3.334 \(\int \frac{x^{5/2}}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=230 \[ -\frac{5 \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{9/4}}-\frac{5}{2 b^2 \sqrt{x}}+\frac{1}{2 b \sqrt{x} \left (b+c x^2\right )} \]

[Out]

-5/(2*b^2*Sqrt[x]) + 1/(2*b*Sqrt[x]*(b + c*x^2)) + (5*c^(1/4)*ArcTan[1 - (Sqrt[2
]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(9/4)) - (5*c^(1/4)*ArcTan[1 + (Sqrt[2
]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(9/4)) - (5*c^(1/4)*Log[Sqrt[b] - Sqrt
[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(9/4)) + (5*c^(1/4)*Log[S
qrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(9/4))

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Rubi [A]  time = 0.386203, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ -\frac{5 \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{9/4}}-\frac{5}{2 b^2 \sqrt{x}}+\frac{1}{2 b \sqrt{x} \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(b*x^2 + c*x^4)^2,x]

[Out]

-5/(2*b^2*Sqrt[x]) + 1/(2*b*Sqrt[x]*(b + c*x^2)) + (5*c^(1/4)*ArcTan[1 - (Sqrt[2
]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(9/4)) - (5*c^(1/4)*ArcTan[1 + (Sqrt[2
]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(9/4)) - (5*c^(1/4)*Log[Sqrt[b] - Sqrt
[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(9/4)) + (5*c^(1/4)*Log[S
qrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(9/4))

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Rubi in Sympy [A]  time = 71.9633, size = 218, normalized size = 0.95 \[ \frac{1}{2 b \sqrt{x} \left (b + c x^{2}\right )} - \frac{5}{2 b^{2} \sqrt{x}} - \frac{5 \sqrt{2} \sqrt [4]{c} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{9}{4}}} + \frac{5 \sqrt{2} \sqrt [4]{c} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{9}{4}}} + \frac{5 \sqrt{2} \sqrt [4]{c} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{9}{4}}} - \frac{5 \sqrt{2} \sqrt [4]{c} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(c*x**4+b*x**2)**2,x)

[Out]

1/(2*b*sqrt(x)*(b + c*x**2)) - 5/(2*b**2*sqrt(x)) - 5*sqrt(2)*c**(1/4)*log(-sqrt
(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(16*b**(9/4)) + 5*sqrt(2)*c
**(1/4)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(16*b**(9/4
)) + 5*sqrt(2)*c**(1/4)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*b**(9/4))
- 5*sqrt(2)*c**(1/4)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*b**(9/4))

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Mathematica [A]  time = 0.415363, size = 212, normalized size = 0.92 \[ \frac{-\frac{8 \sqrt [4]{b} c x^{3/2}}{b+c x^2}-5 \sqrt{2} \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+5 \sqrt{2} \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+10 \sqrt{2} \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-10 \sqrt{2} \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )-\frac{32 \sqrt [4]{b}}{\sqrt{x}}}{16 b^{9/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/(b*x^2 + c*x^4)^2,x]

[Out]

((-32*b^(1/4))/Sqrt[x] - (8*b^(1/4)*c*x^(3/2))/(b + c*x^2) + 10*Sqrt[2]*c^(1/4)*
ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 10*Sqrt[2]*c^(1/4)*ArcTan[1 + (S
qrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 5*Sqrt[2]*c^(1/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/
4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 5*Sqrt[2]*c^(1/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4
)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(16*b^(9/4))

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Maple [A]  time = 0.02, size = 158, normalized size = 0.7 \[ -{\frac{c}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}-{\frac{5\,\sqrt{2}}{16\,{b}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{5\,\sqrt{2}}{8\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{5\,\sqrt{2}}{8\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-2\,{\frac{1}{{b}^{2}\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(c*x^4+b*x^2)^2,x)

[Out]

-1/2/b^2*c*x^(3/2)/(c*x^2+b)-5/16/b^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)))-5/8/b^2/(b
/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-5/8/b^2/(b/c)^(1/4)*2^(1
/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)-2/b^2/x^(1/2)

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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(x^(5/2)/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292784, size = 259, normalized size = 1.13 \[ -\frac{20 \, c x^{2} + 20 \,{\left (b^{2} c x^{2} + b^{3}\right )} \sqrt{x} \left (-\frac{c}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{125 \, b^{7} \left (-\frac{c}{b^{9}}\right )^{\frac{3}{4}}}{125 \, c \sqrt{x} + \sqrt{-15625 \, b^{5} c \sqrt{-\frac{c}{b^{9}}} + 15625 \, c^{2} x}}\right ) + 5 \,{\left (b^{2} c x^{2} + b^{3}\right )} \sqrt{x} \left (-\frac{c}{b^{9}}\right )^{\frac{1}{4}} \log \left (125 \, b^{7} \left (-\frac{c}{b^{9}}\right )^{\frac{3}{4}} + 125 \, c \sqrt{x}\right ) - 5 \,{\left (b^{2} c x^{2} + b^{3}\right )} \sqrt{x} \left (-\frac{c}{b^{9}}\right )^{\frac{1}{4}} \log \left (-125 \, b^{7} \left (-\frac{c}{b^{9}}\right )^{\frac{3}{4}} + 125 \, c \sqrt{x}\right ) + 16 \, b}{8 \,{\left (b^{2} c x^{2} + b^{3}\right )} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

-1/8*(20*c*x^2 + 20*(b^2*c*x^2 + b^3)*sqrt(x)*(-c/b^9)^(1/4)*arctan(125*b^7*(-c/
b^9)^(3/4)/(125*c*sqrt(x) + sqrt(-15625*b^5*c*sqrt(-c/b^9) + 15625*c^2*x))) + 5*
(b^2*c*x^2 + b^3)*sqrt(x)*(-c/b^9)^(1/4)*log(125*b^7*(-c/b^9)^(3/4) + 125*c*sqrt
(x)) - 5*(b^2*c*x^2 + b^3)*sqrt(x)*(-c/b^9)^(1/4)*log(-125*b^7*(-c/b^9)^(3/4) +
125*c*sqrt(x)) + 16*b)/((b^2*c*x^2 + b^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(x**(5/2)/(c*x**4+b*x**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280915, size = 284, normalized size = 1.23 \[ -\frac{5 \, c x^{2} + 4 \, b}{2 \,{\left (c x^{\frac{5}{2}} + b \sqrt{x}\right )} b^{2}} - \frac{5 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{3} c^{2}} - \frac{5 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{3} c^{2}} + \frac{5 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{3} c^{2}} - \frac{5 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{3} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

-1/2*(5*c*x^2 + 4*b)/((c*x^(5/2) + b*sqrt(x))*b^2) - 5/8*sqrt(2)*(b*c^3)^(3/4)*a
rctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/(b^3*c^2) - 5/8
*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^3*c^2) + 5/16*sqrt(2)*(b*c^3)^(3/4)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4)
+ x + sqrt(b/c))/(b^3*c^2) - 5/16*sqrt(2)*(b*c^3)^(3/4)*ln(-sqrt(2)*sqrt(x)*(b/c
)^(1/4) + x + sqrt(b/c))/(b^3*c^2)