3.117 \(\int \frac{(a+b x)^7}{x^{11}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{b^2 (a+b x)^8}{360 a^3 x^8}+\frac{b (a+b x)^8}{45 a^2 x^9}-\frac{(a+b x)^8}{10 a x^{10}} \]

[Out]

-(a + b*x)^8/(10*a*x^10) + (b*(a + b*x)^8)/(45*a^2*x^9) - (b^2*(a + b*x)^8)/(360*a^3*x^8)

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Rubi [A]  time = 0.0098383, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {45, 37} \[ -\frac{b^2 (a+b x)^8}{360 a^3 x^8}+\frac{b (a+b x)^8}{45 a^2 x^9}-\frac{(a+b x)^8}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^7/x^11,x]

[Out]

-(a + b*x)^8/(10*a*x^10) + (b*(a + b*x)^8)/(45*a^2*x^9) - (b^2*(a + b*x)^8)/(360*a^3*x^8)

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 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]

Rubi steps

\begin{align*} \int \frac{(a+b x)^7}{x^{11}} \, dx &=-\frac{(a+b x)^8}{10 a x^{10}}-\frac{b \int \frac{(a+b x)^7}{x^{10}} \, dx}{5 a}\\ &=-\frac{(a+b x)^8}{10 a x^{10}}+\frac{b (a+b x)^8}{45 a^2 x^9}+\frac{b^2 \int \frac{(a+b x)^7}{x^9} \, dx}{45 a^2}\\ &=-\frac{(a+b x)^8}{10 a x^{10}}+\frac{b (a+b x)^8}{45 a^2 x^9}-\frac{b^2 (a+b x)^8}{360 a^3 x^8}\\ \end{align*}

Mathematica [A]  time = 0.00455, size = 93, normalized size = 1.66 \[ -\frac{21 a^5 b^2}{8 x^8}-\frac{5 a^4 b^3}{x^7}-\frac{35 a^3 b^4}{6 x^6}-\frac{21 a^2 b^5}{5 x^5}-\frac{7 a^6 b}{9 x^9}-\frac{a^7}{10 x^{10}}-\frac{7 a b^6}{4 x^4}-\frac{b^7}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^7/x^11,x]

[Out]

-a^7/(10*x^10) - (7*a^6*b)/(9*x^9) - (21*a^5*b^2)/(8*x^8) - (5*a^4*b^3)/x^7 - (35*a^3*b^4)/(6*x^6) - (21*a^2*b
^5)/(5*x^5) - (7*a*b^6)/(4*x^4) - b^7/(3*x^3)

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Maple [A]  time = 0.005, size = 80, normalized size = 1.4 \begin{align*} -{\frac{{a}^{7}}{10\,{x}^{10}}}-{\frac{{b}^{7}}{3\,{x}^{3}}}-{\frac{21\,{a}^{2}{b}^{5}}{5\,{x}^{5}}}-{\frac{7\,a{b}^{6}}{4\,{x}^{4}}}-{\frac{35\,{a}^{3}{b}^{4}}{6\,{x}^{6}}}-{\frac{21\,{a}^{5}{b}^{2}}{8\,{x}^{8}}}-5\,{\frac{{a}^{4}{b}^{3}}{{x}^{7}}}-{\frac{7\,{a}^{6}b}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^7/x^11,x)

[Out]

-1/10*a^7/x^10-1/3*b^7/x^3-21/5*a^2*b^5/x^5-7/4*a*b^6/x^4-35/6*a^3*b^4/x^6-21/8*a^5*b^2/x^8-5*a^4*b^3/x^7-7/9*
a^6*b/x^9

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Maxima [A]  time = 1.05918, size = 107, normalized size = 1.91 \begin{align*} -\frac{120 \, b^{7} x^{7} + 630 \, a b^{6} x^{6} + 1512 \, a^{2} b^{5} x^{5} + 2100 \, a^{3} b^{4} x^{4} + 1800 \, a^{4} b^{3} x^{3} + 945 \, a^{5} b^{2} x^{2} + 280 \, a^{6} b x + 36 \, a^{7}}{360 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^11,x, algorithm="maxima")

[Out]

-1/360*(120*b^7*x^7 + 630*a*b^6*x^6 + 1512*a^2*b^5*x^5 + 2100*a^3*b^4*x^4 + 1800*a^4*b^3*x^3 + 945*a^5*b^2*x^2
+ 280*a^6*b*x + 36*a^7)/x^10

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Fricas [A]  time = 1.77579, size = 190, normalized size = 3.39 \begin{align*} -\frac{120 \, b^{7} x^{7} + 630 \, a b^{6} x^{6} + 1512 \, a^{2} b^{5} x^{5} + 2100 \, a^{3} b^{4} x^{4} + 1800 \, a^{4} b^{3} x^{3} + 945 \, a^{5} b^{2} x^{2} + 280 \, a^{6} b x + 36 \, a^{7}}{360 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^11,x, algorithm="fricas")

[Out]

-1/360*(120*b^7*x^7 + 630*a*b^6*x^6 + 1512*a^2*b^5*x^5 + 2100*a^3*b^4*x^4 + 1800*a^4*b^3*x^3 + 945*a^5*b^2*x^2
+ 280*a^6*b*x + 36*a^7)/x^10

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Sympy [A]  time = 0.949686, size = 85, normalized size = 1.52 \begin{align*} - \frac{36 a^{7} + 280 a^{6} b x + 945 a^{5} b^{2} x^{2} + 1800 a^{4} b^{3} x^{3} + 2100 a^{3} b^{4} x^{4} + 1512 a^{2} b^{5} x^{5} + 630 a b^{6} x^{6} + 120 b^{7} x^{7}}{360 x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**7/x**11,x)

[Out]

-(36*a**7 + 280*a**6*b*x + 945*a**5*b**2*x**2 + 1800*a**4*b**3*x**3 + 2100*a**3*b**4*x**4 + 1512*a**2*b**5*x**
5 + 630*a*b**6*x**6 + 120*b**7*x**7)/(360*x**10)

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Giac [A]  time = 1.20727, size = 107, normalized size = 1.91 \begin{align*} -\frac{120 \, b^{7} x^{7} + 630 \, a b^{6} x^{6} + 1512 \, a^{2} b^{5} x^{5} + 2100 \, a^{3} b^{4} x^{4} + 1800 \, a^{4} b^{3} x^{3} + 945 \, a^{5} b^{2} x^{2} + 280 \, a^{6} b x + 36 \, a^{7}}{360 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^7/x^11,x, algorithm="giac")

[Out]

-1/360*(120*b^7*x^7 + 630*a*b^6*x^6 + 1512*a^2*b^5*x^5 + 2100*a^3*b^4*x^4 + 1800*a^4*b^3*x^3 + 945*a^5*b^2*x^2
+ 280*a^6*b*x + 36*a^7)/x^10