3.247 \(\int \frac{(a+b x)^4}{x^{10}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{6 a^2 b^2}{7 x^7}-\frac{a^3 b}{2 x^8}-\frac{a^4}{9 x^9}-\frac{2 a b^3}{3 x^6}-\frac{b^4}{5 x^5} \]

[Out]

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

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Rubi [A]  time = 0.0191013, antiderivative size = 56, 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} \[ -\frac{6 a^2 b^2}{7 x^7}-\frac{a^3 b}{2 x^8}-\frac{a^4}{9 x^9}-\frac{2 a b^3}{3 x^6}-\frac{b^4}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^4/x^10,x]

[Out]

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

Mathematica [A]  time = 0.0110654, size = 56, normalized size = 1. \[ -\frac{6 a^2 b^2}{7 x^7}-\frac{a^3 b}{2 x^8}-\frac{a^4}{9 x^9}-\frac{2 a b^3}{3 x^6}-\frac{b^4}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/x^10,x]

[Out]

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

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Maple [A]  time = 0.006, size = 47, normalized size = 0.8 \begin{align*} -{\frac{{a}^{4}}{9\,{x}^{9}}}-{\frac{{a}^{3}b}{2\,{x}^{8}}}-{\frac{6\,{b}^{2}{a}^{2}}{7\,{x}^{7}}}-{\frac{2\,a{b}^{3}}{3\,{x}^{6}}}-{\frac{{b}^{4}}{5\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^4/x^10,x)

[Out]

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

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Maxima [A]  time = 1.10312, size = 62, normalized size = 1.11 \begin{align*} -\frac{126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/x^10,x, algorithm="maxima")

[Out]

-1/630*(126*b^4*x^4 + 420*a*b^3*x^3 + 540*a^2*b^2*x^2 + 315*a^3*b*x + 70*a^4)/x^9

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Fricas [A]  time = 1.63862, size = 112, normalized size = 2. \begin{align*} -\frac{126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/x^10,x, algorithm="fricas")

[Out]

-1/630*(126*b^4*x^4 + 420*a*b^3*x^3 + 540*a^2*b^2*x^2 + 315*a^3*b*x + 70*a^4)/x^9

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Sympy [A]  time = 0.710616, size = 49, normalized size = 0.88 \begin{align*} - \frac{70 a^{4} + 315 a^{3} b x + 540 a^{2} b^{2} x^{2} + 420 a b^{3} x^{3} + 126 b^{4} x^{4}}{630 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**4/x**10,x)

[Out]

-(70*a**4 + 315*a**3*b*x + 540*a**2*b**2*x**2 + 420*a*b**3*x**3 + 126*b**4*x**4)/(630*x**9)

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Giac [A]  time = 1.19042, size = 62, normalized size = 1.11 \begin{align*} -\frac{126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^4/x^10,x, algorithm="giac")

[Out]

-1/630*(126*b^4*x^4 + 420*a*b^3*x^3 + 540*a^2*b^2*x^2 + 315*a^3*b*x + 70*a^4)/x^9