3.1.94 \(\int \frac {(a+b x)^5}{x^{11}} \, dx\) [94]

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

[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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Rubi [A]
time = 0.01, antiderivative size = 69, 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 = {45} \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^11,x]

[Out]

-1/10*a^5/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 45

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*}

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Mathematica [A]
time = 0.00, size = 69, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^11,x]

[Out]

-1/10*a^5/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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Mathics [A]
time = 2.17, size = 57, normalized size = 0.83 \begin {gather*} \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 {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^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) / (1260 x ^ 10)

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Maple [A]
time = 0.09, size = 58, normalized size = 0.84

method	result	size
norman	\(\frac {-\frac {1}{5} b^{5} x^{5}-\frac {5}{6} a \,b^{4} x^{4}-\frac {10}{7} a^{2} b^{3} x^{3}-\frac {5}{4} a^{3} b^{2} x^{2}-\frac {5}{9} a^{4} b x -\frac {1}{10} a^{5}}{x^{10}}\)	\(57\)
risch	\(\frac {-\frac {1}{5} b^{5} x^{5}-\frac {5}{6} a \,b^{4} x^{4}-\frac {10}{7} a^{2} b^{3} x^{3}-\frac {5}{4} a^{3} b^{2} x^{2}-\frac {5}{9} a^{4} b x -\frac {1}{10} a^{5}}{x^{10}}\)	\(57\)
gosper	\(-\frac {252 b^{5} x^{5}+1050 a \,b^{4} x^{4}+1800 a^{2} b^{3} x^{3}+1575 a^{3} b^{2} x^{2}+700 a^{4} b x +126 a^{5}}{1260 x^{10}}\)	\(58\)
default	\(-\frac {a^{5}}{10 x^{10}}-\frac {5 a^{4} b}{9 x^{9}}-\frac {5 a^{3} b^{2}}{4 x^{8}}-\frac {10 a^{2} b^{3}}{7 x^{7}}-\frac {5 a \,b^{4}}{6 x^{6}}-\frac {b^{5}}{5 x^{5}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.26, size = 57, normalized size = 0.83 \begin {gather*} -\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 {gather*}

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 = 0.30, size = 57, normalized size = 0.83 \begin {gather*} -\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 {gather*}

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 = 0.23, size = 61, normalized size = 0.88 \begin {gather*} \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 {gather*}

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 = 0.00, size = 65, normalized size = 0.94 \begin {gather*} \frac {-252 x^{5} b^{5}-1050 x^{4} b^{4} a-1800 x^{3} b^{3} a^{2}-1575 x^{2} b^{2} a^{3}-700 x b a^{4}-126 a^{5}}{1260 x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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Mupad [B]
time = 0.08, size = 57, normalized size = 0.83 \begin {gather*} -\frac {\frac {a^5}{10}+\frac {5\,a^4\,b\,x}{9}+\frac {5\,a^3\,b^2\,x^2}{4}+\frac {10\,a^2\,b^3\,x^3}{7}+\frac {5\,a\,b^4\,x^4}{6}+\frac {b^5\,x^5}{5}}{x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/x^11,x)

[Out]

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

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