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]

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

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Rubi [A]  time = 0.020523, antiderivative size = 38, normalized size of antiderivative = 1., 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 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]

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])

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*}

Mathematica [A]  time = 0.0174111, 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]

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

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Maple [B]  time = 0.04, size = 200, normalized size = 5.3 \begin{align*} -{\frac{1}{3\,{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-3}}+{\frac{a}{4\,{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-4}}-{\frac{1}{3\,{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-3}}-{\frac{a}{4\,{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-4}}+{\frac{{a}^{5}}{4\, \left ({b}^{2}x-{a}^{2} \right ) ^{4}{b}^{2}}}-5\,a{b}^{4} \left ( -2/3\,{\frac{{a}^{2}}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{3}}}-1/2\,{\frac{1}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{2}}}-1/4\,{\frac{{a}^{4}}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{4}}} \right ) -10\,{a}^{3}{b}^{2} \left ( -1/3\,{\frac{1}{ \left ({b}^{2}x-{a}^{2} \right ) ^{3}{b}^{4}}}-1/4\,{\frac{{a}^{2}}{{b}^{4} \left ({b}^{2}x-{a}^{2} \right ) ^{4}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.968503, size = 41, normalized size = 1.08 \begin{align*} -\frac{2}{3 \,{\left (b \sqrt{x} + a\right )}^{3} b^{2}} + \frac{a}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{2}} \end{align*}

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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Fricas [B]  time = 1.26191, size = 194, normalized size = 5.11 \begin{align*} \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 )}} \end{align*}

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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Sympy [A]  time = 2.52225, size = 194, normalized size = 5.11 \begin{align*} \begin{cases} \frac{6 a^{2} x}{6 a^{7} + 24 a^{6} b \sqrt{x} + 36 a^{5} b^{2} x + 24 a^{4} b^{3} x^{\frac{3}{2}} + 6 a^{3} b^{4} x^{2}} + \frac{4 a b x^{\frac{3}{2}}}{6 a^{7} + 24 a^{6} b \sqrt{x} + 36 a^{5} b^{2} x + 24 a^{4} b^{3} x^{\frac{3}{2}} + 6 a^{3} b^{4} x^{2}} + \frac{b^{2} x^{2}}{6 a^{7} + 24 a^{6} b \sqrt{x} + 36 a^{5} b^{2} x + 24 a^{4} b^{3} x^{\frac{3}{2}} + 6 a^{3} b^{4} x^{2}} & \text{for}\: a \neq 0 \\- \frac{2}{3 b^{5} x^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((6*a**2*x/(6*a**7 + 24*a**6*b*sqrt(x) + 36*a**5*b**2*x + 24*a**4*b**3*x**(3/2) + 6*a**3*b**4*x**2) +
4*a*b*x**(3/2)/(6*a**7 + 24*a**6*b*sqrt(x) + 36*a**5*b**2*x + 24*a**4*b**3*x**(3/2) + 6*a**3*b**4*x**2) + b**
2*x**2/(6*a**7 + 24*a**6*b*sqrt(x) + 36*a**5*b**2*x + 24*a**4*b**3*x**(3/2) + 6*a**3*b**4*x**2), Ne(a, 0)), (-
2/(3*b**5*x**(3/2)), True))

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Giac [A]  time = 1.11568, size = 30, normalized size = 0.79 \begin{align*} -\frac{4 \, b \sqrt{x} + a}{6 \,{\left (b \sqrt{x} + a\right )}^{4} b^{2}} \end{align*}

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)