3.2228 \(\int \frac{1}{(a+b \sqrt{x})^8} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 a}{7 b^2 \left (a+b \sqrt{x}\right )^7}-\frac{1}{3 b^2 \left (a+b \sqrt{x}\right )^6} \]

[Out]

(2*a)/(7*b^2*(a + b*Sqrt[x])^7) - 1/(3*b^2*(a + b*Sqrt[x])^6)

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Rubi [A]  time = 0.0210688, 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} \[ \frac{2 a}{7 b^2 \left (a+b \sqrt{x}\right )^7}-\frac{1}{3 b^2 \left (a+b \sqrt{x}\right )^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^(-8),x]

[Out]

(2*a)/(7*b^2*(a + b*Sqrt[x])^7) - 1/(3*b^2*(a + b*Sqrt[x])^6)

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 )^8} \, dx &=2 \operatorname{Subst}\left (\int \frac{x}{(a+b x)^8} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^8}+\frac{1}{b (a+b x)^7}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a}{7 b^2 \left (a+b \sqrt{x}\right )^7}-\frac{1}{3 b^2 \left (a+b \sqrt{x}\right )^6}\\ \end{align*}

Mathematica [A]  time = 0.017241, size = 28, normalized size = 0.74 \[ -\frac{a+7 b \sqrt{x}}{21 b^2 \left (a+b \sqrt{x}\right )^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^(-8),x]

[Out]

-(a + 7*b*Sqrt[x])/(21*b^2*(a + b*Sqrt[x])^7)

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Maple [B]  time = 0.064, size = 399, normalized size = 10.5 \begin{align*} -{\frac{{a}^{8}}{7\, \left ({b}^{2}x-{a}^{2} \right ) ^{7}{b}^{2}}}+{b}^{8} \left ( -{\frac{2\,{a}^{6}}{3\,{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{6}}}-{\frac{{a}^{8}}{7\,{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{7}}}-{\frac{{a}^{2}}{{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{4}}}-{\frac{1}{3\,{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{3}}}-{\frac{6\,{a}^{4}}{5\,{b}^{10} \left ({b}^{2}x-{a}^{2} \right ) ^{5}}} \right ) -{\frac{1}{6\,{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-6}}+{\frac{1}{6\,{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-6}}+28\,{b}^{6}{a}^{2} \left ( -1/2\,{\frac{{a}^{4}}{{b}^{8} \left ({b}^{2}x-{a}^{2} \right ) ^{6}}}-1/7\,{\frac{{a}^{6}}{{b}^{8} \left ({b}^{2}x-{a}^{2} \right ) ^{7}}}-1/4\,{\frac{1}{ \left ({b}^{2}x-{a}^{2} \right ) ^{4}{b}^{8}}}-3/5\,{\frac{{a}^{2}}{{b}^{8} \left ({b}^{2}x-{a}^{2} \right ) ^{5}}} \right ) +{\frac{a}{7\,{b}^{2}} \left ( a+b\sqrt{x} \right ) ^{-7}}+{\frac{a}{7\,{b}^{2}} \left ( b\sqrt{x}-a \right ) ^{-7}}+28\,{a}^{6}{b}^{2} \left ( -1/6\,{\frac{1}{{b}^{4} \left ({b}^{2}x-{a}^{2} \right ) ^{6}}}-1/7\,{\frac{{a}^{2}}{{b}^{4} \left ({b}^{2}x-{a}^{2} \right ) ^{7}}} \right ) +70\,{a}^{4}{b}^{4} \left ( -1/3\,{\frac{{a}^{2}}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{6}}}-1/7\,{\frac{{a}^{4}}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{7}}}-1/5\,{\frac{1}{{b}^{6} \left ({b}^{2}x-{a}^{2} \right ) ^{5}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*x^(1/2))^8,x)

[Out]

-1/7*a^8/(b^2*x-a^2)^7/b^2+b^8*(-2/3*a^6/b^10/(b^2*x-a^2)^6-1/7*a^8/b^10/(b^2*x-a^2)^7-a^2/b^10/(b^2*x-a^2)^4-
1/3/b^10/(b^2*x-a^2)^3-6/5*a^4/b^10/(b^2*x-a^2)^5)-1/6/b^2/(a+b*x^(1/2))^6+1/6/b^2/(b*x^(1/2)-a)^6+28*b^6*a^2*
(-1/2*a^4/b^8/(b^2*x-a^2)^6-1/7*a^6/b^8/(b^2*x-a^2)^7-1/4/(b^2*x-a^2)^4/b^8-3/5*a^2/b^8/(b^2*x-a^2)^5)+1/7*a/b
^2/(a+b*x^(1/2))^7+1/7/b^2*a/(b*x^(1/2)-a)^7+28*a^6*b^2*(-1/6/b^4/(b^2*x-a^2)^6-1/7*a^2/b^4/(b^2*x-a^2)^7)+70*
a^4*b^4*(-1/3*a^2/b^6/(b^2*x-a^2)^6-1/7*a^4/b^6/(b^2*x-a^2)^7-1/5/b^6/(b^2*x-a^2)^5)

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Maxima [A]  time = 0.965735, size = 41, normalized size = 1.08 \begin{align*} -\frac{1}{3 \,{\left (b \sqrt{x} + a\right )}^{6} b^{2}} + \frac{2 \, a}{7 \,{\left (b \sqrt{x} + a\right )}^{7} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*x^(1/2))^8,x, algorithm="maxima")

[Out]

-1/3/((b*sqrt(x) + a)^6*b^2) + 2/7*a/((b*sqrt(x) + a)^7*b^2)

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Fricas [B]  time = 1.30682, size = 350, normalized size = 9.21 \begin{align*} -\frac{7 \, b^{8} x^{4} + 140 \, a^{2} b^{6} x^{3} + 210 \, a^{4} b^{4} x^{2} + 28 \, a^{6} b^{2} x - a^{8} - 16 \,{\left (3 \, a b^{7} x^{3} + 14 \, a^{3} b^{5} x^{2} + 7 \, a^{5} b^{3} x\right )} \sqrt{x}}{21 \,{\left (b^{16} x^{7} - 7 \, a^{2} b^{14} x^{6} + 21 \, a^{4} b^{12} x^{5} - 35 \, a^{6} b^{10} x^{4} + 35 \, a^{8} b^{8} x^{3} - 21 \, a^{10} b^{6} x^{2} + 7 \, a^{12} b^{4} x - a^{14} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*x^(1/2))^8,x, algorithm="fricas")

[Out]

-1/21*(7*b^8*x^4 + 140*a^2*b^6*x^3 + 210*a^4*b^4*x^2 + 28*a^6*b^2*x - a^8 - 16*(3*a*b^7*x^3 + 14*a^3*b^5*x^2 +
7*a^5*b^3*x)*sqrt(x))/(b^16*x^7 - 7*a^2*b^14*x^6 + 21*a^4*b^12*x^5 - 35*a^6*b^10*x^4 + 35*a^8*b^8*x^3 - 21*a^
10*b^6*x^2 + 7*a^12*b^4*x - a^14*b^2)

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Sympy [A]  time = 5.7545, size = 199, normalized size = 5.24 \begin{align*} \begin{cases} - \frac{a}{21 a^{7} b^{2} + 147 a^{6} b^{3} \sqrt{x} + 441 a^{5} b^{4} x + 735 a^{4} b^{5} x^{\frac{3}{2}} + 735 a^{3} b^{6} x^{2} + 441 a^{2} b^{7} x^{\frac{5}{2}} + 147 a b^{8} x^{3} + 21 b^{9} x^{\frac{7}{2}}} - \frac{7 b \sqrt{x}}{21 a^{7} b^{2} + 147 a^{6} b^{3} \sqrt{x} + 441 a^{5} b^{4} x + 735 a^{4} b^{5} x^{\frac{3}{2}} + 735 a^{3} b^{6} x^{2} + 441 a^{2} b^{7} x^{\frac{5}{2}} + 147 a b^{8} x^{3} + 21 b^{9} x^{\frac{7}{2}}} & \text{for}\: b \neq 0 \\\frac{x}{a^{8}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*x**(1/2))**8,x)

[Out]

Piecewise((-a/(21*a**7*b**2 + 147*a**6*b**3*sqrt(x) + 441*a**5*b**4*x + 735*a**4*b**5*x**(3/2) + 735*a**3*b**6
*x**2 + 441*a**2*b**7*x**(5/2) + 147*a*b**8*x**3 + 21*b**9*x**(7/2)) - 7*b*sqrt(x)/(21*a**7*b**2 + 147*a**6*b*
*3*sqrt(x) + 441*a**5*b**4*x + 735*a**4*b**5*x**(3/2) + 735*a**3*b**6*x**2 + 441*a**2*b**7*x**(5/2) + 147*a*b*
*8*x**3 + 21*b**9*x**(7/2)), Ne(b, 0)), (x/a**8, True))

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Giac [A]  time = 1.10261, size = 30, normalized size = 0.79 \begin{align*} -\frac{7 \, b \sqrt{x} + a}{21 \,{\left (b \sqrt{x} + a\right )}^{7} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*x^(1/2))^8,x, algorithm="giac")

[Out]

-1/21*(7*b*sqrt(x) + a)/((b*sqrt(x) + a)^7*b^2)