3.93 \(\int \frac{(a+b x)^5}{x^{10}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{10 a^3 b^2}{7 x^7}-\frac{5 a^2 b^3}{3 x^6}-\frac{5 a^4 b}{8 x^8}-\frac{a^5}{9 x^9}-\frac{a b^4}{x^5}-\frac{b^5}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.022074, antiderivative size = 67, 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} \[ -\frac{10 a^3 b^2}{7 x^7}-\frac{5 a^2 b^3}{3 x^6}-\frac{5 a^4 b}{8 x^8}-\frac{a^5}{9 x^9}-\frac{a b^4}{x^5}-\frac{b^5}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^10,x]

[Out]

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

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

Mathematica [A]  time = 0.0116235, size = 67, normalized size = 1. \[ -\frac{10 a^3 b^2}{7 x^7}-\frac{5 a^2 b^3}{3 x^6}-\frac{5 a^4 b}{8 x^8}-\frac{a^5}{9 x^9}-\frac{a b^4}{x^5}-\frac{b^5}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^10,x]

[Out]

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

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Maple [A]  time = 0.006, size = 58, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{9\,{x}^{9}}}-{\frac{5\,{a}^{4}b}{8\,{x}^{8}}}-{\frac{10\,{a}^{3}{b}^{2}}{7\,{x}^{7}}}-{\frac{5\,{a}^{2}{b}^{3}}{3\,{x}^{6}}}-{\frac{a{b}^{4}}{{x}^{5}}}-{\frac{{b}^{5}}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/x^10,x)

[Out]

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

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Maxima [A]  time = 1.05083, size = 77, normalized size = 1.15 \begin{align*} -\frac{126 \, b^{5} x^{5} + 504 \, a b^{4} x^{4} + 840 \, a^{2} b^{3} x^{3} + 720 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b x + 56 \, a^{5}}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^10,x, algorithm="maxima")

[Out]

-1/504*(126*b^5*x^5 + 504*a*b^4*x^4 + 840*a^2*b^3*x^3 + 720*a^3*b^2*x^2 + 315*a^4*b*x + 56*a^5)/x^9

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Fricas [A]  time = 1.57443, size = 136, normalized size = 2.03 \begin{align*} -\frac{126 \, b^{5} x^{5} + 504 \, a b^{4} x^{4} + 840 \, a^{2} b^{3} x^{3} + 720 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b x + 56 \, a^{5}}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^10,x, algorithm="fricas")

[Out]

-1/504*(126*b^5*x^5 + 504*a*b^4*x^4 + 840*a^2*b^3*x^3 + 720*a^3*b^2*x^2 + 315*a^4*b*x + 56*a^5)/x^9

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Sympy [A]  time = 0.757735, size = 61, normalized size = 0.91 \begin{align*} - \frac{56 a^{5} + 315 a^{4} b x + 720 a^{3} b^{2} x^{2} + 840 a^{2} b^{3} x^{3} + 504 a b^{4} x^{4} + 126 b^{5} x^{5}}{504 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/x**10,x)

[Out]

-(56*a**5 + 315*a**4*b*x + 720*a**3*b**2*x**2 + 840*a**2*b**3*x**3 + 504*a*b**4*x**4 + 126*b**5*x**5)/(504*x**
9)

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Giac [A]  time = 1.18753, size = 77, normalized size = 1.15 \begin{align*} -\frac{126 \, b^{5} x^{5} + 504 \, a b^{4} x^{4} + 840 \, a^{2} b^{3} x^{3} + 720 \, a^{3} b^{2} x^{2} + 315 \, a^{4} b x + 56 \, a^{5}}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^10,x, algorithm="giac")

[Out]

-1/504*(126*b^5*x^5 + 504*a*b^4*x^4 + 840*a^2*b^3*x^3 + 720*a^3*b^2*x^2 + 315*a^4*b*x + 56*a^5)/x^9