3.2.40 \(\int \frac {(a+b x)^{10}}{x^6} \, dx\) [140]

Optimal. Leaf size=117 \[ -\frac {a^{10}}{5 x^5}-\frac {5 a^9 b}{2 x^4}-\frac {15 a^8 b^2}{x^3}-\frac {60 a^7 b^3}{x^2}-\frac {210 a^6 b^4}{x}+210 a^4 b^6 x+60 a^3 b^7 x^2+15 a^2 b^8 x^3+\frac {5}{2} a b^9 x^4+\frac {b^{10} x^5}{5}+252 a^5 b^5 \log (x) \]

[Out]

-1/5*a^10/x^5-5/2*a^9*b/x^4-15*a^8*b^2/x^3-60*a^7*b^3/x^2-210*a^6*b^4/x+210*a^4*b^6*x+60*a^3*b^7*x^2+15*a^2*b^
8*x^3+5/2*a*b^9*x^4+1/5*b^10*x^5+252*a^5*b^5*ln(x)

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Rubi [A]
time = 0.03, antiderivative size = 117, 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^{10}}{5 x^5}-\frac {5 a^9 b}{2 x^4}-\frac {15 a^8 b^2}{x^3}-\frac {60 a^7 b^3}{x^2}-\frac {210 a^6 b^4}{x}+252 a^5 b^5 \log (x)+210 a^4 b^6 x+60 a^3 b^7 x^2+15 a^2 b^8 x^3+\frac {5}{2} a b^9 x^4+\frac {b^{10} x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^6,x]

[Out]

-1/5*a^10/x^5 - (5*a^9*b)/(2*x^4) - (15*a^8*b^2)/x^3 - (60*a^7*b^3)/x^2 - (210*a^6*b^4)/x + 210*a^4*b^6*x + 60
*a^3*b^7*x^2 + 15*a^2*b^8*x^3 + (5*a*b^9*x^4)/2 + (b^10*x^5)/5 + 252*a^5*b^5*Log[x]

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)^{10}}{x^6} \, dx &=\int \left (210 a^4 b^6+\frac {a^{10}}{x^6}+\frac {10 a^9 b}{x^5}+\frac {45 a^8 b^2}{x^4}+\frac {120 a^7 b^3}{x^3}+\frac {210 a^6 b^4}{x^2}+\frac {252 a^5 b^5}{x}+120 a^3 b^7 x+45 a^2 b^8 x^2+10 a b^9 x^3+b^{10} x^4\right ) \, dx\\ &=-\frac {a^{10}}{5 x^5}-\frac {5 a^9 b}{2 x^4}-\frac {15 a^8 b^2}{x^3}-\frac {60 a^7 b^3}{x^2}-\frac {210 a^6 b^4}{x}+210 a^4 b^6 x+60 a^3 b^7 x^2+15 a^2 b^8 x^3+\frac {5}{2} a b^9 x^4+\frac {b^{10} x^5}{5}+252 a^5 b^5 \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 117, normalized size = 1.00 \begin {gather*} -\frac {a^{10}}{5 x^5}-\frac {5 a^9 b}{2 x^4}-\frac {15 a^8 b^2}{x^3}-\frac {60 a^7 b^3}{x^2}-\frac {210 a^6 b^4}{x}+210 a^4 b^6 x+60 a^3 b^7 x^2+15 a^2 b^8 x^3+\frac {5}{2} a b^9 x^4+\frac {b^{10} x^5}{5}+252 a^5 b^5 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^6,x]

[Out]

-1/5*a^10/x^5 - (5*a^9*b)/(2*x^4) - (15*a^8*b^2)/x^3 - (60*a^7*b^3)/x^2 - (210*a^6*b^4)/x + 210*a^4*b^6*x + 60
*a^3*b^7*x^2 + 15*a^2*b^8*x^3 + (5*a*b^9*x^4)/2 + (b^10*x^5)/5 + 252*a^5*b^5*Log[x]

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Maple [A]
time = 0.08, size = 110, normalized size = 0.94

method	result	size
default	\(-\frac {a^{10}}{5 x^{5}}-\frac {5 a^{9} b}{2 x^{4}}-\frac {15 a^{8} b^{2}}{x^{3}}-\frac {60 a^{7} b^{3}}{x^{2}}-\frac {210 a^{6} b^{4}}{x}+210 a^{4} b^{6} x +60 a^{3} b^{7} x^{2}+15 a^{2} b^{8} x^{3}+\frac {5 a \,b^{9} x^{4}}{2}+\frac {b^{10} x^{5}}{5}+252 a^{5} b^{5} \ln \left (x \right )\)	\(110\)
risch	\(\frac {b^{10} x^{5}}{5}+\frac {5 a \,b^{9} x^{4}}{2}+15 a^{2} b^{8} x^{3}+60 a^{3} b^{7} x^{2}+210 a^{4} b^{6} x +\frac {-210 a^{6} b^{4} x^{4}-60 a^{7} b^{3} x^{3}-15 a^{8} b^{2} x^{2}-\frac {5}{2} a^{9} b x -\frac {1}{5} a^{10}}{x^{5}}+252 a^{5} b^{5} \ln \left (x \right )\)	\(110\)
norman	\(\frac {-\frac {1}{5} a^{10}+\frac {1}{5} b^{10} x^{10}+\frac {5}{2} a \,b^{9} x^{9}+15 a^{2} b^{8} x^{8}+60 a^{3} b^{7} x^{7}+210 a^{4} b^{6} x^{6}-210 a^{6} b^{4} x^{4}-60 a^{7} b^{3} x^{3}-15 a^{8} b^{2} x^{2}-\frac {5}{2} a^{9} b x}{x^{5}}+252 a^{5} b^{5} \ln \left (x \right )\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10/x^6,x,method=_RETURNVERBOSE)

[Out]

-1/5*a^10/x^5-5/2*a^9*b/x^4-15*a^8*b^2/x^3-60*a^7*b^3/x^2-210*a^6*b^4/x+210*a^4*b^6*x+60*a^3*b^7*x^2+15*a^2*b^
8*x^3+5/2*a*b^9*x^4+1/5*b^10*x^5+252*a^5*b^5*ln(x)

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Maxima [A]
time = 0.35, size = 110, normalized size = 0.94 \begin {gather*} \frac {1}{5} \, b^{10} x^{5} + \frac {5}{2} \, a b^{9} x^{4} + 15 \, a^{2} b^{8} x^{3} + 60 \, a^{3} b^{7} x^{2} + 210 \, a^{4} b^{6} x + 252 \, a^{5} b^{5} \log \left (x\right ) - \frac {2100 \, a^{6} b^{4} x^{4} + 600 \, a^{7} b^{3} x^{3} + 150 \, a^{8} b^{2} x^{2} + 25 \, a^{9} b x + 2 \, a^{10}}{10 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^6,x, algorithm="maxima")

[Out]

1/5*b^10*x^5 + 5/2*a*b^9*x^4 + 15*a^2*b^8*x^3 + 60*a^3*b^7*x^2 + 210*a^4*b^6*x + 252*a^5*b^5*log(x) - 1/10*(21
00*a^6*b^4*x^4 + 600*a^7*b^3*x^3 + 150*a^8*b^2*x^2 + 25*a^9*b*x + 2*a^10)/x^5

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Fricas [A]
time = 0.48, size = 114, normalized size = 0.97 \begin {gather*} \frac {2 \, b^{10} x^{10} + 25 \, a b^{9} x^{9} + 150 \, a^{2} b^{8} x^{8} + 600 \, a^{3} b^{7} x^{7} + 2100 \, a^{4} b^{6} x^{6} + 2520 \, a^{5} b^{5} x^{5} \log \left (x\right ) - 2100 \, a^{6} b^{4} x^{4} - 600 \, a^{7} b^{3} x^{3} - 150 \, a^{8} b^{2} x^{2} - 25 \, a^{9} b x - 2 \, a^{10}}{10 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^6,x, algorithm="fricas")

[Out]

1/10*(2*b^10*x^10 + 25*a*b^9*x^9 + 150*a^2*b^8*x^8 + 600*a^3*b^7*x^7 + 2100*a^4*b^6*x^6 + 2520*a^5*b^5*x^5*log
(x) - 2100*a^6*b^4*x^4 - 600*a^7*b^3*x^3 - 150*a^8*b^2*x^2 - 25*a^9*b*x - 2*a^10)/x^5

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Sympy [A]
time = 0.17, size = 121, normalized size = 1.03 \begin {gather*} 252 a^{5} b^{5} \log {\left (x \right )} + 210 a^{4} b^{6} x + 60 a^{3} b^{7} x^{2} + 15 a^{2} b^{8} x^{3} + \frac {5 a b^{9} x^{4}}{2} + \frac {b^{10} x^{5}}{5} + \frac {- 2 a^{10} - 25 a^{9} b x - 150 a^{8} b^{2} x^{2} - 600 a^{7} b^{3} x^{3} - 2100 a^{6} b^{4} x^{4}}{10 x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10/x**6,x)

[Out]

252*a**5*b**5*log(x) + 210*a**4*b**6*x + 60*a**3*b**7*x**2 + 15*a**2*b**8*x**3 + 5*a*b**9*x**4/2 + b**10*x**5/
5 + (-2*a**10 - 25*a**9*b*x - 150*a**8*b**2*x**2 - 600*a**7*b**3*x**3 - 2100*a**6*b**4*x**4)/(10*x**5)

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Giac [A]
time = 1.58, size = 111, normalized size = 0.95 \begin {gather*} \frac {1}{5} \, b^{10} x^{5} + \frac {5}{2} \, a b^{9} x^{4} + 15 \, a^{2} b^{8} x^{3} + 60 \, a^{3} b^{7} x^{2} + 210 \, a^{4} b^{6} x + 252 \, a^{5} b^{5} \log \left ({\left | x \right |}\right ) - \frac {2100 \, a^{6} b^{4} x^{4} + 600 \, a^{7} b^{3} x^{3} + 150 \, a^{8} b^{2} x^{2} + 25 \, a^{9} b x + 2 \, a^{10}}{10 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^6,x, algorithm="giac")

[Out]

1/5*b^10*x^5 + 5/2*a*b^9*x^4 + 15*a^2*b^8*x^3 + 60*a^3*b^7*x^2 + 210*a^4*b^6*x + 252*a^5*b^5*log(abs(x)) - 1/1
0*(2100*a^6*b^4*x^4 + 600*a^7*b^3*x^3 + 150*a^8*b^2*x^2 + 25*a^9*b*x + 2*a^10)/x^5

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Mupad [B]
time = 0.10, size = 110, normalized size = 0.94 \begin {gather*} \frac {b^{10}\,x^5}{5}-\frac {\frac {a^{10}}{5}+\frac {5\,a^9\,b\,x}{2}+15\,a^8\,b^2\,x^2+60\,a^7\,b^3\,x^3+210\,a^6\,b^4\,x^4}{x^5}+210\,a^4\,b^6\,x+\frac {5\,a\,b^9\,x^4}{2}+60\,a^3\,b^7\,x^2+15\,a^2\,b^8\,x^3+252\,a^5\,b^5\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^10/x^6,x)

[Out]

(b^10*x^5)/5 - (a^10/5 + 15*a^8*b^2*x^2 + 60*a^7*b^3*x^3 + 210*a^6*b^4*x^4 + (5*a^9*b*x)/2)/x^5 + 210*a^4*b^6*
x + (5*a*b^9*x^4)/2 + 60*a^3*b^7*x^2 + 15*a^2*b^8*x^3 + 252*a^5*b^5*log(x)

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