3.2228 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^8 x^2} \, dx\)

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

[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*Sq
rt[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*Lo
g[x])/a^10

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Rubi [A]  time = 0.343844, 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 \[ -\frac{72 b^2 \log \left (a+b \sqrt{x}\right )}{a^{10}}+\frac{36 b^2 \log (x)}{a^{10}}+\frac{56 b^2}{a^9 \left (a+b \sqrt{x}\right )}+\frac{16 b}{a^9 \sqrt{x}}+\frac{21 b^2}{a^8 \left (a+b \sqrt{x}\right )^2}-\frac{1}{a^8 x}+\frac{10 b^2}{a^7 \left (a+b \sqrt{x}\right )^3}+\frac{5 b^2}{a^6 \left (a+b \sqrt{x}\right )^4}+\frac{12 b^2}{5 a^5 \left (a+b \sqrt{x}\right )^5}+\frac{b^2}{a^4 \left (a+b \sqrt{x}\right )^6}+\frac{2 b^2}{7 a^3 \left (a+b \sqrt{x}\right )^7} \]

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*Sq
rt[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*Lo
g[x])/a^10

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(a+b*x**(1/2))**8,x)

[Out]

Timed out

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Mathematica [A]  time = 0.228447, size = 139, normalized size = 0.76 \[ \frac{\frac{a \left (-35 a^8+315 a^7 b \sqrt{x}+6534 a^6 b^2 x+28098 a^5 b^3 x^{3/2}+57834 a^4 b^4 x^2+66990 a^3 b^5 x^{5/2}+44940 a^2 b^6 x^3+16380 a b^7 x^{7/2}+2520 b^8 x^4\right )}{x \left (a+b \sqrt{x}\right )^7}-2520 b^2 \log \left (a+b \sqrt{x}\right )+1260 b^2 \log (x)}{35 a^{10}} \]

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) + 5783
4*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 = 0.022, size = 163, normalized size = 0.9 \[ -{\frac{1}{{a}^{8}x}}+36\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{10}}}-72\,{\frac{{b}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{10}}}+16\,{\frac{b}{{a}^{9}\sqrt{x}}}+{\frac{2\,{b}^{2}}{7\,{a}^{3}} \left ( a+b\sqrt{x} \right ) ^{-7}}+{\frac{{b}^{2}}{{a}^{4}} \left ( a+b\sqrt{x} \right ) ^{-6}}+{\frac{12\,{b}^{2}}{5\,{a}^{5}} \left ( a+b\sqrt{x} \right ) ^{-5}}+5\,{\frac{{b}^{2}}{{a}^{6} \left ( a+b\sqrt{x} \right ) ^{4}}}+10\,{\frac{{b}^{2}}{{a}^{7} \left ( a+b\sqrt{x} \right ) ^{3}}}+21\,{\frac{{b}^{2}}{{a}^{8} \left ( a+b\sqrt{x} \right ) ^{2}}}+56\,{\frac{{b}^{2}}{{a}^{9} \left ( a+b\sqrt{x} \right ) }} \]

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 = 1.46834, size = 265, normalized size = 1.44 \[ \frac{2520 \, b^{8} x^{4} + 16380 \, a b^{7} x^{\frac{7}{2}} + 44940 \, a^{2} b^{6} x^{3} + 66990 \, a^{3} b^{5} x^{\frac{5}{2}} + 57834 \, a^{4} b^{4} x^{2} + 28098 \, a^{5} b^{3} x^{\frac{3}{2}} + 6534 \, a^{6} b^{2} x + 315 \, a^{7} b \sqrt{x} - 35 \, a^{8}}{35 \,{\left (a^{9} b^{7} x^{\frac{9}{2}} + 7 \, a^{10} b^{6} x^{4} + 21 \, a^{11} b^{5} x^{\frac{7}{2}} + 35 \, a^{12} b^{4} x^{3} + 35 \, a^{13} b^{3} x^{\frac{5}{2}} + 21 \, a^{14} b^{2} x^{2} + 7 \, a^{15} b x^{\frac{3}{2}} + a^{16} x\right )}} - \frac{72 \, b^{2} \log \left (b \sqrt{x} + a\right )}{a^{10}} + \frac{36 \, b^{2} \log \left (x\right )}{a^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^8*x^2),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 + 28098*a^5*b^3*x^(3/2) + 6534*a^6*b^2*x + 315*a^7*b*sq
rt(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^1
2*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*log(b*sqrt(x) + a)/a^10 + 36*b^2*log(x)/a^10

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Fricas [A]  time = 0.254873, size = 483, normalized size = 2.62 \[ \frac{2520 \, a b^{8} x^{4} + 44940 \, a^{3} b^{6} x^{3} + 57834 \, a^{5} b^{4} x^{2} + 6534 \, a^{7} b^{2} x - 35 \, a^{9} - 2520 \,{\left (7 \, a b^{8} x^{4} + 35 \, a^{3} b^{6} x^{3} + 21 \, a^{5} b^{4} x^{2} + a^{7} b^{2} x +{\left (b^{9} x^{4} + 21 \, a^{2} b^{7} x^{3} + 35 \, a^{4} b^{5} x^{2} + 7 \, a^{6} b^{3} x\right )} \sqrt{x}\right )} \log \left (b \sqrt{x} + a\right ) + 2520 \,{\left (7 \, a b^{8} x^{4} + 35 \, a^{3} b^{6} x^{3} + 21 \, a^{5} b^{4} x^{2} + a^{7} b^{2} x +{\left (b^{9} x^{4} + 21 \, a^{2} b^{7} x^{3} + 35 \, a^{4} b^{5} x^{2} + 7 \, a^{6} b^{3} x\right )} \sqrt{x}\right )} \log \left (\sqrt{x}\right ) + 21 \,{\left (780 \, a^{2} b^{7} x^{3} + 3190 \, a^{4} b^{5} x^{2} + 1338 \, a^{6} b^{3} x + 15 \, a^{8} b\right )} \sqrt{x}}{35 \,{\left (7 \, a^{11} b^{6} x^{4} + 35 \, a^{13} b^{4} x^{3} + 21 \, a^{15} b^{2} x^{2} + a^{17} x +{\left (a^{10} b^{7} x^{4} + 21 \, a^{12} b^{5} x^{3} + 35 \, a^{14} b^{3} x^{2} + 7 \, a^{16} b x\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/35*(2520*a*b^8*x^4 + 44940*a^3*b^6*x^3 + 57834*a^5*b^4*x^2 + 6534*a^7*b^2*x -
35*a^9 - 2520*(7*a*b^8*x^4 + 35*a^3*b^6*x^3 + 21*a^5*b^4*x^2 + a^7*b^2*x + (b^9*
x^4 + 21*a^2*b^7*x^3 + 35*a^4*b^5*x^2 + 7*a^6*b^3*x)*sqrt(x))*log(b*sqrt(x) + a)
+ 2520*(7*a*b^8*x^4 + 35*a^3*b^6*x^3 + 21*a^5*b^4*x^2 + a^7*b^2*x + (b^9*x^4 +
21*a^2*b^7*x^3 + 35*a^4*b^5*x^2 + 7*a^6*b^3*x)*sqrt(x))*log(sqrt(x)) + 21*(780*a
^2*b^7*x^3 + 3190*a^4*b^5*x^2 + 1338*a^6*b^3*x + 15*a^8*b)*sqrt(x))/(7*a^11*b^6*
x^4 + 35*a^13*b^4*x^3 + 21*a^15*b^2*x^2 + a^17*x + (a^10*b^7*x^4 + 21*a^12*b^5*x
^3 + 35*a^14*b^3*x^2 + 7*a^16*b*x)*sqrt(x))

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

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/XCAS [A]  time = 0.239319, size = 181, normalized size = 0.98 \[ -\frac{72 \, b^{2}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{10}} + \frac{36 \, b^{2}{\rm ln}\left ({\left | x \right |}\right )}{a^{10}} + \frac{2520 \, a b^{8} x^{4} + 16380 \, a^{2} b^{7} x^{\frac{7}{2}} + 44940 \, a^{3} b^{6} x^{3} + 66990 \, a^{4} b^{5} x^{\frac{5}{2}} + 57834 \, a^{5} b^{4} x^{2} + 28098 \, a^{6} b^{3} x^{\frac{3}{2}} + 6534 \, a^{7} b^{2} x + 315 \, a^{8} b \sqrt{x} - 35 \, a^{9}}{35 \,{\left (b \sqrt{x} + a\right )}^{7} a^{10} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-72*b^2*ln(abs(b*sqrt(x) + a))/a^10 + 36*b^2*ln(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)