3.1.95 \(\int \frac {(a+b x)^5}{x^{12}} \, dx\)

Optimal. Leaf size=69 \[ -\frac {a^5}{11 x^{11}}-\frac {a^4 b}{2 x^{10}}-\frac {10 a^3 b^2}{9 x^9}-\frac {5 a^2 b^3}{4 x^8}-\frac {5 a b^4}{7 x^7}-\frac {b^5}{6 x^6} \]

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Rubi [A]  time = 0.02, 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 = {43} \begin {gather*} -\frac {10 a^3 b^2}{9 x^9}-\frac {5 a^2 b^3}{4 x^8}-\frac {a^4 b}{2 x^{10}}-\frac {a^5}{11 x^{11}}-\frac {5 a b^4}{7 x^7}-\frac {b^5}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^5/x^12,x]

[Out]

-a^5/(11*x^11) - (a^4*b)/(2*x^10) - (10*a^3*b^2)/(9*x^9) - (5*a^2*b^3)/(4*x^8) - (5*a*b^4)/(7*x^7) - b^5/(6*x^
6)

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^{12}} \, dx &=\int \left (\frac {a^5}{x^{12}}+\frac {5 a^4 b}{x^{11}}+\frac {10 a^3 b^2}{x^{10}}+\frac {10 a^2 b^3}{x^9}+\frac {5 a b^4}{x^8}+\frac {b^5}{x^7}\right ) \, dx\\ &=-\frac {a^5}{11 x^{11}}-\frac {a^4 b}{2 x^{10}}-\frac {10 a^3 b^2}{9 x^9}-\frac {5 a^2 b^3}{4 x^8}-\frac {5 a b^4}{7 x^7}-\frac {b^5}{6 x^6}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 69, normalized size = 1.00 \begin {gather*} -\frac {a^5}{11 x^{11}}-\frac {a^4 b}{2 x^{10}}-\frac {10 a^3 b^2}{9 x^9}-\frac {5 a^2 b^3}{4 x^8}-\frac {5 a b^4}{7 x^7}-\frac {b^5}{6 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^5/x^12,x]

[Out]

-1/11*a^5/x^11 - (a^4*b)/(2*x^10) - (10*a^3*b^2)/(9*x^9) - (5*a^2*b^3)/(4*x^8) - (5*a*b^4)/(7*x^7) - b^5/(6*x^
6)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^5}{x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^5/x^12,x]

[Out]

IntegrateAlgebraic[(a + b*x)^5/x^12, x]

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fricas [A]  time = 0.83, size = 57, normalized size = 0.83 \begin {gather*} -\frac {462 \, b^{5} x^{5} + 1980 \, a b^{4} x^{4} + 3465 \, a^{2} b^{3} x^{3} + 3080 \, a^{3} b^{2} x^{2} + 1386 \, a^{4} b x + 252 \, a^{5}}{2772 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^12,x, algorithm="fricas")

[Out]

-1/2772*(462*b^5*x^5 + 1980*a*b^4*x^4 + 3465*a^2*b^3*x^3 + 3080*a^3*b^2*x^2 + 1386*a^4*b*x + 252*a^5)/x^11

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giac [A]  time = 1.38, size = 57, normalized size = 0.83 \begin {gather*} -\frac {462 \, b^{5} x^{5} + 1980 \, a b^{4} x^{4} + 3465 \, a^{2} b^{3} x^{3} + 3080 \, a^{3} b^{2} x^{2} + 1386 \, a^{4} b x + 252 \, a^{5}}{2772 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^12,x, algorithm="giac")

[Out]

-1/2772*(462*b^5*x^5 + 1980*a*b^4*x^4 + 3465*a^2*b^3*x^3 + 3080*a^3*b^2*x^2 + 1386*a^4*b*x + 252*a^5)/x^11

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maple [A]  time = 0.01, size = 58, normalized size = 0.84 \begin {gather*} -\frac {b^{5}}{6 x^{6}}-\frac {5 a \,b^{4}}{7 x^{7}}-\frac {5 a^{2} b^{3}}{4 x^{8}}-\frac {10 a^{3} b^{2}}{9 x^{9}}-\frac {a^{4} b}{2 x^{10}}-\frac {a^{5}}{11 x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^5/x^12,x)

[Out]

-1/11*a^5/x^11-1/2*a^4*b/x^10-10/9*a^3*b^2/x^9-5/4*a^2*b^3/x^8-5/7*a*b^4/x^7-1/6*b^5/x^6

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maxima [A]  time = 1.31, size = 57, normalized size = 0.83 \begin {gather*} -\frac {462 \, b^{5} x^{5} + 1980 \, a b^{4} x^{4} + 3465 \, a^{2} b^{3} x^{3} + 3080 \, a^{3} b^{2} x^{2} + 1386 \, a^{4} b x + 252 \, a^{5}}{2772 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^5/x^12,x, algorithm="maxima")

[Out]

-1/2772*(462*b^5*x^5 + 1980*a*b^4*x^4 + 3465*a^2*b^3*x^3 + 3080*a^3*b^2*x^2 + 1386*a^4*b*x + 252*a^5)/x^11

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mupad [B]  time = 0.04, size = 57, normalized size = 0.83 \begin {gather*} -\frac {\frac {a^5}{11}+\frac {a^4\,b\,x}{2}+\frac {10\,a^3\,b^2\,x^2}{9}+\frac {5\,a^2\,b^3\,x^3}{4}+\frac {5\,a\,b^4\,x^4}{7}+\frac {b^5\,x^5}{6}}{x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^5/x^12,x)

[Out]

-(a^5/11 + (b^5*x^5)/6 + (5*a*b^4*x^4)/7 + (10*a^3*b^2*x^2)/9 + (5*a^2*b^3*x^3)/4 + (a^4*b*x)/2)/x^11

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sympy [A]  time = 0.58, size = 61, normalized size = 0.88 \begin {gather*} \frac {- 252 a^{5} - 1386 a^{4} b x - 3080 a^{3} b^{2} x^{2} - 3465 a^{2} b^{3} x^{3} - 1980 a b^{4} x^{4} - 462 b^{5} x^{5}}{2772 x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**5/x**12,x)

[Out]

(-252*a**5 - 1386*a**4*b*x - 3080*a**3*b**2*x**2 - 3465*a**2*b**3*x**3 - 1980*a*b**4*x**4 - 462*b**5*x**5)/(27
72*x**11)

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