3.2229 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^8 x^3} \, dx\)

Optimal. Leaf size=217 \[ -\frac{660 b^4 \log \left (a+b \sqrt{x}\right )}{a^{12}}+\frac{330 b^4 \log (x)}{a^{12}}+\frac{420 b^4}{a^{11} \left (a+b \sqrt{x}\right )}+\frac{240 b^3}{a^{11} \sqrt{x}}+\frac{126 b^4}{a^{10} \left (a+b \sqrt{x}\right )^2}-\frac{36 b^2}{a^{10} x}+\frac{140 b^4}{3 a^9 \left (a+b \sqrt{x}\right )^3}+\frac{16 b}{3 a^9 x^{3/2}}+\frac{35 b^4}{2 a^8 \left (a+b \sqrt{x}\right )^4}-\frac{1}{2 a^8 x^2}+\frac{6 b^4}{a^7 \left (a+b \sqrt{x}\right )^5}+\frac{5 b^4}{3 a^6 \left (a+b \sqrt{x}\right )^6}+\frac{2 b^4}{7 a^5 \left (a+b \sqrt{x}\right )^7} \]

[Out]

(2*b^4)/(7*a^5*(a + b*Sqrt[x])^7) + (5*b^4)/(3*a^6*(a + b*Sqrt[x])^6) + (6*b^4)/
(a^7*(a + b*Sqrt[x])^5) + (35*b^4)/(2*a^8*(a + b*Sqrt[x])^4) + (140*b^4)/(3*a^9*
(a + b*Sqrt[x])^3) + (126*b^4)/(a^10*(a + b*Sqrt[x])^2) + (420*b^4)/(a^11*(a + b
*Sqrt[x])) - 1/(2*a^8*x^2) + (16*b)/(3*a^9*x^(3/2)) - (36*b^2)/(a^10*x) + (240*b
^3)/(a^11*Sqrt[x]) - (660*b^4*Log[a + b*Sqrt[x]])/a^12 + (330*b^4*Log[x])/a^12

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Rubi [A]  time = 0.435843, antiderivative size = 217, 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 \[ -\frac{660 b^4 \log \left (a+b \sqrt{x}\right )}{a^{12}}+\frac{330 b^4 \log (x)}{a^{12}}+\frac{420 b^4}{a^{11} \left (a+b \sqrt{x}\right )}+\frac{240 b^3}{a^{11} \sqrt{x}}+\frac{126 b^4}{a^{10} \left (a+b \sqrt{x}\right )^2}-\frac{36 b^2}{a^{10} x}+\frac{140 b^4}{3 a^9 \left (a+b \sqrt{x}\right )^3}+\frac{16 b}{3 a^9 x^{3/2}}+\frac{35 b^4}{2 a^8 \left (a+b \sqrt{x}\right )^4}-\frac{1}{2 a^8 x^2}+\frac{6 b^4}{a^7 \left (a+b \sqrt{x}\right )^5}+\frac{5 b^4}{3 a^6 \left (a+b \sqrt{x}\right )^6}+\frac{2 b^4}{7 a^5 \left (a+b \sqrt{x}\right )^7} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*Sqrt[x])^8*x^3),x]

[Out]

(2*b^4)/(7*a^5*(a + b*Sqrt[x])^7) + (5*b^4)/(3*a^6*(a + b*Sqrt[x])^6) + (6*b^4)/
(a^7*(a + b*Sqrt[x])^5) + (35*b^4)/(2*a^8*(a + b*Sqrt[x])^4) + (140*b^4)/(3*a^9*
(a + b*Sqrt[x])^3) + (126*b^4)/(a^10*(a + b*Sqrt[x])^2) + (420*b^4)/(a^11*(a + b
*Sqrt[x])) - 1/(2*a^8*x^2) + (16*b)/(3*a^9*x^(3/2)) - (36*b^2)/(a^10*x) + (240*b
^3)/(a^11*Sqrt[x]) - (660*b^4*Log[a + b*Sqrt[x]])/a^12 + (330*b^4*Log[x])/a^12

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

[Out]

Timed out

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Mathematica [A]  time = 0.249116, size = 163, normalized size = 0.75 \[ \frac{\frac{a \left (-21 a^{10}+77 a^9 b \sqrt{x}-385 a^8 b^2 x+3465 a^7 b^3 x^{3/2}+71874 a^6 b^4 x^2+309078 a^5 b^5 x^{5/2}+636174 a^4 b^6 x^3+736890 a^3 b^7 x^{7/2}+494340 a^2 b^8 x^4+180180 a b^9 x^{9/2}+27720 b^{10} x^5\right )}{x^2 \left (a+b \sqrt{x}\right )^7}-27720 b^4 \log \left (a+b \sqrt{x}\right )+13860 b^4 \log (x)}{42 a^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*Sqrt[x])^8*x^3),x]

[Out]

((a*(-21*a^10 + 77*a^9*b*Sqrt[x] - 385*a^8*b^2*x + 3465*a^7*b^3*x^(3/2) + 71874*
a^6*b^4*x^2 + 309078*a^5*b^5*x^(5/2) + 636174*a^4*b^6*x^3 + 736890*a^3*b^7*x^(7/
2) + 494340*a^2*b^8*x^4 + 180180*a*b^9*x^(9/2) + 27720*b^10*x^5))/((a + b*Sqrt[x
])^7*x^2) - 27720*b^4*Log[a + b*Sqrt[x]] + 13860*b^4*Log[x])/(42*a^12)

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Maple [A]  time = 0.022, size = 186, normalized size = 0.9 \[ -{\frac{1}{2\,{a}^{8}{x}^{2}}}+{\frac{16\,b}{3\,{a}^{9}}{x}^{-{\frac{3}{2}}}}-36\,{\frac{{b}^{2}}{x{a}^{10}}}+330\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{12}}}-660\,{\frac{{b}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{12}}}+240\,{\frac{{b}^{3}}{{a}^{11}\sqrt{x}}}+{\frac{2\,{b}^{4}}{7\,{a}^{5}} \left ( a+b\sqrt{x} \right ) ^{-7}}+{\frac{5\,{b}^{4}}{3\,{a}^{6}} \left ( a+b\sqrt{x} \right ) ^{-6}}+6\,{\frac{{b}^{4}}{{a}^{7} \left ( a+b\sqrt{x} \right ) ^{5}}}+{\frac{35\,{b}^{4}}{2\,{a}^{8}} \left ( a+b\sqrt{x} \right ) ^{-4}}+{\frac{140\,{b}^{4}}{3\,{a}^{9}} \left ( a+b\sqrt{x} \right ) ^{-3}}+126\,{\frac{{b}^{4}}{{a}^{10} \left ( a+b\sqrt{x} \right ) ^{2}}}+420\,{\frac{{b}^{4}}{{a}^{11} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a+b*x^(1/2))^8,x)

[Out]

-1/2/a^8/x^2+16/3*b/a^9/x^(3/2)-36*b^2/a^10/x+330*b^4*ln(x)/a^12-660*b^4*ln(a+b*
x^(1/2))/a^12+240*b^3/a^11/x^(1/2)+2/7*b^4/a^5/(a+b*x^(1/2))^7+5/3*b^4/a^6/(a+b*
x^(1/2))^6+6*b^4/a^7/(a+b*x^(1/2))^5+35/2*b^4/a^8/(a+b*x^(1/2))^4+140/3*b^4/a^9/
(a+b*x^(1/2))^3+126*b^4/a^10/(a+b*x^(1/2))^2+420*b^4/a^11/(a+b*x^(1/2))

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Maxima [A]  time = 1.47591, size = 297, normalized size = 1.37 \[ \frac{27720 \, b^{10} x^{5} + 180180 \, a b^{9} x^{\frac{9}{2}} + 494340 \, a^{2} b^{8} x^{4} + 736890 \, a^{3} b^{7} x^{\frac{7}{2}} + 636174 \, a^{4} b^{6} x^{3} + 309078 \, a^{5} b^{5} x^{\frac{5}{2}} + 71874 \, a^{6} b^{4} x^{2} + 3465 \, a^{7} b^{3} x^{\frac{3}{2}} - 385 \, a^{8} b^{2} x + 77 \, a^{9} b \sqrt{x} - 21 \, a^{10}}{42 \,{\left (a^{11} b^{7} x^{\frac{11}{2}} + 7 \, a^{12} b^{6} x^{5} + 21 \, a^{13} b^{5} x^{\frac{9}{2}} + 35 \, a^{14} b^{4} x^{4} + 35 \, a^{15} b^{3} x^{\frac{7}{2}} + 21 \, a^{16} b^{2} x^{3} + 7 \, a^{17} b x^{\frac{5}{2}} + a^{18} x^{2}\right )}} - \frac{660 \, b^{4} \log \left (b \sqrt{x} + a\right )}{a^{12}} + \frac{330 \, b^{4} \log \left (x\right )}{a^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^8*x^3),x, algorithm="maxima")

[Out]

1/42*(27720*b^10*x^5 + 180180*a*b^9*x^(9/2) + 494340*a^2*b^8*x^4 + 736890*a^3*b^
7*x^(7/2) + 636174*a^4*b^6*x^3 + 309078*a^5*b^5*x^(5/2) + 71874*a^6*b^4*x^2 + 34
65*a^7*b^3*x^(3/2) - 385*a^8*b^2*x + 77*a^9*b*sqrt(x) - 21*a^10)/(a^11*b^7*x^(11
/2) + 7*a^12*b^6*x^5 + 21*a^13*b^5*x^(9/2) + 35*a^14*b^4*x^4 + 35*a^15*b^3*x^(7/
2) + 21*a^16*b^2*x^3 + 7*a^17*b*x^(5/2) + a^18*x^2) - 660*b^4*log(b*sqrt(x) + a)
/a^12 + 330*b^4*log(x)/a^12

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Fricas [A]  time = 0.257737, size = 528, normalized size = 2.43 \[ \frac{27720 \, a b^{10} x^{5} + 494340 \, a^{3} b^{8} x^{4} + 636174 \, a^{5} b^{6} x^{3} + 71874 \, a^{7} b^{4} x^{2} - 385 \, a^{9} b^{2} x - 21 \, a^{11} - 27720 \,{\left (7 \, a b^{10} x^{5} + 35 \, a^{3} b^{8} x^{4} + 21 \, a^{5} b^{6} x^{3} + a^{7} b^{4} x^{2} +{\left (b^{11} x^{5} + 21 \, a^{2} b^{9} x^{4} + 35 \, a^{4} b^{7} x^{3} + 7 \, a^{6} b^{5} x^{2}\right )} \sqrt{x}\right )} \log \left (b \sqrt{x} + a\right ) + 27720 \,{\left (7 \, a b^{10} x^{5} + 35 \, a^{3} b^{8} x^{4} + 21 \, a^{5} b^{6} x^{3} + a^{7} b^{4} x^{2} +{\left (b^{11} x^{5} + 21 \, a^{2} b^{9} x^{4} + 35 \, a^{4} b^{7} x^{3} + 7 \, a^{6} b^{5} x^{2}\right )} \sqrt{x}\right )} \log \left (\sqrt{x}\right ) + 77 \,{\left (2340 \, a^{2} b^{9} x^{4} + 9570 \, a^{4} b^{7} x^{3} + 4014 \, a^{6} b^{5} x^{2} + 45 \, a^{8} b^{3} x + a^{10} b\right )} \sqrt{x}}{42 \,{\left (7 \, a^{13} b^{6} x^{5} + 35 \, a^{15} b^{4} x^{4} + 21 \, a^{17} b^{2} x^{3} + a^{19} x^{2} +{\left (a^{12} b^{7} x^{5} + 21 \, a^{14} b^{5} x^{4} + 35 \, a^{16} b^{3} x^{3} + 7 \, a^{18} b x^{2}\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^8*x^3),x, algorithm="fricas")

[Out]

1/42*(27720*a*b^10*x^5 + 494340*a^3*b^8*x^4 + 636174*a^5*b^6*x^3 + 71874*a^7*b^4
*x^2 - 385*a^9*b^2*x - 21*a^11 - 27720*(7*a*b^10*x^5 + 35*a^3*b^8*x^4 + 21*a^5*b
^6*x^3 + a^7*b^4*x^2 + (b^11*x^5 + 21*a^2*b^9*x^4 + 35*a^4*b^7*x^3 + 7*a^6*b^5*x
^2)*sqrt(x))*log(b*sqrt(x) + a) + 27720*(7*a*b^10*x^5 + 35*a^3*b^8*x^4 + 21*a^5*
b^6*x^3 + a^7*b^4*x^2 + (b^11*x^5 + 21*a^2*b^9*x^4 + 35*a^4*b^7*x^3 + 7*a^6*b^5*
x^2)*sqrt(x))*log(sqrt(x)) + 77*(2340*a^2*b^9*x^4 + 9570*a^4*b^7*x^3 + 4014*a^6*
b^5*x^2 + 45*a^8*b^3*x + a^10*b)*sqrt(x))/(7*a^13*b^6*x^5 + 35*a^15*b^4*x^4 + 21
*a^17*b^2*x^3 + a^19*x^2 + (a^12*b^7*x^5 + 21*a^14*b^5*x^4 + 35*a^16*b^3*x^3 + 7
*a^18*b*x^2)*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**3/(a+b*x**(1/2))**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221266, size = 211, normalized size = 0.97 \[ -\frac{660 \, b^{4}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{12}} + \frac{330 \, b^{4}{\rm ln}\left ({\left | x \right |}\right )}{a^{12}} + \frac{27720 \, a b^{10} x^{5} + 180180 \, a^{2} b^{9} x^{\frac{9}{2}} + 494340 \, a^{3} b^{8} x^{4} + 736890 \, a^{4} b^{7} x^{\frac{7}{2}} + 636174 \, a^{5} b^{6} x^{3} + 309078 \, a^{6} b^{5} x^{\frac{5}{2}} + 71874 \, a^{7} b^{4} x^{2} + 3465 \, a^{8} b^{3} x^{\frac{3}{2}} - 385 \, a^{9} b^{2} x + 77 \, a^{10} b \sqrt{x} - 21 \, a^{11}}{42 \,{\left (b \sqrt{x} + a\right )}^{7} a^{12} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^8*x^3),x, algorithm="giac")

[Out]

-660*b^4*ln(abs(b*sqrt(x) + a))/a^12 + 330*b^4*ln(abs(x))/a^12 + 1/42*(27720*a*b
^10*x^5 + 180180*a^2*b^9*x^(9/2) + 494340*a^3*b^8*x^4 + 736890*a^4*b^7*x^(7/2) +
636174*a^5*b^6*x^3 + 309078*a^6*b^5*x^(5/2) + 71874*a^7*b^4*x^2 + 3465*a^8*b^3*
x^(3/2) - 385*a^9*b^2*x + 77*a^10*b*sqrt(x) - 21*a^11)/((b*sqrt(x) + a)^7*a^12*x
^2)