3.2158 \(\int \frac{(a+b \sqrt{x})^{10}}{x} \, dx\)

Optimal. Leaf size=128 \[ 80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac{504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac{240}{7} a^3 b^7 x^{7/2}+\frac{45}{4} a^2 b^8 x^4+45 a^8 b^2 x+20 a^9 b \sqrt{x}+a^{10} \log (x)+\frac{20}{9} a b^9 x^{9/2}+\frac{b^{10} x^5}{5} \]

[Out]

20*a^9*b*Sqrt[x] + 45*a^8*b^2*x + 80*a^7*b^3*x^(3/2) + 105*a^6*b^4*x^2 + (504*a^5*b^5*x^(5/2))/5 + 70*a^4*b^6*
x^3 + (240*a^3*b^7*x^(7/2))/7 + (45*a^2*b^8*x^4)/4 + (20*a*b^9*x^(9/2))/9 + (b^10*x^5)/5 + a^10*Log[x]

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Rubi [A]  time = 0.0656418, antiderivative size = 128, 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} \[ 80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac{504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac{240}{7} a^3 b^7 x^{7/2}+\frac{45}{4} a^2 b^8 x^4+45 a^8 b^2 x+20 a^9 b \sqrt{x}+a^{10} \log (x)+\frac{20}{9} a b^9 x^{9/2}+\frac{b^{10} x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^10/x,x]

[Out]

20*a^9*b*Sqrt[x] + 45*a^8*b^2*x + 80*a^7*b^3*x^(3/2) + 105*a^6*b^4*x^2 + (504*a^5*b^5*x^(5/2))/5 + 70*a^4*b^6*
x^3 + (240*a^3*b^7*x^(7/2))/7 + (45*a^2*b^8*x^4)/4 + (20*a*b^9*x^(9/2))/9 + (b^10*x^5)/5 + a^10*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} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^{10}}{x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (10 a^9 b+\frac{a^{10}}{x}+45 a^8 b^2 x+120 a^7 b^3 x^2+210 a^6 b^4 x^3+252 a^5 b^5 x^4+210 a^4 b^6 x^5+120 a^3 b^7 x^6+45 a^2 b^8 x^7+10 a b^9 x^8+b^{10} x^9\right ) \, dx,x,\sqrt{x}\right )\\ &=20 a^9 b \sqrt{x}+45 a^8 b^2 x+80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac{504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac{240}{7} a^3 b^7 x^{7/2}+\frac{45}{4} a^2 b^8 x^4+\frac{20}{9} a b^9 x^{9/2}+\frac{b^{10} x^5}{5}+a^{10} \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0404464, size = 128, normalized size = 1. \[ 80 a^7 b^3 x^{3/2}+105 a^6 b^4 x^2+\frac{504}{5} a^5 b^5 x^{5/2}+70 a^4 b^6 x^3+\frac{240}{7} a^3 b^7 x^{7/2}+\frac{45}{4} a^2 b^8 x^4+45 a^8 b^2 x+20 a^9 b \sqrt{x}+a^{10} \log (x)+\frac{20}{9} a b^9 x^{9/2}+\frac{b^{10} x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^10/x,x]

[Out]

20*a^9*b*Sqrt[x] + 45*a^8*b^2*x + 80*a^7*b^3*x^(3/2) + 105*a^6*b^4*x^2 + (504*a^5*b^5*x^(5/2))/5 + 70*a^4*b^6*
x^3 + (240*a^3*b^7*x^(7/2))/7 + (45*a^2*b^8*x^4)/4 + (20*a*b^9*x^(9/2))/9 + (b^10*x^5)/5 + a^10*Log[x]

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Maple [A]  time = 0.003, size = 109, normalized size = 0.9 \begin{align*} 45\,{a}^{8}{b}^{2}x+80\,{a}^{7}{b}^{3}{x}^{3/2}+105\,{a}^{6}{b}^{4}{x}^{2}+{\frac{504\,{a}^{5}{b}^{5}}{5}{x}^{{\frac{5}{2}}}}+70\,{a}^{4}{b}^{6}{x}^{3}+{\frac{240\,{a}^{3}{b}^{7}}{7}{x}^{{\frac{7}{2}}}}+{\frac{45\,{a}^{2}{b}^{8}{x}^{4}}{4}}+{\frac{20\,a{b}^{9}}{9}{x}^{{\frac{9}{2}}}}+{\frac{{b}^{10}{x}^{5}}{5}}+{a}^{10}\ln \left ( x \right ) +20\,{a}^{9}b\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

45*a^8*b^2*x+80*a^7*b^3*x^(3/2)+105*a^6*b^4*x^2+504/5*a^5*b^5*x^(5/2)+70*a^4*b^6*x^3+240/7*a^3*b^7*x^(7/2)+45/
4*a^2*b^8*x^4+20/9*a*b^9*x^(9/2)+1/5*b^10*x^5+a^10*ln(x)+20*a^9*b*x^(1/2)

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Maxima [A]  time = 0.946887, size = 146, normalized size = 1.14 \begin{align*} \frac{1}{5} \, b^{10} x^{5} + \frac{20}{9} \, a b^{9} x^{\frac{9}{2}} + \frac{45}{4} \, a^{2} b^{8} x^{4} + \frac{240}{7} \, a^{3} b^{7} x^{\frac{7}{2}} + 70 \, a^{4} b^{6} x^{3} + \frac{504}{5} \, a^{5} b^{5} x^{\frac{5}{2}} + 105 \, a^{6} b^{4} x^{2} + 80 \, a^{7} b^{3} x^{\frac{3}{2}} + 45 \, a^{8} b^{2} x + a^{10} \log \left (x\right ) + 20 \, a^{9} b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^10*x^5 + 20/9*a*b^9*x^(9/2) + 45/4*a^2*b^8*x^4 + 240/7*a^3*b^7*x^(7/2) + 70*a^4*b^6*x^3 + 504/5*a^5*b^5*
x^(5/2) + 105*a^6*b^4*x^2 + 80*a^7*b^3*x^(3/2) + 45*a^8*b^2*x + a^10*log(x) + 20*a^9*b*sqrt(x)

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Fricas [A]  time = 1.53265, size = 277, normalized size = 2.16 \begin{align*} \frac{1}{5} \, b^{10} x^{5} + \frac{45}{4} \, a^{2} b^{8} x^{4} + 70 \, a^{4} b^{6} x^{3} + 105 \, a^{6} b^{4} x^{2} + 45 \, a^{8} b^{2} x + 2 \, a^{10} \log \left (\sqrt{x}\right ) + \frac{4}{315} \,{\left (175 \, a b^{9} x^{4} + 2700 \, a^{3} b^{7} x^{3} + 7938 \, a^{5} b^{5} x^{2} + 6300 \, a^{7} b^{3} x + 1575 \, a^{9} b\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^10*x^5 + 45/4*a^2*b^8*x^4 + 70*a^4*b^6*x^3 + 105*a^6*b^4*x^2 + 45*a^8*b^2*x + 2*a^10*log(sqrt(x)) + 4/31
5*(175*a*b^9*x^4 + 2700*a^3*b^7*x^3 + 7938*a^5*b^5*x^2 + 6300*a^7*b^3*x + 1575*a^9*b)*sqrt(x)

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Sympy [A]  time = 1.79492, size = 131, normalized size = 1.02 \begin{align*} a^{10} \log{\left (x \right )} + 20 a^{9} b \sqrt{x} + 45 a^{8} b^{2} x + 80 a^{7} b^{3} x^{\frac{3}{2}} + 105 a^{6} b^{4} x^{2} + \frac{504 a^{5} b^{5} x^{\frac{5}{2}}}{5} + 70 a^{4} b^{6} x^{3} + \frac{240 a^{3} b^{7} x^{\frac{7}{2}}}{7} + \frac{45 a^{2} b^{8} x^{4}}{4} + \frac{20 a b^{9} x^{\frac{9}{2}}}{9} + \frac{b^{10} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**10*log(x) + 20*a**9*b*sqrt(x) + 45*a**8*b**2*x + 80*a**7*b**3*x**(3/2) + 105*a**6*b**4*x**2 + 504*a**5*b**5
*x**(5/2)/5 + 70*a**4*b**6*x**3 + 240*a**3*b**7*x**(7/2)/7 + 45*a**2*b**8*x**4/4 + 20*a*b**9*x**(9/2)/9 + b**1
0*x**5/5

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Giac [A]  time = 1.08016, size = 147, normalized size = 1.15 \begin{align*} \frac{1}{5} \, b^{10} x^{5} + \frac{20}{9} \, a b^{9} x^{\frac{9}{2}} + \frac{45}{4} \, a^{2} b^{8} x^{4} + \frac{240}{7} \, a^{3} b^{7} x^{\frac{7}{2}} + 70 \, a^{4} b^{6} x^{3} + \frac{504}{5} \, a^{5} b^{5} x^{\frac{5}{2}} + 105 \, a^{6} b^{4} x^{2} + 80 \, a^{7} b^{3} x^{\frac{3}{2}} + 45 \, a^{8} b^{2} x + a^{10} \log \left ({\left | x \right |}\right ) + 20 \, a^{9} b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b^10*x^5 + 20/9*a*b^9*x^(9/2) + 45/4*a^2*b^8*x^4 + 240/7*a^3*b^7*x^(7/2) + 70*a^4*b^6*x^3 + 504/5*a^5*b^5*
x^(5/2) + 105*a^6*b^4*x^2 + 80*a^7*b^3*x^(3/2) + 45*a^8*b^2*x + a^10*log(abs(x)) + 20*a^9*b*sqrt(x)