3.1.39 \(\int \frac {(a+b x) (a c-b c x)^5}{x^8} \, dx\)

Optimal. Leaf size=41 \[ -\frac {c^5 (a-b x)^6}{7 x^7}-\frac {4 b c^5 (a-b x)^6}{21 a x^6} \]

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Rubi [A]  time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {78, 37} \begin {gather*} -\frac {4 b c^5 (a-b x)^6}{21 a x^6}-\frac {c^5 (a-b x)^6}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^8,x]

[Out]

-(c^5*(a - b*x)^6)/(7*x^7) - (4*b*c^5*(a - b*x)^6)/(21*a*x^6)

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 78

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] || LtQ
[p, n]))))

Rubi steps

\begin {align*} \int \frac {(a+b x) (a c-b c x)^5}{x^8} \, dx &=-\frac {c^5 (a-b x)^6}{7 x^7}+\frac {1}{7} (8 b) \int \frac {(a c-b c x)^5}{x^7} \, dx\\ &=-\frac {c^5 (a-b x)^6}{7 x^7}-\frac {4 b c^5 (a-b x)^6}{21 a x^6}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 66, normalized size = 1.61 \begin {gather*} c^5 \left (-\frac {a^6}{7 x^7}+\frac {2 a^5 b}{3 x^6}-\frac {a^4 b^2}{x^5}+\frac {5 a^2 b^4}{3 x^3}-\frac {2 a b^5}{x^2}+\frac {b^6}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^8,x]

[Out]

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

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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)^5}{x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^5)/x^8,x]

[Out]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^5)/x^8, x]

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fricas [A]  time = 1.14, size = 75, normalized size = 1.83 \begin {gather*} \frac {21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^8,x, algorithm="fricas")

[Out]

1/21*(21*b^6*c^5*x^6 - 42*a*b^5*c^5*x^5 + 35*a^2*b^4*c^5*x^4 - 21*a^4*b^2*c^5*x^2 + 14*a^5*b*c^5*x - 3*a^6*c^5
)/x^7

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giac [A]  time = 0.97, size = 75, normalized size = 1.83 \begin {gather*} \frac {21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^8,x, algorithm="giac")

[Out]

1/21*(21*b^6*c^5*x^6 - 42*a*b^5*c^5*x^5 + 35*a^2*b^4*c^5*x^4 - 21*a^4*b^2*c^5*x^2 + 14*a^5*b*c^5*x - 3*a^6*c^5
)/x^7

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maple [A]  time = 0.01, size = 61, normalized size = 1.49 \begin {gather*} \left (\frac {b^{6}}{x}-\frac {2 a \,b^{5}}{x^{2}}+\frac {5 a^{2} b^{4}}{3 x^{3}}-\frac {a^{4} b^{2}}{x^{5}}+\frac {2 a^{5} b}{3 x^{6}}-\frac {a^{6}}{7 x^{7}}\right ) c^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^5/x^8,x)

[Out]

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

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maxima [A]  time = 1.02, size = 75, normalized size = 1.83 \begin {gather*} \frac {21 \, b^{6} c^{5} x^{6} - 42 \, a b^{5} c^{5} x^{5} + 35 \, a^{2} b^{4} c^{5} x^{4} - 21 \, a^{4} b^{2} c^{5} x^{2} + 14 \, a^{5} b c^{5} x - 3 \, a^{6} c^{5}}{21 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^8,x, algorithm="maxima")

[Out]

1/21*(21*b^6*c^5*x^6 - 42*a*b^5*c^5*x^5 + 35*a^2*b^4*c^5*x^4 - 21*a^4*b^2*c^5*x^2 + 14*a^5*b*c^5*x - 3*a^6*c^5
)/x^7

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mupad [B]  time = 0.29, size = 74, normalized size = 1.80 \begin {gather*} -\frac {\frac {a^6\,c^5}{7}-\frac {2\,a^5\,b\,c^5\,x}{3}+a^4\,b^2\,c^5\,x^2-\frac {5\,a^2\,b^4\,c^5\,x^4}{3}+2\,a\,b^5\,c^5\,x^5-b^6\,c^5\,x^6}{x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*c - b*c*x)^5*(a + b*x))/x^8,x)

[Out]

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

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sympy [B]  time = 0.46, size = 82, normalized size = 2.00 \begin {gather*} - \frac {3 a^{6} c^{5} - 14 a^{5} b c^{5} x + 21 a^{4} b^{2} c^{5} x^{2} - 35 a^{2} b^{4} c^{5} x^{4} + 42 a b^{5} c^{5} x^{5} - 21 b^{6} c^{5} x^{6}}{21 x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**8,x)

[Out]

-(3*a**6*c**5 - 14*a**5*b*c**5*x + 21*a**4*b**2*c**5*x**2 - 35*a**2*b**4*c**5*x**4 + 42*a*b**5*c**5*x**5 - 21*
b**6*c**5*x**6)/(21*x**7)

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