3.2161 \(\int \frac{(a+b \sqrt{x})^{10}}{x^4} \, dx\)

Optimal. Leaf size=127 \[ -\frac{45 a^8 b^2}{2 x^2}-\frac{80 a^7 b^3}{x^{3/2}}-\frac{210 a^6 b^4}{x}-\frac{504 a^5 b^5}{\sqrt{x}}+240 a^3 b^7 \sqrt{x}+45 a^2 b^8 x+210 a^4 b^6 \log (x)-\frac{4 a^9 b}{x^{5/2}}-\frac{a^{10}}{3 x^3}+\frac{20}{3} a b^9 x^{3/2}+\frac{b^{10} x^2}{2} \]

[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] + 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]

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Rubi [A]  time = 0.0689496, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{45 a^8 b^2}{2 x^2}-\frac{80 a^7 b^3}{x^{3/2}}-\frac{210 a^6 b^4}{x}-\frac{504 a^5 b^5}{\sqrt{x}}+240 a^3 b^7 \sqrt{x}+45 a^2 b^8 x+210 a^4 b^6 \log (x)-\frac{4 a^9 b}{x^{5/2}}-\frac{a^{10}}{3 x^3}+\frac{20}{3} a b^9 x^{3/2}+\frac{b^{10} x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^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] + 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 266

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]]

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{\left (a+b \sqrt{x}\right )^{10}}{x^4} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^7} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{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]  time = 0.069649, size = 124, normalized size = 0.98 \[ 210 a^4 b^6 \log (x)-\frac{480 a^7 b^3 x^{3/2}+1260 a^6 b^4 x^2+3024 a^5 b^5 x^{5/2}-1440 a^3 b^7 x^{7/2}-270 a^2 b^8 x^4+135 a^8 b^2 x+24 a^9 b \sqrt{x}+2 a^{10}-40 a b^9 x^{9/2}-3 b^{10} x^5}{6 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^10/x^4,x]

[Out]

-(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]

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Maple [A]  time = 0.003, size = 110, normalized size = 0.9 \begin{align*} -{\frac{{a}^{10}}{3\,{x}^{3}}}-4\,{\frac{{a}^{9}b}{{x}^{5/2}}}-{\frac{45\,{a}^{8}{b}^{2}}{2\,{x}^{2}}}-80\,{\frac{{a}^{7}{b}^{3}}{{x}^{3/2}}}-210\,{\frac{{a}^{6}{b}^{4}}{x}}+45\,{a}^{2}{b}^{8}x+{\frac{20\,a{b}^{9}}{3}{x}^{{\frac{3}{2}}}}+{\frac{{b}^{10}{x}^{2}}{2}}+210\,{a}^{4}{b}^{6}\ln \left ( x \right ) -504\,{\frac{{a}^{5}{b}^{5}}{\sqrt{x}}}+240\,{a}^{3}{b}^{7}\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 0.991752, size = 149, normalized size = 1.17 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.57971, size = 274, normalized size = 2.16 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 1.68061, size = 128, normalized size = 1.01 \begin{align*} - \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.08797, size = 150, normalized size = 1.18 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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