3.350 \(\int \frac{\sqrt{x}}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=279 \[ -\frac{221 c^{9/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}+\frac{221 c^{9/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}+\frac{221 c^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{21/4}}-\frac{221 c^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{21/4}}-\frac{221 c^2}{16 b^5 \sqrt{x}}+\frac{221 c}{80 b^4 x^{5/2}}-\frac{221}{144 b^3 x^{9/2}}+\frac{17}{16 b^2 x^{9/2} \left (b+c x^2\right )}+\frac{1}{4 b x^{9/2} \left (b+c x^2\right )^2} \]

[Out]

-221/(144*b^3*x^(9/2)) + (221*c)/(80*b^4*x^(5/2)) - (221*c^2)/(16*b^5*Sqrt[x]) +
1/(4*b*x^(9/2)*(b + c*x^2)^2) + 17/(16*b^2*x^(9/2)*(b + c*x^2)) + (221*c^(9/4)*
ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(21/4)) - (221*c^(9
/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(21/4)) - (221*
c^(9/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*
b^(21/4)) + (221*c^(9/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]
*x])/(64*Sqrt[2]*b^(21/4))

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Rubi [A]  time = 0.548161, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ -\frac{221 c^{9/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}+\frac{221 c^{9/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{21/4}}+\frac{221 c^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{21/4}}-\frac{221 c^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{21/4}}-\frac{221 c^2}{16 b^5 \sqrt{x}}+\frac{221 c}{80 b^4 x^{5/2}}-\frac{221}{144 b^3 x^{9/2}}+\frac{17}{16 b^2 x^{9/2} \left (b+c x^2\right )}+\frac{1}{4 b x^{9/2} \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(b*x^2 + c*x^4)^3,x]

[Out]

-221/(144*b^3*x^(9/2)) + (221*c)/(80*b^4*x^(5/2)) - (221*c^2)/(16*b^5*Sqrt[x]) +
1/(4*b*x^(9/2)*(b + c*x^2)^2) + 17/(16*b^2*x^(9/2)*(b + c*x^2)) + (221*c^(9/4)*
ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(21/4)) - (221*c^(9
/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(21/4)) - (221*
c^(9/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*
b^(21/4)) + (221*c^(9/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]
*x])/(64*Sqrt[2]*b^(21/4))

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Rubi in Sympy [A]  time = 104.615, size = 267, normalized size = 0.96 \[ \frac{1}{4 b x^{\frac{9}{2}} \left (b + c x^{2}\right )^{2}} + \frac{17}{16 b^{2} x^{\frac{9}{2}} \left (b + c x^{2}\right )} - \frac{221}{144 b^{3} x^{\frac{9}{2}}} + \frac{221 c}{80 b^{4} x^{\frac{5}{2}}} - \frac{221 c^{2}}{16 b^{5} \sqrt{x}} - \frac{221 \sqrt{2} c^{\frac{9}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{21}{4}}} + \frac{221 \sqrt{2} c^{\frac{9}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{21}{4}}} + \frac{221 \sqrt{2} c^{\frac{9}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{21}{4}}} - \frac{221 \sqrt{2} c^{\frac{9}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{21}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(c*x**4+b*x**2)**3,x)

[Out]

1/(4*b*x**(9/2)*(b + c*x**2)**2) + 17/(16*b**2*x**(9/2)*(b + c*x**2)) - 221/(144
*b**3*x**(9/2)) + 221*c/(80*b**4*x**(5/2)) - 221*c**2/(16*b**5*sqrt(x)) - 221*sq
rt(2)*c**(9/4)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(12
8*b**(21/4)) + 221*sqrt(2)*c**(9/4)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt
(b) + sqrt(c)*x)/(128*b**(21/4)) + 221*sqrt(2)*c**(9/4)*atan(1 - sqrt(2)*c**(1/4
)*sqrt(x)/b**(1/4))/(64*b**(21/4)) - 221*sqrt(2)*c**(9/4)*atan(1 + sqrt(2)*c**(1
/4)*sqrt(x)/b**(1/4))/(64*b**(21/4))

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Mathematica [A]  time = 0.281639, size = 266, normalized size = 0.95 \[ \frac{-\frac{1440 b^{5/4} c^3 x^{3/2}}{\left (b+c x^2\right )^2}+\frac{6912 b^{5/4} c}{x^{5/2}}-\frac{1280 b^{9/4}}{x^{9/2}}-9945 \sqrt{2} c^{9/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+9945 \sqrt{2} c^{9/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+19890 \sqrt{2} c^{9/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-19890 \sqrt{2} c^{9/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )-\frac{10440 \sqrt [4]{b} c^3 x^{3/2}}{b+c x^2}-\frac{69120 \sqrt [4]{b} c^2}{\sqrt{x}}}{5760 b^{21/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(b*x^2 + c*x^4)^3,x]

[Out]

((-1280*b^(9/4))/x^(9/2) + (6912*b^(5/4)*c)/x^(5/2) - (69120*b^(1/4)*c^2)/Sqrt[x
] - (1440*b^(5/4)*c^3*x^(3/2))/(b + c*x^2)^2 - (10440*b^(1/4)*c^3*x^(3/2))/(b +
c*x^2) + 19890*Sqrt[2]*c^(9/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 1
9890*Sqrt[2]*c^(9/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 9945*Sqrt[2
]*c^(9/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 9945*Sqrt
[2]*c^(9/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(5760*b^
(21/4))

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Maple [A]  time = 0.029, size = 209, normalized size = 0.8 \[ -{\frac{29\,{c}^{4}}{16\,{b}^{5} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{7}{2}}}}-{\frac{33\,{c}^{3}}{16\,{b}^{4} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{221\,{c}^{2}\sqrt{2}}{128\,{b}^{5}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{221\,{c}^{2}\sqrt{2}}{64\,{b}^{5}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{221\,{c}^{2}\sqrt{2}}{64\,{b}^{5}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{2}{9\,{b}^{3}}{x}^{-{\frac{9}{2}}}}-12\,{\frac{{c}^{2}}{{b}^{5}\sqrt{x}}}+{\frac{6\,c}{5\,{b}^{4}}{x}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(c*x^4+b*x^2)^3,x)

[Out]

-29/16*c^4/b^5/(c*x^2+b)^2*x^(7/2)-33/16*c^3/b^4/(c*x^2+b)^2*x^(3/2)-221/128*c^2
/b^5/(b/c)^(1/4)*2^(1/2)*ln((x-(b/c)^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2))/(x+(b/c)
^(1/4)*x^(1/2)*2^(1/2)+(b/c)^(1/2)))-221/64*c^2/b^5/(b/c)^(1/4)*2^(1/2)*arctan(2
^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-221/64*c^2/b^5/(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/
(b/c)^(1/4)*x^(1/2)-1)-2/9/b^3/x^(9/2)-12*c^2/b^5/x^(1/2)+6/5*c/b^4/x^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290753, size = 414, normalized size = 1.48 \[ -\frac{39780 \, c^{4} x^{8} + 71604 \, b c^{3} x^{6} + 28288 \, b^{2} c^{2} x^{4} - 2176 \, b^{3} c x^{2} + 640 \, b^{4} + 39780 \,{\left (b^{5} c^{2} x^{8} + 2 \, b^{6} c x^{6} + b^{7} x^{4}\right )} \sqrt{x} \left (-\frac{c^{9}}{b^{21}}\right )^{\frac{1}{4}} \arctan \left (\frac{10793861 \, b^{16} \left (-\frac{c^{9}}{b^{21}}\right )^{\frac{3}{4}}}{10793861 \, c^{7} \sqrt{x} + \sqrt{-116507435287321 \, b^{11} c^{9} \sqrt{-\frac{c^{9}}{b^{21}}} + 116507435287321 \, c^{14} x}}\right ) + 9945 \,{\left (b^{5} c^{2} x^{8} + 2 \, b^{6} c x^{6} + b^{7} x^{4}\right )} \sqrt{x} \left (-\frac{c^{9}}{b^{21}}\right )^{\frac{1}{4}} \log \left (10793861 \, b^{16} \left (-\frac{c^{9}}{b^{21}}\right )^{\frac{3}{4}} + 10793861 \, c^{7} \sqrt{x}\right ) - 9945 \,{\left (b^{5} c^{2} x^{8} + 2 \, b^{6} c x^{6} + b^{7} x^{4}\right )} \sqrt{x} \left (-\frac{c^{9}}{b^{21}}\right )^{\frac{1}{4}} \log \left (-10793861 \, b^{16} \left (-\frac{c^{9}}{b^{21}}\right )^{\frac{3}{4}} + 10793861 \, c^{7} \sqrt{x}\right )}{2880 \,{\left (b^{5} c^{2} x^{8} + 2 \, b^{6} c x^{6} + b^{7} x^{4}\right )} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

-1/2880*(39780*c^4*x^8 + 71604*b*c^3*x^6 + 28288*b^2*c^2*x^4 - 2176*b^3*c*x^2 +
640*b^4 + 39780*(b^5*c^2*x^8 + 2*b^6*c*x^6 + b^7*x^4)*sqrt(x)*(-c^9/b^21)^(1/4)*
arctan(10793861*b^16*(-c^9/b^21)^(3/4)/(10793861*c^7*sqrt(x) + sqrt(-11650743528
7321*b^11*c^9*sqrt(-c^9/b^21) + 116507435287321*c^14*x))) + 9945*(b^5*c^2*x^8 +
2*b^6*c*x^6 + b^7*x^4)*sqrt(x)*(-c^9/b^21)^(1/4)*log(10793861*b^16*(-c^9/b^21)^(
3/4) + 10793861*c^7*sqrt(x)) - 9945*(b^5*c^2*x^8 + 2*b^6*c*x^6 + b^7*x^4)*sqrt(x
)*(-c^9/b^21)^(1/4)*log(-10793861*b^16*(-c^9/b^21)^(3/4) + 10793861*c^7*sqrt(x))
)/((b^5*c^2*x^8 + 2*b^6*c*x^6 + b^7*x^4)*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(x**(1/2)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282112, size = 312, normalized size = 1.12 \[ -\frac{221 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{6}} - \frac{221 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{6}} + \frac{221 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{6}} - \frac{221 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{6}} - \frac{29 \, c^{4} x^{\frac{7}{2}} + 33 \, b c^{3} x^{\frac{3}{2}}}{16 \,{\left (c x^{2} + b\right )}^{2} b^{5}} - \frac{2 \,{\left (270 \, c^{2} x^{4} - 27 \, b c x^{2} + 5 \, b^{2}\right )}}{45 \, b^{5} x^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

-221/64*sqrt(2)*(b*c^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x
))/(b/c)^(1/4))/b^6 - 221/64*sqrt(2)*(b*c^3)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*
(b/c)^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/b^6 + 221/128*sqrt(2)*(b*c^3)^(3/4)*ln(sqr
t(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^6 - 221/128*sqrt(2)*(b*c^3)^(3/4)*ln
(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^6 - 1/16*(29*c^4*x^(7/2) + 33*b
*c^3*x^(3/2))/((c*x^2 + b)^2*b^5) - 2/45*(270*c^2*x^4 - 27*b*c*x^2 + 5*b^2)/(b^5
*x^(9/2))