3.22.35 \(\int \frac {(a+b \sqrt {x})^3}{x} \, dx\) [2135]

Optimal. Leaf size=37 \[ 6 a^2 b \sqrt {x}+3 a b^2 x+\frac {2}{3} b^3 x^{3/2}+a^3 \log (x) \]

[Out]

3*a*b^2*x+2/3*b^3*x^(3/2)+a^3*ln(x)+6*a^2*b*x^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 37, 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^3 \log (x)+6 a^2 b \sqrt {x}+3 a b^2 x+\frac {2}{3} b^3 x^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^3/x,x]

[Out]

6*a^2*b*Sqrt[x] + 3*a*b^2*x + (2*b^3*x^(3/2))/3 + a^3*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 )^3}{x} \, dx &=2 \text {Subst}\left (\int \frac {(a+b x)^3}{x} \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (3 a^2 b+\frac {a^3}{x}+3 a b^2 x+b^3 x^2\right ) \, dx,x,\sqrt {x}\right )\\ &=6 a^2 b \sqrt {x}+3 a b^2 x+\frac {2}{3} b^3 x^{3/2}+a^3 \log (x)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 37, normalized size = 1.00 \begin {gather*} 6 a^2 b \sqrt {x}+3 a b^2 x+\frac {2}{3} b^3 x^{3/2}+a^3 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^3/x,x]

[Out]

6*a^2*b*Sqrt[x] + 3*a*b^2*x + (2*b^3*x^(3/2))/3 + a^3*Log[x]

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Maple [A]
time = 0.19, size = 32, normalized size = 0.86

method	result	size
derivativedivides	\(3 a \,b^{2} x +\frac {2 b^{3} x^{\frac {3}{2}}}{3}+a^{3} \ln \left (x \right )+6 a^{2} b \sqrt {x}\)	\(32\)
default	\(3 a \,b^{2} x +\frac {2 b^{3} x^{\frac {3}{2}}}{3}+a^{3} \ln \left (x \right )+6 a^{2} b \sqrt {x}\)	\(32\)
trager	\(3 a \,b^{2} \left (x -1\right )+\frac {2 \left (b^{2} x +9 a^{2}\right ) b \sqrt {x}}{3}-a^{3} \ln \left (\frac {1}{x}\right )\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*x^(1/2))^3/x,x,method=_RETURNVERBOSE)

[Out]

3*a*b^2*x+2/3*b^3*x^(3/2)+a^3*ln(x)+6*a^2*b*x^(1/2)

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Maxima [A]
time = 0.30, size = 31, normalized size = 0.84 \begin {gather*} \frac {2}{3} \, b^{3} x^{\frac {3}{2}} + 3 \, a b^{2} x + a^{3} \log \left (x\right ) + 6 \, a^{2} b \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x,x, algorithm="maxima")

[Out]

2/3*b^3*x^(3/2) + 3*a*b^2*x + a^3*log(x) + 6*a^2*b*sqrt(x)

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Fricas [A]
time = 0.39, size = 34, normalized size = 0.92 \begin {gather*} 3 \, a b^{2} x + 2 \, a^{3} \log \left (\sqrt {x}\right ) + \frac {2}{3} \, {\left (b^{3} x + 9 \, a^{2} b\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x,x, algorithm="fricas")

[Out]

3*a*b^2*x + 2*a^3*log(sqrt(x)) + 2/3*(b^3*x + 9*a^2*b)*sqrt(x)

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Sympy [A]
time = 0.08, size = 37, normalized size = 1.00 \begin {gather*} a^{3} \log {\left (x \right )} + 6 a^{2} b \sqrt {x} + 3 a b^{2} x + \frac {2 b^{3} x^{\frac {3}{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**3/x,x)

[Out]

a**3*log(x) + 6*a**2*b*sqrt(x) + 3*a*b**2*x + 2*b**3*x**(3/2)/3

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Giac [A]
time = 0.53, size = 32, normalized size = 0.86 \begin {gather*} \frac {2}{3} \, b^{3} x^{\frac {3}{2}} + 3 \, a b^{2} x + a^{3} \log \left ({\left | x \right |}\right ) + 6 \, a^{2} b \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x,x, algorithm="giac")

[Out]

2/3*b^3*x^(3/2) + 3*a*b^2*x + a^3*log(abs(x)) + 6*a^2*b*sqrt(x)

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Mupad [B]
time = 0.04, size = 34, normalized size = 0.92 \begin {gather*} 2\,a^3\,\ln \left (\sqrt {x}\right )+\frac {2\,b^3\,x^{3/2}}{3}+6\,a^2\,b\,\sqrt {x}+3\,a\,b^2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^3/x,x)

[Out]

2*a^3*log(x^(1/2)) + (2*b^3*x^(3/2))/3 + 6*a^2*b*x^(1/2) + 3*a*b^2*x

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