3.2146 \(\int \frac{(a+b \sqrt{x})^5}{x} \, dx\)

Optimal. Leaf size=65 \[ \frac{20}{3} a^2 b^3 x^{3/2}+10 a^3 b^2 x+10 a^4 b \sqrt{x}+a^5 \log (x)+\frac{5}{2} a b^4 x^2+\frac{2}{5} b^5 x^{5/2} \]

[Out]

10*a^4*b*Sqrt[x] + 10*a^3*b^2*x + (20*a^2*b^3*x^(3/2))/3 + (5*a*b^4*x^2)/2 + (2*b^5*x^(5/2))/5 + a^5*Log[x]

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Rubi [A]  time = 0.0311541, antiderivative size = 65, 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} \[ \frac{20}{3} a^2 b^3 x^{3/2}+10 a^3 b^2 x+10 a^4 b \sqrt{x}+a^5 \log (x)+\frac{5}{2} a b^4 x^2+\frac{2}{5} b^5 x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^5/x,x]

[Out]

10*a^4*b*Sqrt[x] + 10*a^3*b^2*x + (20*a^2*b^3*x^(3/2))/3 + (5*a*b^4*x^2)/2 + (2*b^5*x^(5/2))/5 + a^5*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 )^5}{x} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (5 a^4 b+\frac{a^5}{x}+10 a^3 b^2 x+10 a^2 b^3 x^2+5 a b^4 x^3+b^5 x^4\right ) \, dx,x,\sqrt{x}\right )\\ &=10 a^4 b \sqrt{x}+10 a^3 b^2 x+\frac{20}{3} a^2 b^3 x^{3/2}+\frac{5}{2} a b^4 x^2+\frac{2}{5} b^5 x^{5/2}+a^5 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0211607, size = 65, normalized size = 1. \[ \frac{20}{3} a^2 b^3 x^{3/2}+10 a^3 b^2 x+10 a^4 b \sqrt{x}+a^5 \log (x)+\frac{5}{2} a b^4 x^2+\frac{2}{5} b^5 x^{5/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^5/x,x]

[Out]

10*a^4*b*Sqrt[x] + 10*a^3*b^2*x + (20*a^2*b^3*x^(3/2))/3 + (5*a*b^4*x^2)/2 + (2*b^5*x^(5/2))/5 + a^5*Log[x]

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Maple [A]  time = 0.003, size = 54, normalized size = 0.8 \begin{align*} 10\,{a}^{3}{b}^{2}x+{\frac{20\,{a}^{2}{b}^{3}}{3}{x}^{{\frac{3}{2}}}}+{\frac{5\,a{b}^{4}{x}^{2}}{2}}+{\frac{2\,{b}^{5}}{5}{x}^{{\frac{5}{2}}}}+{a}^{5}\ln \left ( x \right ) +10\,{a}^{4}b\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^5/x,x)

[Out]

10*a^3*b^2*x+20/3*a^2*b^3*x^(3/2)+5/2*a*b^4*x^2+2/5*b^5*x^(5/2)+a^5*ln(x)+10*a^4*b*x^(1/2)

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Maxima [A]  time = 0.975073, size = 72, normalized size = 1.11 \begin{align*} \frac{2}{5} \, b^{5} x^{\frac{5}{2}} + \frac{5}{2} \, a b^{4} x^{2} + \frac{20}{3} \, a^{2} b^{3} x^{\frac{3}{2}} + 10 \, a^{3} b^{2} x + a^{5} \log \left (x\right ) + 10 \, a^{4} b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5/x,x, algorithm="maxima")

[Out]

2/5*b^5*x^(5/2) + 5/2*a*b^4*x^2 + 20/3*a^2*b^3*x^(3/2) + 10*a^3*b^2*x + a^5*log(x) + 10*a^4*b*sqrt(x)

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Fricas [A]  time = 1.48419, size = 140, normalized size = 2.15 \begin{align*} \frac{5}{2} \, a b^{4} x^{2} + 10 \, a^{3} b^{2} x + 2 \, a^{5} \log \left (\sqrt{x}\right ) + \frac{2}{15} \,{\left (3 \, b^{5} x^{2} + 50 \, a^{2} b^{3} x + 75 \, a^{4} b\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5/x,x, algorithm="fricas")

[Out]

5/2*a*b^4*x^2 + 10*a^3*b^2*x + 2*a^5*log(sqrt(x)) + 2/15*(3*b^5*x^2 + 50*a^2*b^3*x + 75*a^4*b)*sqrt(x)

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Sympy [A]  time = 0.566302, size = 66, normalized size = 1.02 \begin{align*} a^{5} \log{\left (x \right )} + 10 a^{4} b \sqrt{x} + 10 a^{3} b^{2} x + \frac{20 a^{2} b^{3} x^{\frac{3}{2}}}{3} + \frac{5 a b^{4} x^{2}}{2} + \frac{2 b^{5} x^{\frac{5}{2}}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**5/x,x)

[Out]

a**5*log(x) + 10*a**4*b*sqrt(x) + 10*a**3*b**2*x + 20*a**2*b**3*x**(3/2)/3 + 5*a*b**4*x**2/2 + 2*b**5*x**(5/2)
/5

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Giac [A]  time = 1.11264, size = 73, normalized size = 1.12 \begin{align*} \frac{2}{5} \, b^{5} x^{\frac{5}{2}} + \frac{5}{2} \, a b^{4} x^{2} + \frac{20}{3} \, a^{2} b^{3} x^{\frac{3}{2}} + 10 \, a^{3} b^{2} x + a^{5} \log \left ({\left | x \right |}\right ) + 10 \, a^{4} b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^5/x,x, algorithm="giac")

[Out]

2/5*b^5*x^(5/2) + 5/2*a*b^4*x^2 + 20/3*a^2*b^3*x^(3/2) + 10*a^3*b^2*x + a^5*log(abs(x)) + 10*a^4*b*sqrt(x)