3.94 \(\int \frac{(a+b x)^5}{x^{11}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{5 a^3 b^2}{4 x^8}-\frac{10 a^2 b^3}{7 x^7}-\frac{5 a^4 b}{9 x^9}-\frac{a^5}{10 x^{10}}-\frac{5 a b^4}{6 x^6}-\frac{b^5}{5 x^5} \]

[Out]

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

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Rubi [A]  time = 0.0205153, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ -\frac{5 a^3 b^2}{4 x^8}-\frac{10 a^2 b^3}{7 x^7}-\frac{5 a^4 b}{9 x^9}-\frac{a^5}{10 x^{10}}-\frac{5 a b^4}{6 x^6}-\frac{b^5}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^11,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^5}{x^{11}} \, dx &=\int \left (\frac{a^5}{x^{11}}+\frac{5 a^4 b}{x^{10}}+\frac{10 a^3 b^2}{x^9}+\frac{10 a^2 b^3}{x^8}+\frac{5 a b^4}{x^7}+\frac{b^5}{x^6}\right ) \, dx\\ &=-\frac{a^5}{10 x^{10}}-\frac{5 a^4 b}{9 x^9}-\frac{5 a^3 b^2}{4 x^8}-\frac{10 a^2 b^3}{7 x^7}-\frac{5 a b^4}{6 x^6}-\frac{b^5}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0061435, size = 69, normalized size = 1. \[ -\frac{5 a^3 b^2}{4 x^8}-\frac{10 a^2 b^3}{7 x^7}-\frac{5 a^4 b}{9 x^9}-\frac{a^5}{10 x^{10}}-\frac{5 a b^4}{6 x^6}-\frac{b^5}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^11,x]

[Out]

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

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Maple [A]  time = 0.006, size = 58, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}}{10\,{x}^{10}}}-{\frac{5\,{a}^{4}b}{9\,{x}^{9}}}-{\frac{5\,{a}^{3}{b}^{2}}{4\,{x}^{8}}}-{\frac{10\,{a}^{2}{b}^{3}}{7\,{x}^{7}}}-{\frac{5\,a{b}^{4}}{6\,{x}^{6}}}-{\frac{{b}^{5}}{5\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/x^11,x)

[Out]

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

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Maxima [A]  time = 1.05087, size = 77, normalized size = 1.12 \begin{align*} -\frac{252 \, b^{5} x^{5} + 1050 \, a b^{4} x^{4} + 1800 \, a^{2} b^{3} x^{3} + 1575 \, a^{3} b^{2} x^{2} + 700 \, a^{4} b x + 126 \, a^{5}}{1260 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^11,x, algorithm="maxima")

[Out]

-1/1260*(252*b^5*x^5 + 1050*a*b^4*x^4 + 1800*a^2*b^3*x^3 + 1575*a^3*b^2*x^2 + 700*a^4*b*x + 126*a^5)/x^10

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Fricas [A]  time = 1.31939, size = 144, normalized size = 2.09 \begin{align*} -\frac{252 \, b^{5} x^{5} + 1050 \, a b^{4} x^{4} + 1800 \, a^{2} b^{3} x^{3} + 1575 \, a^{3} b^{2} x^{2} + 700 \, a^{4} b x + 126 \, a^{5}}{1260 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^11,x, algorithm="fricas")

[Out]

-1/1260*(252*b^5*x^5 + 1050*a*b^4*x^4 + 1800*a^2*b^3*x^3 + 1575*a^3*b^2*x^2 + 700*a^4*b*x + 126*a^5)/x^10

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Sympy [A]  time = 0.738283, size = 61, normalized size = 0.88 \begin{align*} - \frac{126 a^{5} + 700 a^{4} b x + 1575 a^{3} b^{2} x^{2} + 1800 a^{2} b^{3} x^{3} + 1050 a b^{4} x^{4} + 252 b^{5} x^{5}}{1260 x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/x**11,x)

[Out]

-(126*a**5 + 700*a**4*b*x + 1575*a**3*b**2*x**2 + 1800*a**2*b**3*x**3 + 1050*a*b**4*x**4 + 252*b**5*x**5)/(126
0*x**10)

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Giac [A]  time = 1.17538, size = 77, normalized size = 1.12 \begin{align*} -\frac{252 \, b^{5} x^{5} + 1050 \, a b^{4} x^{4} + 1800 \, a^{2} b^{3} x^{3} + 1575 \, a^{3} b^{2} x^{2} + 700 \, a^{4} b x + 126 \, a^{5}}{1260 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^11,x, algorithm="giac")

[Out]

-1/1260*(252*b^5*x^5 + 1050*a*b^4*x^4 + 1800*a^2*b^3*x^3 + 1575*a^3*b^2*x^2 + 700*a^4*b*x + 126*a^5)/x^10