3.1.26 \(\int \frac {(a+b x) (a c-b c x)^4}{x^9} \, dx\)

Optimal. Leaf size=87 \[ -\frac {a^5 c^4}{8 x^8}+\frac {3 a^4 b c^4}{7 x^7}-\frac {a^3 b^2 c^4}{3 x^6}-\frac {2 a^2 b^3 c^4}{5 x^5}+\frac {3 a b^4 c^4}{4 x^4}-\frac {b^5 c^4}{3 x^3} \]

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Rubi [A]  time = 0.04, antiderivative size = 87, 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 {a^3 b^2 c^4}{3 x^6}-\frac {2 a^2 b^3 c^4}{5 x^5}+\frac {3 a^4 b c^4}{7 x^7}-\frac {a^5 c^4}{8 x^8}+\frac {3 a b^4 c^4}{4 x^4}-\frac {b^5 c^4}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^4)/x^9,x]

[Out]

-(a^5*c^4)/(8*x^8) + (3*a^4*b*c^4)/(7*x^7) - (a^3*b^2*c^4)/(3*x^6) - (2*a^2*b^3*c^4)/(5*x^5) + (3*a*b^4*c^4)/(
4*x^4) - (b^5*c^4)/(3*x^3)

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^9} \, dx &=\int \left (\frac {a^5 c^4}{x^9}-\frac {3 a^4 b c^4}{x^8}+\frac {2 a^3 b^2 c^4}{x^7}+\frac {2 a^2 b^3 c^4}{x^6}-\frac {3 a b^4 c^4}{x^5}+\frac {b^5 c^4}{x^4}\right ) \, dx\\ &=-\frac {a^5 c^4}{8 x^8}+\frac {3 a^4 b c^4}{7 x^7}-\frac {a^3 b^2 c^4}{3 x^6}-\frac {2 a^2 b^3 c^4}{5 x^5}+\frac {3 a b^4 c^4}{4 x^4}-\frac {b^5 c^4}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 87, normalized size = 1.00 \begin {gather*} -\frac {a^5 c^4}{8 x^8}+\frac {3 a^4 b c^4}{7 x^7}-\frac {a^3 b^2 c^4}{3 x^6}-\frac {2 a^2 b^3 c^4}{5 x^5}+\frac {3 a b^4 c^4}{4 x^4}-\frac {b^5 c^4}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^4)/x^9,x]

[Out]

-1/8*(a^5*c^4)/x^8 + (3*a^4*b*c^4)/(7*x^7) - (a^3*b^2*c^4)/(3*x^6) - (2*a^2*b^3*c^4)/(5*x^5) + (3*a*b^4*c^4)/(
4*x^4) - (b^5*c^4)/(3*x^3)

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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^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^4)/x^9,x]

[Out]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^4)/x^9, x]

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fricas [A]  time = 1.33, size = 75, normalized size = 0.86 \begin {gather*} -\frac {280 \, b^{5} c^{4} x^{5} - 630 \, a b^{4} c^{4} x^{4} + 336 \, a^{2} b^{3} c^{4} x^{3} + 280 \, a^{3} b^{2} c^{4} x^{2} - 360 \, a^{4} b c^{4} x + 105 \, a^{5} c^{4}}{840 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^9,x, algorithm="fricas")

[Out]

-1/840*(280*b^5*c^4*x^5 - 630*a*b^4*c^4*x^4 + 336*a^2*b^3*c^4*x^3 + 280*a^3*b^2*c^4*x^2 - 360*a^4*b*c^4*x + 10
5*a^5*c^4)/x^8

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giac [A]  time = 1.11, size = 75, normalized size = 0.86 \begin {gather*} -\frac {280 \, b^{5} c^{4} x^{5} - 630 \, a b^{4} c^{4} x^{4} + 336 \, a^{2} b^{3} c^{4} x^{3} + 280 \, a^{3} b^{2} c^{4} x^{2} - 360 \, a^{4} b c^{4} x + 105 \, a^{5} c^{4}}{840 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^9,x, algorithm="giac")

[Out]

-1/840*(280*b^5*c^4*x^5 - 630*a*b^4*c^4*x^4 + 336*a^2*b^3*c^4*x^3 + 280*a^3*b^2*c^4*x^2 - 360*a^4*b*c^4*x + 10
5*a^5*c^4)/x^8

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maple [A]  time = 0.01, size = 62, normalized size = 0.71 \begin {gather*} \left (-\frac {b^{5}}{3 x^{3}}+\frac {3 a \,b^{4}}{4 x^{4}}-\frac {2 a^{2} b^{3}}{5 x^{5}}-\frac {a^{3} b^{2}}{3 x^{6}}+\frac {3 a^{4} b}{7 x^{7}}-\frac {a^{5}}{8 x^{8}}\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^9,x)

[Out]

c^4*(-1/3*a^3*b^2/x^6-2/5*a^2*b^3/x^5-1/8*a^5/x^8-1/3*b^5/x^3+3/7*a^4*b/x^7+3/4*a*b^4/x^4)

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maxima [A]  time = 1.02, size = 75, normalized size = 0.86 \begin {gather*} -\frac {280 \, b^{5} c^{4} x^{5} - 630 \, a b^{4} c^{4} x^{4} + 336 \, a^{2} b^{3} c^{4} x^{3} + 280 \, a^{3} b^{2} c^{4} x^{2} - 360 \, a^{4} b c^{4} x + 105 \, a^{5} c^{4}}{840 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^4/x^9,x, algorithm="maxima")

[Out]

-1/840*(280*b^5*c^4*x^5 - 630*a*b^4*c^4*x^4 + 336*a^2*b^3*c^4*x^3 + 280*a^3*b^2*c^4*x^2 - 360*a^4*b*c^4*x + 10
5*a^5*c^4)/x^8

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mupad [B]  time = 0.29, size = 75, normalized size = 0.86 \begin {gather*} -\frac {\frac {a^5\,c^4}{8}-\frac {3\,a^4\,b\,c^4\,x}{7}+\frac {a^3\,b^2\,c^4\,x^2}{3}+\frac {2\,a^2\,b^3\,c^4\,x^3}{5}-\frac {3\,a\,b^4\,c^4\,x^4}{4}+\frac {b^5\,c^4\,x^5}{3}}{x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*c - b*c*x)^4*(a + b*x))/x^9,x)

[Out]

-((a^5*c^4)/8 + (b^5*c^4*x^5)/3 - (3*a*b^4*c^4*x^4)/4 + (a^3*b^2*c^4*x^2)/3 + (2*a^2*b^3*c^4*x^3)/5 - (3*a^4*b
*c^4*x)/7)/x^8

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sympy [A]  time = 0.46, size = 80, normalized size = 0.92 \begin {gather*} \frac {- 105 a^{5} c^{4} + 360 a^{4} b c^{4} x - 280 a^{3} b^{2} c^{4} x^{2} - 336 a^{2} b^{3} c^{4} x^{3} + 630 a b^{4} c^{4} x^{4} - 280 b^{5} c^{4} x^{5}}{840 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**4/x**9,x)

[Out]

(-105*a**5*c**4 + 360*a**4*b*c**4*x - 280*a**3*b**2*c**4*x**2 - 336*a**2*b**3*c**4*x**3 + 630*a*b**4*c**4*x**4
- 280*b**5*c**4*x**5)/(840*x**8)

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