3.1.46 \(\int \frac {(a+b x) (a c-b c x)^6}{x^{10}} \, dx\)

Optimal. Leaf size=65 \[ -\frac {11 b^2 c^6 (a-b x)^7}{504 a^2 x^7}-\frac {c^6 (a-b x)^7}{9 x^9}-\frac {11 b c^6 (a-b x)^7}{72 a x^8} \]

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Rubi [A]  time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {78, 45, 37} \begin {gather*} -\frac {11 b^2 c^6 (a-b x)^7}{504 a^2 x^7}-\frac {11 b c^6 (a-b x)^7}{72 a x^8}-\frac {c^6 (a-b x)^7}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a*c - b*c*x)^6)/x^10,x]

[Out]

-(c^6*(a - b*x)^7)/(9*x^9) - (11*b*c^6*(a - b*x)^7)/(72*a*x^8) - (11*b^2*c^6*(a - b*x)^7)/(504*a^2*x^7)

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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 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)^6}{x^{10}} \, dx &=-\frac {c^6 (a-b x)^7}{9 x^9}+\frac {1}{9} (11 b) \int \frac {(a c-b c x)^6}{x^9} \, dx\\ &=-\frac {c^6 (a-b x)^7}{9 x^9}-\frac {11 b c^6 (a-b x)^7}{72 a x^8}+\frac {\left (11 b^2\right ) \int \frac {(a c-b c x)^6}{x^8} \, dx}{72 a}\\ &=-\frac {c^6 (a-b x)^7}{9 x^9}-\frac {11 b c^6 (a-b x)^7}{72 a x^8}-\frac {11 b^2 c^6 (a-b x)^7}{504 a^2 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 116, normalized size = 1.78 \begin {gather*} -\frac {a^7 c^6}{9 x^9}+\frac {5 a^6 b c^6}{8 x^8}-\frac {9 a^5 b^2 c^6}{7 x^7}+\frac {5 a^4 b^3 c^6}{6 x^6}+\frac {a^3 b^4 c^6}{x^5}-\frac {9 a^2 b^5 c^6}{4 x^4}+\frac {5 a b^6 c^6}{3 x^3}-\frac {b^7 c^6}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^6)/x^10,x]

[Out]

-1/9*(a^7*c^6)/x^9 + (5*a^6*b*c^6)/(8*x^8) - (9*a^5*b^2*c^6)/(7*x^7) + (5*a^4*b^3*c^6)/(6*x^6) + (a^3*b^4*c^6)
/x^5 - (9*a^2*b^5*c^6)/(4*x^4) + (5*a*b^6*c^6)/(3*x^3) - (b^7*c^6)/(2*x^2)

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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)^6}{x^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^6)/x^10,x]

[Out]

IntegrateAlgebraic[((a + b*x)*(a*c - b*c*x)^6)/x^10, x]

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fricas [A]  time = 1.11, size = 103, normalized size = 1.58 \begin {gather*} -\frac {252 \, b^{7} c^{6} x^{7} - 840 \, a b^{6} c^{6} x^{6} + 1134 \, a^{2} b^{5} c^{6} x^{5} - 504 \, a^{3} b^{4} c^{6} x^{4} - 420 \, a^{4} b^{3} c^{6} x^{3} + 648 \, a^{5} b^{2} c^{6} x^{2} - 315 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{504 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^10,x, algorithm="fricas")

[Out]

-1/504*(252*b^7*c^6*x^7 - 840*a*b^6*c^6*x^6 + 1134*a^2*b^5*c^6*x^5 - 504*a^3*b^4*c^6*x^4 - 420*a^4*b^3*c^6*x^3
+ 648*a^5*b^2*c^6*x^2 - 315*a^6*b*c^6*x + 56*a^7*c^6)/x^9

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giac [A]  time = 0.97, size = 103, normalized size = 1.58 \begin {gather*} -\frac {252 \, b^{7} c^{6} x^{7} - 840 \, a b^{6} c^{6} x^{6} + 1134 \, a^{2} b^{5} c^{6} x^{5} - 504 \, a^{3} b^{4} c^{6} x^{4} - 420 \, a^{4} b^{3} c^{6} x^{3} + 648 \, a^{5} b^{2} c^{6} x^{2} - 315 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{504 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^10,x, algorithm="giac")

[Out]

-1/504*(252*b^7*c^6*x^7 - 840*a*b^6*c^6*x^6 + 1134*a^2*b^5*c^6*x^5 - 504*a^3*b^4*c^6*x^4 - 420*a^4*b^3*c^6*x^3
+ 648*a^5*b^2*c^6*x^2 - 315*a^6*b*c^6*x + 56*a^7*c^6)/x^9

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maple [A]  time = 0.00, size = 83, normalized size = 1.28 \begin {gather*} \left (-\frac {b^{7}}{2 x^{2}}+\frac {5 a \,b^{6}}{3 x^{3}}-\frac {9 a^{2} b^{5}}{4 x^{4}}+\frac {a^{3} b^{4}}{x^{5}}+\frac {5 a^{4} b^{3}}{6 x^{6}}-\frac {9 a^{5} b^{2}}{7 x^{7}}+\frac {5 a^{6} b}{8 x^{8}}-\frac {a^{7}}{9 x^{9}}\right ) c^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(-b*c*x+a*c)^6/x^10,x)

[Out]

c^6*(5/6*a^4*b^3/x^6+a^3*b^4/x^5-1/2*b^7/x^2+5/8*a^6*b/x^8-1/9*a^7/x^9-9/7*a^5*b^2/x^7+5/3*a*b^6/x^3-9/4*a^2*b
^5/x^4)

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maxima [A]  time = 1.04, size = 103, normalized size = 1.58 \begin {gather*} -\frac {252 \, b^{7} c^{6} x^{7} - 840 \, a b^{6} c^{6} x^{6} + 1134 \, a^{2} b^{5} c^{6} x^{5} - 504 \, a^{3} b^{4} c^{6} x^{4} - 420 \, a^{4} b^{3} c^{6} x^{3} + 648 \, a^{5} b^{2} c^{6} x^{2} - 315 \, a^{6} b c^{6} x + 56 \, a^{7} c^{6}}{504 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)^6/x^10,x, algorithm="maxima")

[Out]

-1/504*(252*b^7*c^6*x^7 - 840*a*b^6*c^6*x^6 + 1134*a^2*b^5*c^6*x^5 - 504*a^3*b^4*c^6*x^4 - 420*a^4*b^3*c^6*x^3
+ 648*a^5*b^2*c^6*x^2 - 315*a^6*b*c^6*x + 56*a^7*c^6)/x^9

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mupad [B]  time = 0.32, size = 103, normalized size = 1.58 \begin {gather*} -\frac {\frac {a^7\,c^6}{9}-\frac {5\,a^6\,b\,c^6\,x}{8}+\frac {9\,a^5\,b^2\,c^6\,x^2}{7}-\frac {5\,a^4\,b^3\,c^6\,x^3}{6}-a^3\,b^4\,c^6\,x^4+\frac {9\,a^2\,b^5\,c^6\,x^5}{4}-\frac {5\,a\,b^6\,c^6\,x^6}{3}+\frac {b^7\,c^6\,x^7}{2}}{x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*c - b*c*x)^6*(a + b*x))/x^10,x)

[Out]

-((a^7*c^6)/9 + (b^7*c^6*x^7)/2 - (5*a*b^6*c^6*x^6)/3 + (9*a^5*b^2*c^6*x^2)/7 - (5*a^4*b^3*c^6*x^3)/6 - a^3*b^
4*c^6*x^4 + (9*a^2*b^5*c^6*x^5)/4 - (5*a^6*b*c^6*x)/8)/x^9

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sympy [A]  time = 0.64, size = 110, normalized size = 1.69 \begin {gather*} \frac {- 56 a^{7} c^{6} + 315 a^{6} b c^{6} x - 648 a^{5} b^{2} c^{6} x^{2} + 420 a^{4} b^{3} c^{6} x^{3} + 504 a^{3} b^{4} c^{6} x^{4} - 1134 a^{2} b^{5} c^{6} x^{5} + 840 a b^{6} c^{6} x^{6} - 252 b^{7} c^{6} x^{7}}{504 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(-b*c*x+a*c)**6/x**10,x)

[Out]

(-56*a**7*c**6 + 315*a**6*b*c**6*x - 648*a**5*b**2*c**6*x**2 + 420*a**4*b**3*c**6*x**3 + 504*a**3*b**4*c**6*x*
*4 - 1134*a**2*b**5*c**6*x**5 + 840*a*b**6*c**6*x**6 - 252*b**7*c**6*x**7)/(504*x**9)

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