3.2209 \(\int \frac {x}{(a+b \sqrt {x})^3} \, dx\)

Optimal. Leaf size=64 \[ \frac {a^3}{b^4 \left (a+b \sqrt {x}\right )^2}-\frac {6 a^2}{b^4 \left (a+b \sqrt {x}\right )}-\frac {6 a \log \left (a+b \sqrt {x}\right )}{b^4}+\frac {2 \sqrt {x}}{b^3} \]

[Out]

-6*a*ln(a+b*x^(1/2))/b^4+2*x^(1/2)/b^3+a^3/b^4/(a+b*x^(1/2))^2-6*a^2/b^4/(a+b*x^(1/2))

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Rubi [A]  time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {a^3}{b^4 \left (a+b \sqrt {x}\right )^2}-\frac {6 a^2}{b^4 \left (a+b \sqrt {x}\right )}-\frac {6 a \log \left (a+b \sqrt {x}\right )}{b^4}+\frac {2 \sqrt {x}}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Sqrt[x])^3,x]

[Out]

a^3/(b^4*(a + b*Sqrt[x])^2) - (6*a^2)/(b^4*(a + b*Sqrt[x])) + (2*Sqrt[x])/b^3 - (6*a*Log[a + b*Sqrt[x]])/b^4

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])

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]]

Rubi steps

\begin {align*} \int \frac {x}{\left (a+b \sqrt {x}\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3}{(a+b x)^3} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {1}{b^3}-\frac {a^3}{b^3 (a+b x)^3}+\frac {3 a^2}{b^3 (a+b x)^2}-\frac {3 a}{b^3 (a+b x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {a^3}{b^4 \left (a+b \sqrt {x}\right )^2}-\frac {6 a^2}{b^4 \left (a+b \sqrt {x}\right )}+\frac {2 \sqrt {x}}{b^3}-\frac {6 a \log \left (a+b \sqrt {x}\right )}{b^4}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 57, normalized size = 0.89 \[ \frac {\frac {a^3}{\left (a+b \sqrt {x}\right )^2}-\frac {6 a^2}{a+b \sqrt {x}}-6 a \log \left (a+b \sqrt {x}\right )+2 b \sqrt {x}}{b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*Sqrt[x])^3,x]

[Out]

(a^3/(a + b*Sqrt[x])^2 - (6*a^2)/(a + b*Sqrt[x]) + 2*b*Sqrt[x] - 6*a*Log[a + b*Sqrt[x]])/b^4

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fricas [A]  time = 0.89, size = 101, normalized size = 1.58 \[ \frac {7 \, a^{3} b^{2} x - 5 \, a^{5} - 6 \, {\left (a b^{4} x^{2} - 2 \, a^{3} b^{2} x + a^{5}\right )} \log \left (b \sqrt {x} + a\right ) + 2 \, {\left (b^{5} x^{2} - 5 \, a^{2} b^{3} x + 3 \, a^{4} b\right )} \sqrt {x}}{b^{8} x^{2} - 2 \, a^{2} b^{6} x + a^{4} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7*a^3*b^2*x - 5*a^5 - 6*(a*b^4*x^2 - 2*a^3*b^2*x + a^5)*log(b*sqrt(x) + a) + 2*(b^5*x^2 - 5*a^2*b^3*x + 3*a^4
*b)*sqrt(x))/(b^8*x^2 - 2*a^2*b^6*x + a^4*b^4)

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giac [A]  time = 0.15, size = 53, normalized size = 0.83 \[ -\frac {6 \, a \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{4}} + \frac {2 \, \sqrt {x}}{b^{3}} - \frac {6 \, a^{2} b \sqrt {x} + 5 \, a^{3}}{{\left (b \sqrt {x} + a\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*a*log(abs(b*sqrt(x) + a))/b^4 + 2*sqrt(x)/b^3 - (6*a^2*b*sqrt(x) + 5*a^3)/((b*sqrt(x) + a)^2*b^4)

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maple [A]  time = 0.01, size = 57, normalized size = 0.89 \[ \frac {a^{3}}{\left (b \sqrt {x}+a \right )^{2} b^{4}}-\frac {6 a^{2}}{\left (b \sqrt {x}+a \right ) b^{4}}-\frac {6 a \ln \left (b \sqrt {x}+a \right )}{b^{4}}+\frac {2 \sqrt {x}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*a*ln(b*x^(1/2)+a)/b^4+2*x^(1/2)/b^3+a^3/b^4/(b*x^(1/2)+a)^2-6*a^2/b^4/(b*x^(1/2)+a)

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maxima [A]  time = 0.85, size = 60, normalized size = 0.94 \[ -\frac {6 \, a \log \left (b \sqrt {x} + a\right )}{b^{4}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}}{b^{4}} - \frac {6 \, a^{2}}{{\left (b \sqrt {x} + a\right )} b^{4}} + \frac {a^{3}}{{\left (b \sqrt {x} + a\right )}^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*a*log(b*sqrt(x) + a)/b^4 + 2*(b*sqrt(x) + a)/b^4 - 6*a^2/((b*sqrt(x) + a)*b^4) + a^3/((b*sqrt(x) + a)^2*b^4
)

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mupad [B]  time = 1.15, size = 66, normalized size = 1.03 \[ \frac {2\,\sqrt {x}}{b^3}-\frac {\frac {5\,a^3}{b}+6\,a^2\,\sqrt {x}}{b^5\,x+a^2\,b^3+2\,a\,b^4\,\sqrt {x}}-\frac {6\,a\,\ln \left (a+b\,\sqrt {x}\right )}{b^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^(1/2))^3,x)

[Out]

(2*x^(1/2))/b^3 - ((5*a^3)/b + 6*a^2*x^(1/2))/(b^5*x + a^2*b^3 + 2*a*b^4*x^(1/2)) - (6*a*log(a + b*x^(1/2)))/b
^4

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sympy [A]  time = 0.95, size = 233, normalized size = 3.64 \[ \begin {cases} - \frac {6 a^{3} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt {x} + b^{6} x} - \frac {9 a^{3}}{a^{2} b^{4} + 2 a b^{5} \sqrt {x} + b^{6} x} - \frac {12 a^{2} b \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt {x} + b^{6} x} - \frac {12 a^{2} b \sqrt {x}}{a^{2} b^{4} + 2 a b^{5} \sqrt {x} + b^{6} x} - \frac {6 a b^{2} x \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{a^{2} b^{4} + 2 a b^{5} \sqrt {x} + b^{6} x} + \frac {2 b^{3} x^{\frac {3}{2}}}{a^{2} b^{4} + 2 a b^{5} \sqrt {x} + b^{6} x} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{3}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-6*a**3*log(a/b + sqrt(x))/(a**2*b**4 + 2*a*b**5*sqrt(x) + b**6*x) - 9*a**3/(a**2*b**4 + 2*a*b**5*s
qrt(x) + b**6*x) - 12*a**2*b*sqrt(x)*log(a/b + sqrt(x))/(a**2*b**4 + 2*a*b**5*sqrt(x) + b**6*x) - 12*a**2*b*sq
rt(x)/(a**2*b**4 + 2*a*b**5*sqrt(x) + b**6*x) - 6*a*b**2*x*log(a/b + sqrt(x))/(a**2*b**4 + 2*a*b**5*sqrt(x) +
b**6*x) + 2*b**3*x**(3/2)/(a**2*b**4 + 2*a*b**5*sqrt(x) + b**6*x), Ne(b, 0)), (x**2/(2*a**3), True))

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