3.337 \(\int \frac{1}{\sqrt{x} \left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=243 \[ -\frac{11 c^{7/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{15/4}}+\frac{11 c^{7/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{15/4}}-\frac{11 c^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{15/4}}+\frac{11 c^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{15/4}}+\frac{11 c}{6 b^3 x^{3/2}}-\frac{11}{14 b^2 x^{7/2}}+\frac{1}{2 b x^{7/2} \left (b+c x^2\right )} \]

[Out]

-11/(14*b^2*x^(7/2)) + (11*c)/(6*b^3*x^(3/2)) + 1/(2*b*x^(7/2)*(b + c*x^2)) - (1
1*c^(7/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(15/4)) +
(11*c^(7/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(15/4))
- (11*c^(7/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqr
t[2]*b^(15/4)) + (11*c^(7/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqr
t[c]*x])/(8*Sqrt[2]*b^(15/4))

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Rubi [A]  time = 0.433978, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ -\frac{11 c^{7/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{15/4}}+\frac{11 c^{7/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{15/4}}-\frac{11 c^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{15/4}}+\frac{11 c^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{15/4}}+\frac{11 c}{6 b^3 x^{3/2}}-\frac{11}{14 b^2 x^{7/2}}+\frac{1}{2 b x^{7/2} \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[x]*(b*x^2 + c*x^4)^2),x]

[Out]

-11/(14*b^2*x^(7/2)) + (11*c)/(6*b^3*x^(3/2)) + 1/(2*b*x^(7/2)*(b + c*x^2)) - (1
1*c^(7/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(15/4)) +
(11*c^(7/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(15/4))
- (11*c^(7/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqr
t[2]*b^(15/4)) + (11*c^(7/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqr
t[c]*x])/(8*Sqrt[2]*b^(15/4))

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Rubi in Sympy [A]  time = 78.5541, size = 231, normalized size = 0.95 \[ \frac{1}{2 b x^{\frac{7}{2}} \left (b + c x^{2}\right )} - \frac{11}{14 b^{2} x^{\frac{7}{2}}} + \frac{11 c}{6 b^{3} x^{\frac{3}{2}}} - \frac{11 \sqrt{2} c^{\frac{7}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{15}{4}}} + \frac{11 \sqrt{2} c^{\frac{7}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{15}{4}}} - \frac{11 \sqrt{2} c^{\frac{7}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{15}{4}}} + \frac{11 \sqrt{2} c^{\frac{7}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{15}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*b*x**(7/2)*(b + c*x**2)) - 11/(14*b**2*x**(7/2)) + 11*c/(6*b**3*x**(3/2)) -
11*sqrt(2)*c**(7/4)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*
x)/(16*b**(15/4)) + 11*sqrt(2)*c**(7/4)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) +
sqrt(b) + sqrt(c)*x)/(16*b**(15/4)) - 11*sqrt(2)*c**(7/4)*atan(1 - sqrt(2)*c**(1
/4)*sqrt(x)/b**(1/4))/(8*b**(15/4)) + 11*sqrt(2)*c**(7/4)*atan(1 + sqrt(2)*c**(1
/4)*sqrt(x)/b**(1/4))/(8*b**(15/4))

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Mathematica [A]  time = 0.429004, size = 227, normalized size = 0.93 \[ \frac{\frac{168 b^{3/4} c^2 \sqrt{x}}{b+c x^2}+\frac{448 b^{3/4} c}{x^{3/2}}-\frac{96 b^{7/4}}{x^{7/2}}-231 \sqrt{2} c^{7/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+231 \sqrt{2} c^{7/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-462 \sqrt{2} c^{7/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+462 \sqrt{2} c^{7/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{336 b^{15/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[x]*(b*x^2 + c*x^4)^2),x]

[Out]

((-96*b^(7/4))/x^(7/2) + (448*b^(3/4)*c)/x^(3/2) + (168*b^(3/4)*c^2*Sqrt[x])/(b
+ c*x^2) - 462*Sqrt[2]*c^(7/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 4
62*Sqrt[2]*c^(7/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 231*Sqrt[2]*c
^(7/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 231*Sqrt[2]*
c^(7/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(336*b^(15/4
))

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Maple [A]  time = 0.022, size = 178, normalized size = 0.7 \[{\frac{{c}^{2}}{2\,{b}^{3} \left ( c{x}^{2}+b \right ) }\sqrt{x}}+{\frac{11\,{c}^{2}\sqrt{2}}{16\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{11\,{c}^{2}\sqrt{2}}{8\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{11\,{c}^{2}\sqrt{2}}{8\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{2}{7\,{b}^{2}}{x}^{-{\frac{7}{2}}}}+{\frac{4\,c}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c*x^4+b*x^2)^2/x^(1/2),x)

[Out]

1/2/b^3*c^2*x^(1/2)/(c*x^2+b)+11/16/b^4*c^2*(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)))+11/8
/b^4*c^2*(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+11/8/b^4*c^2*
(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)-2/7/b^2/x^(7/2)+4/3*c/
b^3/x^(3/2)

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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(1/((c*x^4 + b*x^2)^2*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289265, size = 316, normalized size = 1.3 \[ \frac{308 \, c^{2} x^{4} + 176 \, b c x^{2} - 924 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )} \sqrt{x} \left (-\frac{c^{7}}{b^{15}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{4} \left (-\frac{c^{7}}{b^{15}}\right )^{\frac{1}{4}}}{c^{2} \sqrt{x} + \sqrt{b^{8} \sqrt{-\frac{c^{7}}{b^{15}}} + c^{4} x}}\right ) + 231 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )} \sqrt{x} \left (-\frac{c^{7}}{b^{15}}\right )^{\frac{1}{4}} \log \left (11 \, b^{4} \left (-\frac{c^{7}}{b^{15}}\right )^{\frac{1}{4}} + 11 \, c^{2} \sqrt{x}\right ) - 231 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )} \sqrt{x} \left (-\frac{c^{7}}{b^{15}}\right )^{\frac{1}{4}} \log \left (-11 \, b^{4} \left (-\frac{c^{7}}{b^{15}}\right )^{\frac{1}{4}} + 11 \, c^{2} \sqrt{x}\right ) - 48 \, b^{2}}{168 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)^2*sqrt(x)),x, algorithm="fricas")

[Out]

1/168*(308*c^2*x^4 + 176*b*c*x^2 - 924*(b^3*c*x^5 + b^4*x^3)*sqrt(x)*(-c^7/b^15)
^(1/4)*arctan(b^4*(-c^7/b^15)^(1/4)/(c^2*sqrt(x) + sqrt(b^8*sqrt(-c^7/b^15) + c^
4*x))) + 231*(b^3*c*x^5 + b^4*x^3)*sqrt(x)*(-c^7/b^15)^(1/4)*log(11*b^4*(-c^7/b^
15)^(1/4) + 11*c^2*sqrt(x)) - 231*(b^3*c*x^5 + b^4*x^3)*sqrt(x)*(-c^7/b^15)^(1/4
)*log(-11*b^4*(-c^7/b^15)^(1/4) + 11*c^2*sqrt(x)) - 48*b^2)/((b^3*c*x^5 + b^4*x^
3)*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/(c*x**4+b*x**2)**2/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276142, size = 286, normalized size = 1.18 \[ \frac{11 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{4}} + \frac{11 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{4}} + \frac{11 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{4}} - \frac{11 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} c{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{4}} + \frac{c^{2} \sqrt{x}}{2 \,{\left (c x^{2} + b\right )} b^{3}} + \frac{2 \,{\left (14 \, c x^{2} - 3 \, b\right )}}{21 \, b^{3} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)^2*sqrt(x)),x, algorithm="giac")

[Out]

11/8*sqrt(2)*(b*c^3)^(1/4)*c*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x)
)/(b/c)^(1/4))/b^4 + 11/8*sqrt(2)*(b*c^3)^(1/4)*c*arctan(-1/2*sqrt(2)*(sqrt(2)*(
b/c)^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/b^4 + 11/16*sqrt(2)*(b*c^3)^(1/4)*c*ln(sqrt
(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^4 - 11/16*sqrt(2)*(b*c^3)^(1/4)*c*ln(
-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^4 + 1/2*c^2*sqrt(x)/((c*x^2 + b)
*b^3) + 2/21*(14*c*x^2 - 3*b)/(b^3*x^(7/2))