3.22.46 \(\int \frac {(a+b \sqrt {x})^5}{x} \, dx\) [2146]

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

[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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Rubi [A]
time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, 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 = {272, 45} \begin {gather*} a^5 \log (x)+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} \end {gather*}

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 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*} \int \frac {\left (a+b \sqrt {x}\right )^5}{x} \, dx &=2 \text {Subst}\left (\int \frac {(a+b x)^5}{x} \, dx,x,\sqrt {x}\right )\\ &=2 \text {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*}

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Mathematica [A]
time = 0.02, size = 67, normalized size = 1.03 \begin {gather*} \frac {1}{30} b \sqrt {x} \left (300 a^4+300 a^3 b \sqrt {x}+200 a^2 b^2 x+75 a b^3 x^{3/2}+12 b^4 x^2\right )+2 a^5 \log \left (\sqrt {x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^5/x,x]

[Out]

(b*Sqrt[x]*(300*a^4 + 300*a^3*b*Sqrt[x] + 200*a^2*b^2*x + 75*a*b^3*x^(3/2) + 12*b^4*x^2))/30 + 2*a^5*Log[Sqrt[
x]]

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Maple [A]
time = 0.19, size = 54, normalized size = 0.83

method	result	size
derivativedivides	\(10 a^{3} b^{2} x +\frac {20 a^{2} b^{3} x^{\frac {3}{2}}}{3}+\frac {5 a \,b^{4} x^{2}}{2}+\frac {2 b^{5} x^{\frac {5}{2}}}{5}+a^{5} \ln \left (x \right )+10 a^{4} b \sqrt {x}\)	\(54\)
default	\(10 a^{3} b^{2} x +\frac {20 a^{2} b^{3} x^{\frac {3}{2}}}{3}+\frac {5 a \,b^{4} x^{2}}{2}+\frac {2 b^{5} x^{\frac {5}{2}}}{5}+a^{5} \ln \left (x \right )+10 a^{4} b \sqrt {x}\)	\(54\)
trager	\(\frac {5 \left (x -1\right ) \left (b^{2} x +4 a^{2}+b^{2}\right ) a \,b^{2}}{2}+\frac {2 b \left (3 b^{4} x^{2}+50 a^{2} b^{2} x +75 a^{4}\right ) \sqrt {x}}{15}-a^{5} \ln \left (\frac {1}{x}\right )\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.29, size = 53, normalized size = 0.82 \begin {gather*} \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 {gather*}

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 = 0.37, size = 57, normalized size = 0.88 \begin {gather*} \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 {gather*}

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 = 0.12, size = 66, normalized size = 1.02 \begin {gather*} 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 {gather*}

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 = 0.46, size = 54, normalized size = 0.83 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.03, size = 56, normalized size = 0.86 \begin {gather*} 2\,a^5\,\ln \left (\sqrt {x}\right )+\frac {2\,b^5\,x^{5/2}}{5}+10\,a^3\,b^2\,x+\frac {5\,a\,b^4\,x^2}{2}+10\,a^4\,b\,\sqrt {x}+\frac {20\,a^2\,b^3\,x^{3/2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^5/x,x)

[Out]

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

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