3.2219 \(\int \frac {1}{(a+b \sqrt {x})^5} \, dx\)

Optimal. Leaf size=38 \[ \frac {a}{2 b^2 \left (a+b \sqrt {x}\right )^4}-\frac {2}{3 b^2 \left (a+b \sqrt {x}\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {190, 43} \[ \frac {a}{2 b^2 \left (a+b \sqrt {x}\right )^4}-\frac {2}{3 b^2 \left (a+b \sqrt {x}\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^(-5),x]

[Out]

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

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 190

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(1/n - 1)*(a + b*x)^p, x], x, x^n], x] /
; FreeQ[{a, b, p}, x] && FractionQ[n] && IntegerQ[1/n]

Rubi steps

\begin {align*} \int \frac {1}{\left (a+b \sqrt {x}\right )^5} \, dx &=2 \operatorname {Subst}\left (\int \frac {x}{(a+b x)^5} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^5}+\frac {1}{b (a+b x)^4}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {a}{2 b^2 \left (a+b \sqrt {x}\right )^4}-\frac {2}{3 b^2 \left (a+b \sqrt {x}\right )^3}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 28, normalized size = 0.74 \[ -\frac {a+4 b \sqrt {x}}{6 b^2 \left (a+b \sqrt {x}\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^(-5),x]

[Out]

-1/6*(a + 4*b*Sqrt[x])/(b^2*(a + b*Sqrt[x])^4)

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fricas [B]  time = 1.01, size = 96, normalized size = 2.53 \[ \frac {15 \, a b^{4} x^{2} + 10 \, a^{3} b^{2} x - a^{5} - 4 \, {\left (b^{5} x^{2} + 5 \, a^{2} b^{3} x\right )} \sqrt {x}}{6 \, {\left (b^{10} x^{4} - 4 \, a^{2} b^{8} x^{3} + 6 \, a^{4} b^{6} x^{2} - 4 \, a^{6} b^{4} x + a^{8} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(15*a*b^4*x^2 + 10*a^3*b^2*x - a^5 - 4*(b^5*x^2 + 5*a^2*b^3*x)*sqrt(x))/(b^10*x^4 - 4*a^2*b^8*x^3 + 6*a^4*
b^6*x^2 - 4*a^6*b^4*x + a^8*b^2)

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giac [A]  time = 0.16, size = 22, normalized size = 0.58 \[ -\frac {4 \, b \sqrt {x} + a}{6 \, {\left (b \sqrt {x} + a\right )}^{4} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.04, size = 200, normalized size = 5.26 \[ -10 \left (-\frac {a^{2}}{4 \left (b^{2} x -a^{2}\right )^{4} b^{4}}-\frac {1}{3 \left (b^{2} x -a^{2}\right )^{3} b^{4}}\right ) a^{3} b^{2}-5 \left (-\frac {a^{4}}{4 \left (b^{2} x -a^{2}\right )^{4} b^{6}}-\frac {2 a^{2}}{3 \left (b^{2} x -a^{2}\right )^{3} b^{6}}-\frac {1}{2 \left (b^{2} x -a^{2}\right )^{2} b^{6}}\right ) a \,b^{4}+\frac {a^{5}}{4 \left (b^{2} x -a^{2}\right )^{4} b^{2}}+\frac {a}{4 \left (b \sqrt {x}+a \right )^{4} b^{2}}-\frac {a}{4 \left (b \sqrt {x}-a \right )^{4} b^{2}}-\frac {1}{3 \left (b \sqrt {x}+a \right )^{3} b^{2}}-\frac {1}{3 \left (b \sqrt {x}-a \right )^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.86, size = 30, normalized size = 0.79 \[ -\frac {2}{3 \, {\left (b \sqrt {x} + a\right )}^{3} b^{2}} + \frac {a}{2 \, {\left (b \sqrt {x} + a\right )}^{4} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.20, size = 57, normalized size = 1.50 \[ -\frac {\frac {a}{6\,b^2}+\frac {2\,\sqrt {x}}{3\,b}}{a^4+b^4\,x^2+6\,a^2\,b^2\,x+4\,a^3\,b\,\sqrt {x}+4\,a\,b^3\,x^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*x^(1/2))^5,x)

[Out]

-(a/(6*b^2) + (2*x^(1/2))/(3*b))/(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 = 1.81, size = 121, normalized size = 3.18 \[ \begin {cases} - \frac {a}{6 a^{4} b^{2} + 24 a^{3} b^{3} \sqrt {x} + 36 a^{2} b^{4} x + 24 a b^{5} x^{\frac {3}{2}} + 6 b^{6} x^{2}} - \frac {4 b \sqrt {x}}{6 a^{4} b^{2} + 24 a^{3} b^{3} \sqrt {x} + 36 a^{2} b^{4} x + 24 a b^{5} x^{\frac {3}{2}} + 6 b^{6} x^{2}} & \text {for}\: b \neq 0 \\\frac {x}{a^{5}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a/(6*a**4*b**2 + 24*a**3*b**3*sqrt(x) + 36*a**2*b**4*x + 24*a*b**5*x**(3/2) + 6*b**6*x**2) - 4*b*s
qrt(x)/(6*a**4*b**2 + 24*a**3*b**3*sqrt(x) + 36*a**2*b**4*x + 24*a*b**5*x**(3/2) + 6*b**6*x**2), Ne(b, 0)), (x
/a**5, True))

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