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

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

[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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Rubi [A]  time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 56, normalized size = 1.00 \[ -\frac {a^4}{9 x^9}-\frac {a^3 b}{2 x^8}-\frac {6 a^2 b^2}{7 x^7}-\frac {2 a b^3}{3 x^6}-\frac {b^4}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^4/x^10,x]

[Out]

-1/9*a^4/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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fricas [A]  time = 0.46, size = 46, normalized size = 0.82 \[ -\frac {126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \]

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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giac [A]  time = 1.25, size = 46, normalized size = 0.82 \[ -\frac {126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \]

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

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maple [A]  time = 0.01, size = 47, normalized size = 0.84 \[ -\frac {b^{4}}{5 x^{5}}-\frac {2 a \,b^{3}}{3 x^{6}}-\frac {6 a^{2} b^{2}}{7 x^{7}}-\frac {a^{3} b}{2 x^{8}}-\frac {a^{4}}{9 x^{9}} \]

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 = 1.34, size = 46, normalized size = 0.82 \[ -\frac {126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \]

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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mupad [B]  time = 0.03, size = 46, normalized size = 0.82 \[ -\frac {\frac {a^4}{9}+\frac {a^3\,b\,x}{2}+\frac {6\,a^2\,b^2\,x^2}{7}+\frac {2\,a\,b^3\,x^3}{3}+\frac {b^4\,x^4}{5}}{x^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^4/x^10,x)

[Out]

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

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sympy [A]  time = 0.50, size = 49, normalized size = 0.88 \[ \frac {- 70 a^{4} - 315 a^{3} b x - 540 a^{2} b^{2} x^{2} - 420 a b^{3} x^{3} - 126 b^{4} x^{4}}{630 x^{9}} \]

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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