3.2218 \(\int \frac {x}{(a+b \sqrt {x})^5} \, dx\)

Optimal. Leaf size=21 \[ \frac {x^2}{2 a \left (a+b \sqrt {x}\right )^4} \]

[Out]

1/2*x^2/a/(a+b*x^(1/2))^4

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Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {264} \[ \frac {x^2}{2 a \left (a+b \sqrt {x}\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*Sqrt[x])^5,x]

[Out]

x^2/(2*a*(a + b*Sqrt[x])^4)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {x}{\left (a+b \sqrt {x}\right )^5} \, dx &=\frac {x^2}{2 a \left (a+b \sqrt {x}\right )^4}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ \frac {x^2}{2 a \left (a+b \sqrt {x}\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*Sqrt[x])^5,x]

[Out]

x^2/(2*a*(a + b*Sqrt[x])^4)

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fricas [B]  time = 1.13, size = 108, normalized size = 5.14 \[ \frac {10 \, a b^{6} x^{3} - 5 \, a^{3} b^{4} x^{2} + 4 \, a^{5} b^{2} x - a^{7} - 4 \, {\left (b^{7} x^{3} + a^{2} b^{5} x^{2}\right )} \sqrt {x}}{2 \, {\left (b^{12} x^{4} - 4 \, a^{2} b^{10} x^{3} + 6 \, a^{4} b^{8} x^{2} - 4 \, a^{6} b^{6} x + a^{8} b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(10*a*b^6*x^3 - 5*a^3*b^4*x^2 + 4*a^5*b^2*x - a^7 - 4*(b^7*x^3 + a^2*b^5*x^2)*sqrt(x))/(b^12*x^4 - 4*a^2*b
^10*x^3 + 6*a^4*b^8*x^2 - 4*a^6*b^6*x + a^8*b^4)

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giac [B]  time = 0.15, size = 42, normalized size = 2.00 \[ -\frac {4 \, b^{3} x^{\frac {3}{2}} + 6 \, a b^{2} x + 4 \, a^{2} b \sqrt {x} + a^{3}}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 65, normalized size = 3.10 \[ \frac {a^{3}}{2 \left (b \sqrt {x}+a \right )^{4} b^{4}}-\frac {2 a^{2}}{\left (b \sqrt {x}+a \right )^{3} b^{4}}+\frac {3 a}{\left (b \sqrt {x}+a \right )^{2} b^{4}}-\frac {2}{\left (b \sqrt {x}+a \right ) b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.93, size = 64, normalized size = 3.05 \[ -\frac {2}{{\left (b \sqrt {x} + a\right )} b^{4}} + \frac {3 \, a}{{\left (b \sqrt {x} + a\right )}^{2} b^{4}} - \frac {2 \, a^{2}}{{\left (b \sqrt {x} + a\right )}^{3} b^{4}} + \frac {a^{3}}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.19, size = 77, normalized size = 3.67 \[ -\frac {\frac {a^3}{2\,b^4}+\frac {2\,x^{3/2}}{b}+\frac {2\,a^2\,\sqrt {x}}{b^3}+\frac {3\,a\,x}{b^2}}{a^4+b^4\,x^2+6\,a^2\,b^2\,x+4\,a^3\,b\,\sqrt {x}+4\,a\,b^3\,x^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^(1/2))^5,x)

[Out]

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

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sympy [A]  time = 1.85, size = 253, normalized size = 12.05 \[ \begin {cases} - \frac {a^{3}}{2 a^{4} b^{4} + 8 a^{3} b^{5} \sqrt {x} + 12 a^{2} b^{6} x + 8 a b^{7} x^{\frac {3}{2}} + 2 b^{8} x^{2}} - \frac {4 a^{2} b \sqrt {x}}{2 a^{4} b^{4} + 8 a^{3} b^{5} \sqrt {x} + 12 a^{2} b^{6} x + 8 a b^{7} x^{\frac {3}{2}} + 2 b^{8} x^{2}} - \frac {6 a b^{2} x}{2 a^{4} b^{4} + 8 a^{3} b^{5} \sqrt {x} + 12 a^{2} b^{6} x + 8 a b^{7} x^{\frac {3}{2}} + 2 b^{8} x^{2}} - \frac {4 b^{3} x^{\frac {3}{2}}}{2 a^{4} b^{4} + 8 a^{3} b^{5} \sqrt {x} + 12 a^{2} b^{6} x + 8 a b^{7} x^{\frac {3}{2}} + 2 b^{8} x^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{5}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**3/(2*a**4*b**4 + 8*a**3*b**5*sqrt(x) + 12*a**2*b**6*x + 8*a*b**7*x**(3/2) + 2*b**8*x**2) - 4*a*
*2*b*sqrt(x)/(2*a**4*b**4 + 8*a**3*b**5*sqrt(x) + 12*a**2*b**6*x + 8*a*b**7*x**(3/2) + 2*b**8*x**2) - 6*a*b**2
*x/(2*a**4*b**4 + 8*a**3*b**5*sqrt(x) + 12*a**2*b**6*x + 8*a*b**7*x**(3/2) + 2*b**8*x**2) - 4*b**3*x**(3/2)/(2
*a**4*b**4 + 8*a**3*b**5*sqrt(x) + 12*a**2*b**6*x + 8*a*b**7*x**(3/2) + 2*b**8*x**2), Ne(b, 0)), (x**2/(2*a**5
), True))

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