\(\int \frac {(a+b \sqrt {x})^{10}}{x^4} \, dx\) [2161]

Optimal result

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

[Out]

-1/3*a^10/x^3-4*a^9*b/x^(5/2)-45/2*a^8*b^2/x^2-80*a^7*b^3/x^(3/2)-210*a^6*b^4/x+45*a^2*b^8*x+20/3*a*b^9*x^(3/2
)+1/2*b^10*x^2+210*a^4*b^6*ln(x)-504*a^5*b^5/x^(1/2)+240*a^3*b^7*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} \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^4} \, dx=-\frac {a^{10}}{3 x^3}-\frac {4 a^9 b}{x^{5/2}}-\frac {45 a^8 b^2}{2 x^2}-\frac {80 a^7 b^3}{x^{3/2}}-\frac {210 a^6 b^4}{x}-\frac {504 a^5 b^5}{\sqrt {x}}+210 a^4 b^6 \log (x)+240 a^3 b^7 \sqrt {x}+45 a^2 b^8 x+\frac {20}{3} a b^9 x^{3/2}+\frac {b^{10} x^2}{2} \]

[In]

Int[(a + b*Sqrt[x])^10/x^4,x]

[Out]

-1/3*a^10/x^3 - (4*a^9*b)/x^(5/2) - (45*a^8*b^2)/(2*x^2) - (80*a^7*b^3)/x^(3/2) - (210*a^6*b^4)/x - (504*a^5*b
^5)/Sqrt[x] + 240*a^3*b^7*Sqrt[x] + 45*a^2*b^8*x + (20*a*b^9*x^(3/2))/3 + (b^10*x^2)/2 + 210*a^4*b^6*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^7} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (120 a^3 b^7+\frac {a^{10}}{x^7}+\frac {10 a^9 b}{x^6}+\frac {45 a^8 b^2}{x^5}+\frac {120 a^7 b^3}{x^4}+\frac {210 a^6 b^4}{x^3}+\frac {252 a^5 b^5}{x^2}+\frac {210 a^4 b^6}{x}+45 a^2 b^8 x+10 a b^9 x^2+b^{10} x^3\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {a^{10}}{3 x^3}-\frac {4 a^9 b}{x^{5/2}}-\frac {45 a^8 b^2}{2 x^2}-\frac {80 a^7 b^3}{x^{3/2}}-\frac {210 a^6 b^4}{x}-\frac {504 a^5 b^5}{\sqrt {x}}+240 a^3 b^7 \sqrt {x}+45 a^2 b^8 x+\frac {20}{3} a b^9 x^{3/2}+\frac {b^{10} x^2}{2}+210 a^4 b^6 \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^4} \, dx=-\frac {2 a^{10}+24 a^9 b \sqrt {x}+135 a^8 b^2 x+480 a^7 b^3 x^{3/2}+1260 a^6 b^4 x^2+3024 a^5 b^5 x^{5/2}-1440 a^3 b^7 x^{7/2}-270 a^2 b^8 x^4-40 a b^9 x^{9/2}-3 b^{10} x^5}{6 x^3}+210 a^4 b^6 \log (x) \]

[In]

Integrate[(a + b*Sqrt[x])^10/x^4,x]

[Out]

-1/6*(2*a^10 + 24*a^9*b*Sqrt[x] + 135*a^8*b^2*x + 480*a^7*b^3*x^(3/2) + 1260*a^6*b^4*x^2 + 3024*a^5*b^5*x^(5/2
) - 1440*a^3*b^7*x^(7/2) - 270*a^2*b^8*x^4 - 40*a*b^9*x^(9/2) - 3*b^10*x^5)/x^3 + 210*a^4*b^6*Log[x]

Maple [A] (verified)

Time = 3.52 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(-\frac {a^{10}}{3 x^{3}}-\frac {4 a^{9} b}{x^{\frac {5}{2}}}-\frac {45 a^{8} b^{2}}{2 x^{2}}-\frac {80 a^{7} b^{3}}{x^{\frac {3}{2}}}-\frac {210 a^{6} b^{4}}{x}+45 b^{8} x \,a^{2}+\frac {20 a \,b^{9} x^{\frac {3}{2}}}{3}+\frac {x^{2} b^{10}}{2}+210 a^{4} b^{6} \ln \left (x \right )-\frac {504 a^{5} b^{5}}{\sqrt {x}}+240 a^{3} b^{7} \sqrt {x}\)	\(110\)
default	\(-\frac {a^{10}}{3 x^{3}}-\frac {4 a^{9} b}{x^{\frac {5}{2}}}-\frac {45 a^{8} b^{2}}{2 x^{2}}-\frac {80 a^{7} b^{3}}{x^{\frac {3}{2}}}-\frac {210 a^{6} b^{4}}{x}+45 b^{8} x \,a^{2}+\frac {20 a \,b^{9} x^{\frac {3}{2}}}{3}+\frac {x^{2} b^{10}}{2}+210 a^{4} b^{6} \ln \left (x \right )-\frac {504 a^{5} b^{5}}{\sqrt {x}}+240 a^{3} b^{7} \sqrt {x}\)	\(110\)
trager	\(\frac {\left (-1+x \right ) \left (3 b^{10} x^{4}+270 a^{2} b^{8} x^{3}+3 b^{10} x^{3}+2 a^{10} x^{2}+135 a^{8} b^{2} x^{2}+1260 x^{2} a^{6} b^{4}+2 a^{10} x +135 a^{8} b^{2} x +2 a^{10}\right )}{6 x^{3}}-\frac {4 \left (-5 b^{8} x^{4}-180 a^{2} b^{6} x^{3}+378 a^{4} b^{4} x^{2}+60 a^{6} b^{2} x +3 a^{8}\right ) a b}{3 x^{\frac {5}{2}}}+210 a^{4} b^{6} \ln \left (x \right )\)	\(150\)

[In]

int((a+b*x^(1/2))^10/x^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^4} \, dx=\frac {3 \, b^{10} x^{5} + 270 \, a^{2} b^{8} x^{4} + 2520 \, a^{4} b^{6} x^{3} \log \left (\sqrt {x}\right ) - 1260 \, a^{6} b^{4} x^{2} - 135 \, a^{8} b^{2} x - 2 \, a^{10} + 8 \, {\left (5 \, a b^{9} x^{4} + 180 \, a^{3} b^{7} x^{3} - 378 \, a^{5} b^{5} x^{2} - 60 \, a^{7} b^{3} x - 3 \, a^{9} b\right )} \sqrt {x}}{6 \, x^{3}} \]

[In]

integrate((a+b*x^(1/2))^10/x^4,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

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

[In]

integrate((a+b*x**(1/2))**10/x**4,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^4} \, dx=\frac {1}{2} \, b^{10} x^{2} + \frac {20}{3} \, a b^{9} x^{\frac {3}{2}} + 45 \, a^{2} b^{8} x + 210 \, a^{4} b^{6} \log \left (x\right ) + 240 \, a^{3} b^{7} \sqrt {x} - \frac {3024 \, a^{5} b^{5} x^{\frac {5}{2}} + 1260 \, a^{6} b^{4} x^{2} + 480 \, a^{7} b^{3} x^{\frac {3}{2}} + 135 \, a^{8} b^{2} x + 24 \, a^{9} b \sqrt {x} + 2 \, a^{10}}{6 \, x^{3}} \]

[In]

integrate((a+b*x^(1/2))^10/x^4,x, algorithm="maxima")

[Out]

1/2*b^10*x^2 + 20/3*a*b^9*x^(3/2) + 45*a^2*b^8*x + 210*a^4*b^6*log(x) + 240*a^3*b^7*sqrt(x) - 1/6*(3024*a^5*b^
5*x^(5/2) + 1260*a^6*b^4*x^2 + 480*a^7*b^3*x^(3/2) + 135*a^8*b^2*x + 24*a^9*b*sqrt(x) + 2*a^10)/x^3

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.87 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^4} \, dx=\frac {1}{2} \, b^{10} x^{2} + \frac {20}{3} \, a b^{9} x^{\frac {3}{2}} + 45 \, a^{2} b^{8} x + 210 \, a^{4} b^{6} \log \left ({\left | x \right |}\right ) + 240 \, a^{3} b^{7} \sqrt {x} - \frac {3024 \, a^{5} b^{5} x^{\frac {5}{2}} + 1260 \, a^{6} b^{4} x^{2} + 480 \, a^{7} b^{3} x^{\frac {3}{2}} + 135 \, a^{8} b^{2} x + 24 \, a^{9} b \sqrt {x} + 2 \, a^{10}}{6 \, x^{3}} \]

[In]

integrate((a+b*x^(1/2))^10/x^4,x, algorithm="giac")

[Out]

1/2*b^10*x^2 + 20/3*a*b^9*x^(3/2) + 45*a^2*b^8*x + 210*a^4*b^6*log(abs(x)) + 240*a^3*b^7*sqrt(x) - 1/6*(3024*a
^5*b^5*x^(5/2) + 1260*a^6*b^4*x^2 + 480*a^7*b^3*x^(3/2) + 135*a^8*b^2*x + 24*a^9*b*sqrt(x) + 2*a^10)/x^3

Mupad [B] (verification not implemented)

Time = 5.76 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+b \sqrt {x}\right )^{10}}{x^4} \, dx=\frac {b^{10}\,x^2}{2}-\frac {\frac {a^{10}}{3}+\frac {45\,a^8\,b^2\,x}{2}+4\,a^9\,b\,\sqrt {x}+210\,a^6\,b^4\,x^2+80\,a^7\,b^3\,x^{3/2}+504\,a^5\,b^5\,x^{5/2}}{x^3}+420\,a^4\,b^6\,\ln \left (\sqrt {x}\right )+45\,a^2\,b^8\,x+\frac {20\,a\,b^9\,x^{3/2}}{3}+240\,a^3\,b^7\,\sqrt {x} \]

[In]

int((a + b*x^(1/2))^10/x^4,x)

[Out]

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