3.2.39 \(\int \frac {(a+b x)^{10}}{x^5} \, dx\)

Optimal. Leaf size=119 \[ -\frac {a^{10}}{4 x^4}-\frac {10 a^9 b}{3 x^3}-\frac {45 a^8 b^2}{2 x^2}-\frac {120 a^7 b^3}{x}+210 a^6 b^4 \log (x)+252 a^5 b^5 x+105 a^4 b^6 x^2+40 a^3 b^7 x^3+\frac {45}{4} a^2 b^8 x^4+2 a b^9 x^5+\frac {b^{10} x^6}{6} \]

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 119, 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 {45 a^8 b^2}{2 x^2}+105 a^4 b^6 x^2+40 a^3 b^7 x^3+\frac {45}{4} a^2 b^8 x^4-\frac {120 a^7 b^3}{x}+252 a^5 b^5 x+210 a^6 b^4 \log (x)-\frac {10 a^9 b}{3 x^3}-\frac {a^{10}}{4 x^4}+2 a b^9 x^5+\frac {b^{10} x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^10/x^5,x]

[Out]

-a^10/(4*x^4) - (10*a^9*b)/(3*x^3) - (45*a^8*b^2)/(2*x^2) - (120*a^7*b^3)/x + 252*a^5*b^5*x + 105*a^4*b^6*x^2
+ 40*a^3*b^7*x^3 + (45*a^2*b^8*x^4)/4 + 2*a*b^9*x^5 + (b^10*x^6)/6 + 210*a^6*b^4*Log[x]

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)^{10}}{x^5} \, dx &=\int \left (252 a^5 b^5+\frac {a^{10}}{x^5}+\frac {10 a^9 b}{x^4}+\frac {45 a^8 b^2}{x^3}+\frac {120 a^7 b^3}{x^2}+\frac {210 a^6 b^4}{x}+210 a^4 b^6 x+120 a^3 b^7 x^2+45 a^2 b^8 x^3+10 a b^9 x^4+b^{10} x^5\right ) \, dx\\ &=-\frac {a^{10}}{4 x^4}-\frac {10 a^9 b}{3 x^3}-\frac {45 a^8 b^2}{2 x^2}-\frac {120 a^7 b^3}{x}+252 a^5 b^5 x+105 a^4 b^6 x^2+40 a^3 b^7 x^3+\frac {45}{4} a^2 b^8 x^4+2 a b^9 x^5+\frac {b^{10} x^6}{6}+210 a^6 b^4 \log (x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 119, normalized size = 1.00 \begin {gather*} -\frac {a^{10}}{4 x^4}-\frac {10 a^9 b}{3 x^3}-\frac {45 a^8 b^2}{2 x^2}-\frac {120 a^7 b^3}{x}+210 a^6 b^4 \log (x)+252 a^5 b^5 x+105 a^4 b^6 x^2+40 a^3 b^7 x^3+\frac {45}{4} a^2 b^8 x^4+2 a b^9 x^5+\frac {b^{10} x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^10/x^5,x]

[Out]

-1/4*a^10/x^4 - (10*a^9*b)/(3*x^3) - (45*a^8*b^2)/(2*x^2) - (120*a^7*b^3)/x + 252*a^5*b^5*x + 105*a^4*b^6*x^2
+ 40*a^3*b^7*x^3 + (45*a^2*b^8*x^4)/4 + 2*a*b^9*x^5 + (b^10*x^6)/6 + 210*a^6*b^4*Log[x]

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^{10}}{x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^10/x^5,x]

[Out]

IntegrateAlgebraic[(a + b*x)^10/x^5, x]

________________________________________________________________________________________

fricas [A]  time = 1.56, size = 114, normalized size = 0.96 \begin {gather*} \frac {2 \, b^{10} x^{10} + 24 \, a b^{9} x^{9} + 135 \, a^{2} b^{8} x^{8} + 480 \, a^{3} b^{7} x^{7} + 1260 \, a^{4} b^{6} x^{6} + 3024 \, a^{5} b^{5} x^{5} + 2520 \, a^{6} b^{4} x^{4} \log \relax (x) - 1440 \, a^{7} b^{3} x^{3} - 270 \, a^{8} b^{2} x^{2} - 40 \, a^{9} b x - 3 \, a^{10}}{12 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^5,x, algorithm="fricas")

[Out]

1/12*(2*b^10*x^10 + 24*a*b^9*x^9 + 135*a^2*b^8*x^8 + 480*a^3*b^7*x^7 + 1260*a^4*b^6*x^6 + 3024*a^5*b^5*x^5 + 2
520*a^6*b^4*x^4*log(x) - 1440*a^7*b^3*x^3 - 270*a^8*b^2*x^2 - 40*a^9*b*x - 3*a^10)/x^4

________________________________________________________________________________________

giac [A]  time = 1.12, size = 111, normalized size = 0.93 \begin {gather*} \frac {1}{6} \, b^{10} x^{6} + 2 \, a b^{9} x^{5} + \frac {45}{4} \, a^{2} b^{8} x^{4} + 40 \, a^{3} b^{7} x^{3} + 105 \, a^{4} b^{6} x^{2} + 252 \, a^{5} b^{5} x + 210 \, a^{6} b^{4} \log \left ({\left | x \right |}\right ) - \frac {1440 \, a^{7} b^{3} x^{3} + 270 \, a^{8} b^{2} x^{2} + 40 \, a^{9} b x + 3 \, a^{10}}{12 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^5,x, algorithm="giac")

[Out]

1/6*b^10*x^6 + 2*a*b^9*x^5 + 45/4*a^2*b^8*x^4 + 40*a^3*b^7*x^3 + 105*a^4*b^6*x^2 + 252*a^5*b^5*x + 210*a^6*b^4
*log(abs(x)) - 1/12*(1440*a^7*b^3*x^3 + 270*a^8*b^2*x^2 + 40*a^9*b*x + 3*a^10)/x^4

________________________________________________________________________________________

maple [A]  time = 0.01, size = 110, normalized size = 0.92 \begin {gather*} \frac {b^{10} x^{6}}{6}+2 a \,b^{9} x^{5}+\frac {45 a^{2} b^{8} x^{4}}{4}+40 a^{3} b^{7} x^{3}+105 a^{4} b^{6} x^{2}+210 a^{6} b^{4} \ln \relax (x )+252 a^{5} b^{5} x -\frac {120 a^{7} b^{3}}{x}-\frac {45 a^{8} b^{2}}{2 x^{2}}-\frac {10 a^{9} b}{3 x^{3}}-\frac {a^{10}}{4 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10/x^5,x)

[Out]

-1/4*a^10/x^4-10/3*a^9*b/x^3-45/2*a^8*b^2/x^2-120*a^7*b^3/x+252*a^5*b^5*x+105*a^4*b^6*x^2+40*a^3*b^7*x^3+45/4*
a^2*b^8*x^4+2*a*b^9*x^5+1/6*b^10*x^6+210*a^6*b^4*ln(x)

________________________________________________________________________________________

maxima [A]  time = 1.27, size = 110, normalized size = 0.92 \begin {gather*} \frac {1}{6} \, b^{10} x^{6} + 2 \, a b^{9} x^{5} + \frac {45}{4} \, a^{2} b^{8} x^{4} + 40 \, a^{3} b^{7} x^{3} + 105 \, a^{4} b^{6} x^{2} + 252 \, a^{5} b^{5} x + 210 \, a^{6} b^{4} \log \relax (x) - \frac {1440 \, a^{7} b^{3} x^{3} + 270 \, a^{8} b^{2} x^{2} + 40 \, a^{9} b x + 3 \, a^{10}}{12 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10/x^5,x, algorithm="maxima")

[Out]

1/6*b^10*x^6 + 2*a*b^9*x^5 + 45/4*a^2*b^8*x^4 + 40*a^3*b^7*x^3 + 105*a^4*b^6*x^2 + 252*a^5*b^5*x + 210*a^6*b^4
*log(x) - 1/12*(1440*a^7*b^3*x^3 + 270*a^8*b^2*x^2 + 40*a^9*b*x + 3*a^10)/x^4

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 110, normalized size = 0.92 \begin {gather*} \frac {b^{10}\,x^6}{6}-\frac {\frac {a^{10}}{4}+\frac {10\,a^9\,b\,x}{3}+\frac {45\,a^8\,b^2\,x^2}{2}+120\,a^7\,b^3\,x^3}{x^4}+252\,a^5\,b^5\,x+2\,a\,b^9\,x^5+105\,a^4\,b^6\,x^2+40\,a^3\,b^7\,x^3+\frac {45\,a^2\,b^8\,x^4}{4}+210\,a^6\,b^4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^10/x^5,x)

[Out]

(b^10*x^6)/6 - (a^10/4 + (45*a^8*b^2*x^2)/2 + 120*a^7*b^3*x^3 + (10*a^9*b*x)/3)/x^4 + 252*a^5*b^5*x + 2*a*b^9*
x^5 + 105*a^4*b^6*x^2 + 40*a^3*b^7*x^3 + (45*a^2*b^8*x^4)/4 + 210*a^6*b^4*log(x)

________________________________________________________________________________________

sympy [A]  time = 0.44, size = 121, normalized size = 1.02 \begin {gather*} 210 a^{6} b^{4} \log {\relax (x )} + 252 a^{5} b^{5} x + 105 a^{4} b^{6} x^{2} + 40 a^{3} b^{7} x^{3} + \frac {45 a^{2} b^{8} x^{4}}{4} + 2 a b^{9} x^{5} + \frac {b^{10} x^{6}}{6} + \frac {- 3 a^{10} - 40 a^{9} b x - 270 a^{8} b^{2} x^{2} - 1440 a^{7} b^{3} x^{3}}{12 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10/x**5,x)

[Out]

210*a**6*b**4*log(x) + 252*a**5*b**5*x + 105*a**4*b**6*x**2 + 40*a**3*b**7*x**3 + 45*a**2*b**8*x**4/4 + 2*a*b*
*9*x**5 + b**10*x**6/6 + (-3*a**10 - 40*a**9*b*x - 270*a**8*b**2*x**2 - 1440*a**7*b**3*x**3)/(12*x**4)

________________________________________________________________________________________