3.97 \(\int \frac{(a+b x)^5}{x^{14}} \, dx\)

Optimal. Leaf size=67 \[ -\frac{10 a^3 b^2}{11 x^{11}}-\frac{a^2 b^3}{x^{10}}-\frac{5 a^4 b}{12 x^{12}}-\frac{a^5}{13 x^{13}}-\frac{5 a b^4}{9 x^9}-\frac{b^5}{8 x^8} \]

[Out]

-a^5/(13*x^13) - (5*a^4*b)/(12*x^12) - (10*a^3*b^2)/(11*x^11) - (a^2*b^3)/x^10 - (5*a*b^4)/(9*x^9) - b^5/(8*x^
8)

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Rubi [A]  time = 0.0215932, 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} \[ -\frac{10 a^3 b^2}{11 x^{11}}-\frac{a^2 b^3}{x^{10}}-\frac{5 a^4 b}{12 x^{12}}-\frac{a^5}{13 x^{13}}-\frac{5 a b^4}{9 x^9}-\frac{b^5}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^14,x]

[Out]

-a^5/(13*x^13) - (5*a^4*b)/(12*x^12) - (10*a^3*b^2)/(11*x^11) - (a^2*b^3)/x^10 - (5*a*b^4)/(9*x^9) - b^5/(8*x^
8)

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^{14}} \, dx &=\int \left (\frac{a^5}{x^{14}}+\frac{5 a^4 b}{x^{13}}+\frac{10 a^3 b^2}{x^{12}}+\frac{10 a^2 b^3}{x^{11}}+\frac{5 a b^4}{x^{10}}+\frac{b^5}{x^9}\right ) \, dx\\ &=-\frac{a^5}{13 x^{13}}-\frac{5 a^4 b}{12 x^{12}}-\frac{10 a^3 b^2}{11 x^{11}}-\frac{a^2 b^3}{x^{10}}-\frac{5 a b^4}{9 x^9}-\frac{b^5}{8 x^8}\\ \end{align*}

Mathematica [A]  time = 0.0044323, size = 67, normalized size = 1. \[ -\frac{10 a^3 b^2}{11 x^{11}}-\frac{a^2 b^3}{x^{10}}-\frac{5 a^4 b}{12 x^{12}}-\frac{a^5}{13 x^{13}}-\frac{5 a b^4}{9 x^9}-\frac{b^5}{8 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^14,x]

[Out]

-a^5/(13*x^13) - (5*a^4*b)/(12*x^12) - (10*a^3*b^2)/(11*x^11) - (a^2*b^3)/x^10 - (5*a*b^4)/(9*x^9) - b^5/(8*x^
8)

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Maple [A]  time = 0.006, size = 58, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{13\,{x}^{13}}}-{\frac{5\,{a}^{4}b}{12\,{x}^{12}}}-{\frac{10\,{a}^{3}{b}^{2}}{11\,{x}^{11}}}-{\frac{{a}^{2}{b}^{3}}{{x}^{10}}}-{\frac{5\,a{b}^{4}}{9\,{x}^{9}}}-{\frac{{b}^{5}}{8\,{x}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/x^14,x)

[Out]

-1/13*a^5/x^13-5/12*a^4*b/x^12-10/11*a^3*b^2/x^11-a^2*b^3/x^10-5/9*a*b^4/x^9-1/8*b^5/x^8

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Maxima [A]  time = 1.06149, size = 77, normalized size = 1.15 \begin{align*} -\frac{1287 \, b^{5} x^{5} + 5720 \, a b^{4} x^{4} + 10296 \, a^{2} b^{3} x^{3} + 9360 \, a^{3} b^{2} x^{2} + 4290 \, a^{4} b x + 792 \, a^{5}}{10296 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^14,x, algorithm="maxima")

[Out]

-1/10296*(1287*b^5*x^5 + 5720*a*b^4*x^4 + 10296*a^2*b^3*x^3 + 9360*a^3*b^2*x^2 + 4290*a^4*b*x + 792*a^5)/x^13

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Fricas [A]  time = 1.42976, size = 150, normalized size = 2.24 \begin{align*} -\frac{1287 \, b^{5} x^{5} + 5720 \, a b^{4} x^{4} + 10296 \, a^{2} b^{3} x^{3} + 9360 \, a^{3} b^{2} x^{2} + 4290 \, a^{4} b x + 792 \, a^{5}}{10296 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^14,x, algorithm="fricas")

[Out]

-1/10296*(1287*b^5*x^5 + 5720*a*b^4*x^4 + 10296*a^2*b^3*x^3 + 9360*a^3*b^2*x^2 + 4290*a^4*b*x + 792*a^5)/x^13

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Sympy [A]  time = 0.930171, size = 61, normalized size = 0.91 \begin{align*} - \frac{792 a^{5} + 4290 a^{4} b x + 9360 a^{3} b^{2} x^{2} + 10296 a^{2} b^{3} x^{3} + 5720 a b^{4} x^{4} + 1287 b^{5} x^{5}}{10296 x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/x**14,x)

[Out]

-(792*a**5 + 4290*a**4*b*x + 9360*a**3*b**2*x**2 + 10296*a**2*b**3*x**3 + 5720*a*b**4*x**4 + 1287*b**5*x**5)/(
10296*x**13)

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Giac [A]  time = 1.20667, size = 77, normalized size = 1.15 \begin{align*} -\frac{1287 \, b^{5} x^{5} + 5720 \, a b^{4} x^{4} + 10296 \, a^{2} b^{3} x^{3} + 9360 \, a^{3} b^{2} x^{2} + 4290 \, a^{4} b x + 792 \, a^{5}}{10296 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^14,x, algorithm="giac")

[Out]

-1/10296*(1287*b^5*x^5 + 5720*a*b^4*x^4 + 10296*a^2*b^3*x^3 + 9360*a^3*b^2*x^2 + 4290*a^4*b*x + 792*a^5)/x^13