\(\int \frac {(a+b \sqrt {x})^{10}}{x^3} \, dx\) [2160]

Optimal result

Integrand size = 15, antiderivative size = 127 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=-\frac {a^{10}}{2 x^2}-\frac {20 a^9 b}{3 x^{3/2}}-\frac {45 a^8 b^2}{x}-\frac {240 a^7 b^3}{\sqrt {x}}+504 a^5 b^5 \sqrt {x}+210 a^4 b^6 x+80 a^3 b^7 x^{3/2}+\frac {45}{2} a^2 b^8 x^2+4 a b^9 x^{5/2}+\frac {b^{10} x^3}{3}+210 a^6 b^4 \log (x) \]

[Out]

-1/2*a^10/x^2-20/3*a^9*b/x^(3/2)-45*a^8*b^2/x+210*a^4*b^6*x+80*a^3*b^7*x^(3/2)+45/2*a^2*b^8*x^2+4*a*b^9*x^(5/2
)+1/3*b^10*x^3+210*a^6*b^4*ln(x)-240*a^7*b^3/x^(1/2)+504*a^5*b^5*x^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=-\frac {a^{10}}{2 x^2}-\frac {20 a^9 b}{3 x^{3/2}}-\frac {45 a^8 b^2}{x}-\frac {240 a^7 b^3}{\sqrt {x}}+210 a^6 b^4 \log (x)+504 a^5 b^5 \sqrt {x}+210 a^4 b^6 x+80 a^3 b^7 x^{3/2}+\frac {45}{2} a^2 b^8 x^2+4 a b^9 x^{5/2}+\frac {b^{10} x^3}{3} \]

[In]

Int[(a + b*Sqrt[x])^10/x^3,x]

[Out]

-1/2*a^10/x^2 - (20*a^9*b)/(3*x^(3/2)) - (45*a^8*b^2)/x - (240*a^7*b^3)/Sqrt[x] + 504*a^5*b^5*Sqrt[x] + 210*a^
4*b^6*x + 80*a^3*b^7*x^(3/2) + (45*a^2*b^8*x^2)/2 + 4*a*b^9*x^(5/2) + (b^10*x^3)/3 + 210*a^6*b^4*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])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {(a+b x)^{10}}{x^5} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\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,x,\sqrt {x}\right ) \\ & = -\frac {a^{10}}{2 x^2}-\frac {20 a^9 b}{3 x^{3/2}}-\frac {45 a^8 b^2}{x}-\frac {240 a^7 b^3}{\sqrt {x}}+504 a^5 b^5 \sqrt {x}+210 a^4 b^6 x+80 a^3 b^7 x^{3/2}+\frac {45}{2} a^2 b^8 x^2+4 a b^9 x^{5/2}+\frac {b^{10} x^3}{3}+210 a^6 b^4 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=\frac {-3 a^{10}-40 a^9 b \sqrt {x}-270 a^8 b^2 x-1440 a^7 b^3 x^{3/2}+3024 a^5 b^5 x^{5/2}+1260 a^4 b^6 x^3+480 a^3 b^7 x^{7/2}+135 a^2 b^8 x^4+24 a b^9 x^{9/2}+2 b^{10} x^5}{6 x^2}+420 a^6 b^4 \log \left (\sqrt {x}\right ) \]

[In]

Integrate[(a + b*Sqrt[x])^10/x^3,x]

[Out]

(-3*a^10 - 40*a^9*b*Sqrt[x] - 270*a^8*b^2*x - 1440*a^7*b^3*x^(3/2) + 3024*a^5*b^5*x^(5/2) + 1260*a^4*b^6*x^3 +
480*a^3*b^7*x^(7/2) + 135*a^2*b^8*x^4 + 24*a*b^9*x^(9/2) + 2*b^10*x^5)/(6*x^2) + 420*a^6*b^4*Log[Sqrt[x]]

Maple [A] (verified)

Time = 3.49 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(-\frac {a^{10}}{2 x^{2}}-\frac {20 a^{9} b}{3 x^{\frac {3}{2}}}-\frac {45 a^{8} b^{2}}{x}+210 a^{4} b^{6} x +80 a^{3} b^{7} x^{\frac {3}{2}}+\frac {45 b^{8} x^{2} a^{2}}{2}+4 a \,b^{9} x^{\frac {5}{2}}+\frac {b^{10} x^{3}}{3}+210 a^{6} b^{4} \ln \left (x \right )-\frac {240 a^{7} b^{3}}{\sqrt {x}}+504 a^{5} b^{5} \sqrt {x}\)	\(110\)
default	\(-\frac {a^{10}}{2 x^{2}}-\frac {20 a^{9} b}{3 x^{\frac {3}{2}}}-\frac {45 a^{8} b^{2}}{x}+210 a^{4} b^{6} x +80 a^{3} b^{7} x^{\frac {3}{2}}+\frac {45 b^{8} x^{2} a^{2}}{2}+4 a \,b^{9} x^{\frac {5}{2}}+\frac {b^{10} x^{3}}{3}+210 a^{6} b^{4} \ln \left (x \right )-\frac {240 a^{7} b^{3}}{\sqrt {x}}+504 a^{5} b^{5} \sqrt {x}\)	\(110\)
trager	\(\frac {\left (-1+x \right ) \left (2 b^{10} x^{4}+135 a^{2} b^{8} x^{3}+2 b^{10} x^{3}+1260 a^{4} b^{6} x^{2}+135 b^{8} x^{2} a^{2}+2 x^{2} b^{10}+3 a^{10} x +270 a^{8} b^{2} x +3 a^{10}\right )}{6 x^{2}}-\frac {4 \left (-3 b^{8} x^{4}-60 a^{2} b^{6} x^{3}-378 a^{4} b^{4} x^{2}+180 a^{6} b^{2} x +5 a^{8}\right ) a b}{3 x^{\frac {3}{2}}}-210 a^{6} b^{4} \ln \left (\frac {1}{x}\right )\)	\(152\)

[In]

int((a+b*x^(1/2))^10/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*a^10/x^2-20/3*a^9*b/x^(3/2)-45*a^8*b^2/x+210*a^4*b^6*x+80*a^3*b^7*x^(3/2)+45/2*b^8*x^2*a^2+4*a*b^9*x^(5/2
)+1/3*b^10*x^3+210*a^6*b^4*ln(x)-240*a^7*b^3/x^(1/2)+504*a^5*b^5*x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=\frac {2 \, b^{10} x^{5} + 135 \, a^{2} b^{8} x^{4} + 1260 \, a^{4} b^{6} x^{3} + 2520 \, a^{6} b^{4} x^{2} \log \left (\sqrt {x}\right ) - 270 \, a^{8} b^{2} x - 3 \, a^{10} + 8 \, {\left (3 \, a b^{9} x^{4} + 60 \, a^{3} b^{7} x^{3} + 378 \, a^{5} b^{5} x^{2} - 180 \, a^{7} b^{3} x - 5 \, a^{9} b\right )} \sqrt {x}}{6 \, x^{2}} \]

[In]

integrate((a+b*x^(1/2))^10/x^3,x, algorithm="fricas")

[Out]

1/6*(2*b^10*x^5 + 135*a^2*b^8*x^4 + 1260*a^4*b^6*x^3 + 2520*a^6*b^4*x^2*log(sqrt(x)) - 270*a^8*b^2*x - 3*a^10
+ 8*(3*a*b^9*x^4 + 60*a^3*b^7*x^3 + 378*a^5*b^5*x^2 - 180*a^7*b^3*x - 5*a^9*b)*sqrt(x))/x^2

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=- \frac {a^{10}}{2 x^{2}} - \frac {20 a^{9} b}{3 x^{\frac {3}{2}}} - \frac {45 a^{8} b^{2}}{x} - \frac {240 a^{7} b^{3}}{\sqrt {x}} + 210 a^{6} b^{4} \log {\left (x \right )} + 504 a^{5} b^{5} \sqrt {x} + 210 a^{4} b^{6} x + 80 a^{3} b^{7} x^{\frac {3}{2}} + \frac {45 a^{2} b^{8} x^{2}}{2} + 4 a b^{9} x^{\frac {5}{2}} + \frac {b^{10} x^{3}}{3} \]

[In]

integrate((a+b*x**(1/2))**10/x**3,x)

[Out]

-a**10/(2*x**2) - 20*a**9*b/(3*x**(3/2)) - 45*a**8*b**2/x - 240*a**7*b**3/sqrt(x) + 210*a**6*b**4*log(x) + 504
*a**5*b**5*sqrt(x) + 210*a**4*b**6*x + 80*a**3*b**7*x**(3/2) + 45*a**2*b**8*x**2/2 + 4*a*b**9*x**(5/2) + b**10
*x**3/3

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=\frac {1}{3} \, b^{10} x^{3} + 4 \, a b^{9} x^{\frac {5}{2}} + \frac {45}{2} \, a^{2} b^{8} x^{2} + 80 \, a^{3} b^{7} x^{\frac {3}{2}} + 210 \, a^{4} b^{6} x + 210 \, a^{6} b^{4} \log \left (x\right ) + 504 \, a^{5} b^{5} \sqrt {x} - \frac {1440 \, a^{7} b^{3} x^{\frac {3}{2}} + 270 \, a^{8} b^{2} x + 40 \, a^{9} b \sqrt {x} + 3 \, a^{10}}{6 \, x^{2}} \]

[In]

integrate((a+b*x^(1/2))^10/x^3,x, algorithm="maxima")

[Out]

1/3*b^10*x^3 + 4*a*b^9*x^(5/2) + 45/2*a^2*b^8*x^2 + 80*a^3*b^7*x^(3/2) + 210*a^4*b^6*x + 210*a^6*b^4*log(x) +
504*a^5*b^5*sqrt(x) - 1/6*(1440*a^7*b^3*x^(3/2) + 270*a^8*b^2*x + 40*a^9*b*sqrt(x) + 3*a^10)/x^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=\frac {1}{3} \, b^{10} x^{3} + 4 \, a b^{9} x^{\frac {5}{2}} + \frac {45}{2} \, a^{2} b^{8} x^{2} + 80 \, a^{3} b^{7} x^{\frac {3}{2}} + 210 \, a^{4} b^{6} x + 210 \, a^{6} b^{4} \log \left ({\left | x \right |}\right ) + 504 \, a^{5} b^{5} \sqrt {x} - \frac {1440 \, a^{7} b^{3} x^{\frac {3}{2}} + 270 \, a^{8} b^{2} x + 40 \, a^{9} b \sqrt {x} + 3 \, a^{10}}{6 \, x^{2}} \]

[In]

integrate((a+b*x^(1/2))^10/x^3,x, algorithm="giac")

[Out]

1/3*b^10*x^3 + 4*a*b^9*x^(5/2) + 45/2*a^2*b^8*x^2 + 80*a^3*b^7*x^(3/2) + 210*a^4*b^6*x + 210*a^6*b^4*log(abs(x
)) + 504*a^5*b^5*sqrt(x) - 1/6*(1440*a^7*b^3*x^(3/2) + 270*a^8*b^2*x + 40*a^9*b*sqrt(x) + 3*a^10)/x^2

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^3} \, dx=\frac {b^{10}\,x^3}{3}-\frac {\frac {a^{10}}{2}+45\,a^8\,b^2\,x+\frac {20\,a^9\,b\,\sqrt {x}}{3}+240\,a^7\,b^3\,x^{3/2}}{x^2}+420\,a^6\,b^4\,\ln \left (\sqrt {x}\right )+210\,a^4\,b^6\,x+4\,a\,b^9\,x^{5/2}+\frac {45\,a^2\,b^8\,x^2}{2}+504\,a^5\,b^5\,\sqrt {x}+80\,a^3\,b^7\,x^{3/2} \]

[In]

int((a + b*x^(1/2))^10/x^3,x)

[Out]

(b^10*x^3)/3 - (a^10/2 + 45*a^8*b^2*x + (20*a^9*b*x^(1/2))/3 + 240*a^7*b^3*x^(3/2))/x^2 + 420*a^6*b^4*log(x^(1
/2)) + 210*a^4*b^6*x + 4*a*b^9*x^(5/2) + (45*a^2*b^8*x^2)/2 + 504*a^5*b^5*x^(1/2) + 80*a^3*b^7*x^(3/2)