3.2230 \(\int \frac{1}{(a+b \sqrt{x})^8 x^2} \, dx\)

Optimal. Leaf size=184 \[ \frac{56 b^2}{a^9 \left (a+b \sqrt{x}\right )}+\frac{21 b^2}{a^8 \left (a+b \sqrt{x}\right )^2}+\frac{10 b^2}{a^7 \left (a+b \sqrt{x}\right )^3}+\frac{5 b^2}{a^6 \left (a+b \sqrt{x}\right )^4}+\frac{12 b^2}{5 a^5 \left (a+b \sqrt{x}\right )^5}+\frac{b^2}{a^4 \left (a+b \sqrt{x}\right )^6}+\frac{2 b^2}{7 a^3 \left (a+b \sqrt{x}\right )^7}-\frac{72 b^2 \log \left (a+b \sqrt{x}\right )}{a^{10}}+\frac{36 b^2 \log (x)}{a^{10}}+\frac{16 b}{a^9 \sqrt{x}}-\frac{1}{a^8 x} \]

[Out]

(2*b^2)/(7*a^3*(a + b*Sqrt[x])^7) + b^2/(a^4*(a + b*Sqrt[x])^6) + (12*b^2)/(5*a^5*(a + b*Sqrt[x])^5) + (5*b^2)
/(a^6*(a + b*Sqrt[x])^4) + (10*b^2)/(a^7*(a + b*Sqrt[x])^3) + (21*b^2)/(a^8*(a + b*Sqrt[x])^2) + (56*b^2)/(a^9
*(a + b*Sqrt[x])) - 1/(a^8*x) + (16*b)/(a^9*Sqrt[x]) - (72*b^2*Log[a + b*Sqrt[x]])/a^10 + (36*b^2*Log[x])/a^10

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Rubi [A]  time = 0.158164, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 44} \[ \frac{56 b^2}{a^9 \left (a+b \sqrt{x}\right )}+\frac{21 b^2}{a^8 \left (a+b \sqrt{x}\right )^2}+\frac{10 b^2}{a^7 \left (a+b \sqrt{x}\right )^3}+\frac{5 b^2}{a^6 \left (a+b \sqrt{x}\right )^4}+\frac{12 b^2}{5 a^5 \left (a+b \sqrt{x}\right )^5}+\frac{b^2}{a^4 \left (a+b \sqrt{x}\right )^6}+\frac{2 b^2}{7 a^3 \left (a+b \sqrt{x}\right )^7}-\frac{72 b^2 \log \left (a+b \sqrt{x}\right )}{a^{10}}+\frac{36 b^2 \log (x)}{a^{10}}+\frac{16 b}{a^9 \sqrt{x}}-\frac{1}{a^8 x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])^8*x^2),x]

[Out]

(2*b^2)/(7*a^3*(a + b*Sqrt[x])^7) + b^2/(a^4*(a + b*Sqrt[x])^6) + (12*b^2)/(5*a^5*(a + b*Sqrt[x])^5) + (5*b^2)
/(a^6*(a + b*Sqrt[x])^4) + (10*b^2)/(a^7*(a + b*Sqrt[x])^3) + (21*b^2)/(a^8*(a + b*Sqrt[x])^2) + (56*b^2)/(a^9
*(a + b*Sqrt[x])) - 1/(a^8*x) + (16*b)/(a^9*Sqrt[x]) - (72*b^2*Log[a + b*Sqrt[x]])/a^10 + (36*b^2*Log[x])/a^10

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]]

Rule 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right )^8 x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^8} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^8 x^3}-\frac{8 b}{a^9 x^2}+\frac{36 b^2}{a^{10} x}-\frac{b^3}{a^3 (a+b x)^8}-\frac{3 b^3}{a^4 (a+b x)^7}-\frac{6 b^3}{a^5 (a+b x)^6}-\frac{10 b^3}{a^6 (a+b x)^5}-\frac{15 b^3}{a^7 (a+b x)^4}-\frac{21 b^3}{a^8 (a+b x)^3}-\frac{28 b^3}{a^9 (a+b x)^2}-\frac{36 b^3}{a^{10} (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 b^2}{7 a^3 \left (a+b \sqrt{x}\right )^7}+\frac{b^2}{a^4 \left (a+b \sqrt{x}\right )^6}+\frac{12 b^2}{5 a^5 \left (a+b \sqrt{x}\right )^5}+\frac{5 b^2}{a^6 \left (a+b \sqrt{x}\right )^4}+\frac{10 b^2}{a^7 \left (a+b \sqrt{x}\right )^3}+\frac{21 b^2}{a^8 \left (a+b \sqrt{x}\right )^2}+\frac{56 b^2}{a^9 \left (a+b \sqrt{x}\right )}-\frac{1}{a^8 x}+\frac{16 b}{a^9 \sqrt{x}}-\frac{72 b^2 \log \left (a+b \sqrt{x}\right )}{a^{10}}+\frac{36 b^2 \log (x)}{a^{10}}\\ \end{align*}

Mathematica [A]  time = 0.20598, size = 139, normalized size = 0.76 \[ \frac{\frac{a \left (28098 a^5 b^3 x^{3/2}+57834 a^4 b^4 x^2+66990 a^3 b^5 x^{5/2}+44940 a^2 b^6 x^3+6534 a^6 b^2 x+315 a^7 b \sqrt{x}-35 a^8+16380 a b^7 x^{7/2}+2520 b^8 x^4\right )}{x \left (a+b \sqrt{x}\right )^7}-2520 b^2 \log \left (a+b \sqrt{x}\right )+1260 b^2 \log (x)}{35 a^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sqrt[x])^8*x^2),x]

[Out]

((a*(-35*a^8 + 315*a^7*b*Sqrt[x] + 6534*a^6*b^2*x + 28098*a^5*b^3*x^(3/2) + 57834*a^4*b^4*x^2 + 66990*a^3*b^5*
x^(5/2) + 44940*a^2*b^6*x^3 + 16380*a*b^7*x^(7/2) + 2520*b^8*x^4))/((a + b*Sqrt[x])^7*x) - 2520*b^2*Log[a + b*
Sqrt[x]] + 1260*b^2*Log[x])/(35*a^10)

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Maple [A]  time = 0.013, size = 163, normalized size = 0.9 \begin{align*} -{\frac{1}{{a}^{8}x}}+36\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{10}}}-72\,{\frac{{b}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{10}}}+16\,{\frac{b}{{a}^{9}\sqrt{x}}}+{\frac{2\,{b}^{2}}{7\,{a}^{3}} \left ( a+b\sqrt{x} \right ) ^{-7}}+{\frac{{b}^{2}}{{a}^{4}} \left ( a+b\sqrt{x} \right ) ^{-6}}+{\frac{12\,{b}^{2}}{5\,{a}^{5}} \left ( a+b\sqrt{x} \right ) ^{-5}}+5\,{\frac{{b}^{2}}{{a}^{6} \left ( a+b\sqrt{x} \right ) ^{4}}}+10\,{\frac{{b}^{2}}{{a}^{7} \left ( a+b\sqrt{x} \right ) ^{3}}}+21\,{\frac{{b}^{2}}{{a}^{8} \left ( a+b\sqrt{x} \right ) ^{2}}}+56\,{\frac{{b}^{2}}{{a}^{9} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a^8/x+36*b^2*ln(x)/a^10-72*b^2*ln(a+b*x^(1/2))/a^10+16*b/a^9/x^(1/2)+2/7*b^2/a^3/(a+b*x^(1/2))^7+b^2/a^4/(a
+b*x^(1/2))^6+12/5*b^2/a^5/(a+b*x^(1/2))^5+5*b^2/a^6/(a+b*x^(1/2))^4+10*b^2/a^7/(a+b*x^(1/2))^3+21*b^2/a^8/(a+
b*x^(1/2))^2+56*b^2/a^9/(a+b*x^(1/2))

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Maxima [A]  time = 1.03509, size = 265, normalized size = 1.44 \begin{align*} \frac{2520 \, b^{8} x^{4} + 16380 \, a b^{7} x^{\frac{7}{2}} + 44940 \, a^{2} b^{6} x^{3} + 66990 \, a^{3} b^{5} x^{\frac{5}{2}} + 57834 \, a^{4} b^{4} x^{2} + 28098 \, a^{5} b^{3} x^{\frac{3}{2}} + 6534 \, a^{6} b^{2} x + 315 \, a^{7} b \sqrt{x} - 35 \, a^{8}}{35 \,{\left (a^{9} b^{7} x^{\frac{9}{2}} + 7 \, a^{10} b^{6} x^{4} + 21 \, a^{11} b^{5} x^{\frac{7}{2}} + 35 \, a^{12} b^{4} x^{3} + 35 \, a^{13} b^{3} x^{\frac{5}{2}} + 21 \, a^{14} b^{2} x^{2} + 7 \, a^{15} b x^{\frac{3}{2}} + a^{16} x\right )}} - \frac{72 \, b^{2} \log \left (b \sqrt{x} + a\right )}{a^{10}} + \frac{36 \, b^{2} \log \left (x\right )}{a^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*(2520*b^8*x^4 + 16380*a*b^7*x^(7/2) + 44940*a^2*b^6*x^3 + 66990*a^3*b^5*x^(5/2) + 57834*a^4*b^4*x^2 + 280
98*a^5*b^3*x^(3/2) + 6534*a^6*b^2*x + 315*a^7*b*sqrt(x) - 35*a^8)/(a^9*b^7*x^(9/2) + 7*a^10*b^6*x^4 + 21*a^11*
b^5*x^(7/2) + 35*a^12*b^4*x^3 + 35*a^13*b^3*x^(5/2) + 21*a^14*b^2*x^2 + 7*a^15*b*x^(3/2) + a^16*x) - 72*b^2*lo
g(b*sqrt(x) + a)/a^10 + 36*b^2*log(x)/a^10

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Fricas [B]  time = 1.37995, size = 1015, normalized size = 5.52 \begin{align*} -\frac{1260 \, a^{2} b^{14} x^{7} - 8190 \, a^{4} b^{12} x^{6} + 22470 \, a^{6} b^{10} x^{5} - 33495 \, a^{8} b^{8} x^{4} + 28924 \, a^{10} b^{6} x^{3} - 13888 \, a^{12} b^{4} x^{2} + 3594 \, a^{14} b^{2} x - 35 \, a^{16} + 2520 \,{\left (b^{16} x^{8} - 7 \, a^{2} b^{14} x^{7} + 21 \, a^{4} b^{12} x^{6} - 35 \, a^{6} b^{10} x^{5} + 35 \, a^{8} b^{8} x^{4} - 21 \, a^{10} b^{6} x^{3} + 7 \, a^{12} b^{4} x^{2} - a^{14} b^{2} x\right )} \log \left (b \sqrt{x} + a\right ) - 2520 \,{\left (b^{16} x^{8} - 7 \, a^{2} b^{14} x^{7} + 21 \, a^{4} b^{12} x^{6} - 35 \, a^{6} b^{10} x^{5} + 35 \, a^{8} b^{8} x^{4} - 21 \, a^{10} b^{6} x^{3} + 7 \, a^{12} b^{4} x^{2} - a^{14} b^{2} x\right )} \log \left (\sqrt{x}\right ) - 8 \,{\left (315 \, a b^{15} x^{7} - 2100 \, a^{3} b^{13} x^{6} + 5943 \, a^{5} b^{11} x^{5} - 9216 \, a^{7} b^{9} x^{4} + 8393 \, a^{9} b^{7} x^{3} - 4410 \, a^{11} b^{5} x^{2} + 1225 \, a^{13} b^{3} x - 70 \, a^{15} b\right )} \sqrt{x}}{35 \,{\left (a^{10} b^{14} x^{8} - 7 \, a^{12} b^{12} x^{7} + 21 \, a^{14} b^{10} x^{6} - 35 \, a^{16} b^{8} x^{5} + 35 \, a^{18} b^{6} x^{4} - 21 \, a^{20} b^{4} x^{3} + 7 \, a^{22} b^{2} x^{2} - a^{24} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(1260*a^2*b^14*x^7 - 8190*a^4*b^12*x^6 + 22470*a^6*b^10*x^5 - 33495*a^8*b^8*x^4 + 28924*a^10*b^6*x^3 - 1
3888*a^12*b^4*x^2 + 3594*a^14*b^2*x - 35*a^16 + 2520*(b^16*x^8 - 7*a^2*b^14*x^7 + 21*a^4*b^12*x^6 - 35*a^6*b^1
0*x^5 + 35*a^8*b^8*x^4 - 21*a^10*b^6*x^3 + 7*a^12*b^4*x^2 - a^14*b^2*x)*log(b*sqrt(x) + a) - 2520*(b^16*x^8 -
7*a^2*b^14*x^7 + 21*a^4*b^12*x^6 - 35*a^6*b^10*x^5 + 35*a^8*b^8*x^4 - 21*a^10*b^6*x^3 + 7*a^12*b^4*x^2 - a^14*
b^2*x)*log(sqrt(x)) - 8*(315*a*b^15*x^7 - 2100*a^3*b^13*x^6 + 5943*a^5*b^11*x^5 - 9216*a^7*b^9*x^4 + 8393*a^9*
b^7*x^3 - 4410*a^11*b^5*x^2 + 1225*a^13*b^3*x - 70*a^15*b)*sqrt(x))/(a^10*b^14*x^8 - 7*a^12*b^12*x^7 + 21*a^14
*b^10*x^6 - 35*a^16*b^8*x^5 + 35*a^18*b^6*x^4 - 21*a^20*b^4*x^3 + 7*a^22*b^2*x^2 - a^24*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.13045, size = 181, normalized size = 0.98 \begin{align*} -\frac{72 \, b^{2} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{10}} + \frac{36 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{10}} + \frac{2520 \, a b^{8} x^{4} + 16380 \, a^{2} b^{7} x^{\frac{7}{2}} + 44940 \, a^{3} b^{6} x^{3} + 66990 \, a^{4} b^{5} x^{\frac{5}{2}} + 57834 \, a^{5} b^{4} x^{2} + 28098 \, a^{6} b^{3} x^{\frac{3}{2}} + 6534 \, a^{7} b^{2} x + 315 \, a^{8} b \sqrt{x} - 35 \, a^{9}}{35 \,{\left (b \sqrt{x} + a\right )}^{7} a^{10} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-72*b^2*log(abs(b*sqrt(x) + a))/a^10 + 36*b^2*log(abs(x))/a^10 + 1/35*(2520*a*b^8*x^4 + 16380*a^2*b^7*x^(7/2)
+ 44940*a^3*b^6*x^3 + 66990*a^4*b^5*x^(5/2) + 57834*a^5*b^4*x^2 + 28098*a^6*b^3*x^(3/2) + 6534*a^7*b^2*x + 315
*a^8*b*sqrt(x) - 35*a^9)/((b*sqrt(x) + a)^7*a^10*x)