3.2160 \(\int \frac{(a+b \sqrt{x})^{10}}{x^3} \, dx\)

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

[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] + 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]

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Rubi [A]  time = 0.070199, 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} \[ 80 a^3 b^7 x^{3/2}+\frac{45}{2} a^2 b^8 x^2-\frac{45 a^8 b^2}{x}-\frac{240 a^7 b^3}{\sqrt{x}}+504 a^5 b^5 \sqrt{x}+210 a^4 b^6 x+210 a^6 b^4 \log (x)-\frac{20 a^9 b}{3 x^{3/2}}-\frac{a^{10}}{2 x^2}+4 a b^9 x^{5/2}+\frac{b^{10} x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^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] + 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 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^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x^5} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{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]  time = 0.0563367, size = 127, normalized size = 1. \[ 80 a^3 b^7 x^{3/2}+\frac{45}{2} a^2 b^8 x^2-\frac{45 a^8 b^2}{x}-\frac{240 a^7 b^3}{\sqrt{x}}+504 a^5 b^5 \sqrt{x}+210 a^4 b^6 x+210 a^6 b^4 \log (x)-\frac{20 a^9 b}{3 x^{3/2}}-\frac{a^{10}}{2 x^2}+4 a b^9 x^{5/2}+\frac{b^{10} x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^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] + 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]

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Maple [A]  time = 0.004, size = 110, normalized size = 0.9 \begin{align*} -{\frac{{a}^{10}}{2\,{x}^{2}}}-{\frac{20\,{a}^{9}b}{3}{x}^{-{\frac{3}{2}}}}-45\,{\frac{{a}^{8}{b}^{2}}{x}}+210\,{a}^{4}{b}^{6}x+80\,{a}^{3}{b}^{7}{x}^{3/2}+{\frac{45\,{a}^{2}{b}^{8}{x}^{2}}{2}}+4\,a{b}^{9}{x}^{5/2}+{\frac{{b}^{10}{x}^{3}}{3}}+210\,{a}^{6}{b}^{4}\ln \left ( x \right ) -240\,{\frac{{a}^{7}{b}^{3}}{\sqrt{x}}}+504\,{a}^{5}{b}^{5}\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [A]  time = 0.948798, size = 149, normalized size = 1.17 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.55754, size = 274, normalized size = 2.16 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 1.72337, size = 128, normalized size = 1.01 \begin{align*} - \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [A]  time = 1.08175, size = 150, normalized size = 1.18 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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