^8} \, dx &=\int \left (\frac {a^5 c^4}{x^8}-\frac {3 a^4 b c^4}{x^7}+\frac {2 a^3 b^2 c^4}{x^6}+\frac {2 a^2 b^3 c^4}{x^5}-\frac {3 a b^4 c^4}{x^4}+\frac {b^5 c^4}{x^3}\right ) \, dx\\ &=-\frac {a^5 c^4}{7 x^7}+\frac {a^4 b c^4}{2 x^6}-\frac {2 a^3 b^2 c^4}{5 x^5}-\frac {a^2 b^3 c^4}{2 x^4}+\frac {a b^4 c^4}{x^3}-\frac {b^5 c^4}{2 x^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 84, normalized size = 1.00 \begin {gather*} -\frac {a^5 c^4}{7 x^7}+\frac {a^4 b c^4}{2 x^6}-\frac {2 a^3 b^2 c^4}{5 x^5}-\frac {a^2 b^3 c^4}{2 x^4}+\frac {a b^4 c^4}{x^3}-\frac {b^5 c^4}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^8,x]

[Out]

-1/7*(a^5*c^4)/x^7 + (a^4*b*c^4)/(2*x^6) - (2*a^3*b^2*c^4)/(5*x^5) - (a^2*b^3*c^4)/(2*x^4) + (a*b^4*c^4)/x^3 -
(b^5*c^4)/(2*x^2)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x) (a c-b c x)^4}{x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^4)/x^8,x]

[Out]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^4)/x^8, x]

________________________________________________________________________________________

fricas [A]  time = 1.32, size = 75, normalized size = 0.89 \begin {gather*} -\frac {35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^8,x, algorithm="fricas")

[Out]

-1/70*(35*b^5*c^4*x^5 - 70*a*b^4*c^4*x^4 + 35*a^2*b^3*c^4*x^3 + 28*a^3*b^2*c^4*x^2 - 35*a^4*b*c^4*x + 10*a^5*c
^4)/x^7

________________________________________________________________________________________

giac [A]  time = 0.93, size = 75, normalized size = 0.89 \begin {gather*} -\frac {35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^8,x, algorithm="giac")

[Out]

-1/70*(35*b^5*c^4*x^5 - 70*a*b^4*c^4*x^4 + 35*a^2*b^3*c^4*x^3 + 28*a^3*b^2*c^4*x^2 - 35*a^4*b*c^4*x + 10*a^5*c
^4)/x^7

________________________________________________________________________________________

maple [A]  time = 0.00, size = 61, normalized size = 0.73 \begin {gather*} \left (-\frac {b^{5}}{2 x^{2}}+\frac {a \,b^{4}}{x^{3}}-\frac {a^{2} b^{3}}{2 x^{4}}-\frac {2 a^{3} b^{2}}{5 x^{5}}+\frac {a^{4} b}{2 x^{6}}-\frac {a^{5}}{7 x^{7}}\right ) c^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^4/x^8,x)

[Out]

c^4*(1/2*a^4*b/x^6-2/5*a^3*b^2/x^5-1/2*b^5/x^2+a*b^4/x^3-1/7*a^5/x^7-1/2*a^2*b^3/x^4)

________________________________________________________________________________________

maxima [A]  time = 1.03, size = 75, normalized size = 0.89 \begin {gather*} -\frac {35 \, b^{5} c^{4} x^{5} - 70 \, a b^{4} c^{4} x^{4} + 35 \, a^{2} b^{3} c^{4} x^{3} + 28 \, a^{3} b^{2} c^{4} x^{2} - 35 \, a^{4} b c^{4} x + 10 \, a^{5} c^{4}}{70 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^8,x, algorithm="maxima")

[Out]

-1/70*(35*b^5*c^4*x^5 - 70*a*b^4*c^4*x^4 + 35*a^2*b^3*c^4*x^3 + 28*a^3*b^2*c^4*x^2 - 35*a^4*b*c^4*x + 10*a^5*c
^4)/x^7

________________________________________________________________________________________

mupad [B]  time = 0.29, size = 75, normalized size = 0.89 \begin {gather*} -\frac {\frac {a^5\,c^4}{7}-\frac {a^4\,b\,c^4\,x}{2}+\frac {2\,a^3\,b^2\,c^4\,x^2}{5}+\frac {a^2\,b^3\,c^4\,x^3}{2}-a\,b^4\,c^4\,x^4+\frac {b^5\,c^4\,x^5}{2}}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*c - b*c*x)^4*(a + b*x))/x^8,x)

[Out]

-((a^5*c^4)/7 + (b^5*c^4*x^5)/2 - a*b^4*c^4*x^4 + (2*a^3*b^2*c^4*x^2)/5 + (a^2*b^3*c^4*x^3)/2 - (a^4*b*c^4*x)/
2)/x^7

________________________________________________________________________________________

sympy [A]  time = 0.44, size = 80, normalized size = 0.95 \begin {gather*} \frac {- 10 a^{5} c^{4} + 35 a^{4} b c^{4} x - 28 a^{3} b^{2} c^{4} x^{2} - 35 a^{2} b^{3} c^{4} x^{3} + 70 a b^{4} c^{4} x^{4} - 35 b^{5} c^{4} x^{5}}{70 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**4/x**8,x)

[Out]

(-10*a**5*c**4 + 35*a**4*b*c**4*x - 28*a**3*b**2*c**4*x**2 - 35*a**2*b**3*c**4*x**3 + 70*a*b**4*c**4*x**4 - 35
*b**5*c**4*x**5)/(70*x**7)

________________________________________________________________________________________