3.332 \(\int \frac{x^{9/2}}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=218 \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{5/4} c^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{5/4} c^{3/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{5/4} c^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{5/4} c^{3/4}}+\frac{x^{3/2}}{2 b \left (b+c x^2\right )} \]

[Out]

x^(3/2)/(2*b*(b + c*x^2)) - ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]/(4*Sqr
t[2]*b^(5/4)*c^(3/4)) + ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]/(4*Sqrt[2]
*b^(5/4)*c^(3/4)) + Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]/(
8*Sqrt[2]*b^(5/4)*c^(3/4)) - Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqr
t[c]*x]/(8*Sqrt[2]*b^(5/4)*c^(3/4))

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Rubi [A]  time = 0.341862, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ \frac{\log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{5/4} c^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{5/4} c^{3/4}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{5/4} c^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{5/4} c^{3/4}}+\frac{x^{3/2}}{2 b \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^(9/2)/(b*x^2 + c*x^4)^2,x]

[Out]

x^(3/2)/(2*b*(b + c*x^2)) - ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]/(4*Sqr
t[2]*b^(5/4)*c^(3/4)) + ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)]/(4*Sqrt[2]
*b^(5/4)*c^(3/4)) + Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]/(
8*Sqrt[2]*b^(5/4)*c^(3/4)) - Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqr
t[c]*x]/(8*Sqrt[2]*b^(5/4)*c^(3/4))

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Rubi in Sympy [A]  time = 65.8037, size = 197, normalized size = 0.9 \[ \frac{x^{\frac{3}{2}}}{2 b \left (b + c x^{2}\right )} + \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{5}{4}} c^{\frac{3}{4}}} - \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{5}{4}} c^{\frac{3}{4}}} - \frac{\sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{5}{4}} c^{\frac{3}{4}}} + \frac{\sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{5}{4}} c^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(9/2)/(c*x**4+b*x**2)**2,x)

[Out]

x**(3/2)/(2*b*(b + c*x**2)) + sqrt(2)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + s
qrt(b) + sqrt(c)*x)/(16*b**(5/4)*c**(3/4)) - sqrt(2)*log(sqrt(2)*b**(1/4)*c**(1/
4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(16*b**(5/4)*c**(3/4)) - sqrt(2)*atan(1 - sqrt
(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*b**(5/4)*c**(3/4)) + sqrt(2)*atan(1 + sqrt(2)*
c**(1/4)*sqrt(x)/b**(1/4))/(8*b**(5/4)*c**(3/4))

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Mathematica [A]  time = 0.264392, size = 198, normalized size = 0.91 \[ \frac{\frac{\sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{c^{3/4}}-\frac{\sqrt{2} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{c^{3/4}}-\frac{2 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{c^{3/4}}+\frac{2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{c^{3/4}}+\frac{8 \sqrt [4]{b} x^{3/2}}{b+c x^2}}{16 b^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(9/2)/(b*x^2 + c*x^4)^2,x]

[Out]

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

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Maple [A]  time = 0.012, size = 158, normalized size = 0.7 \[{\frac{1}{2\,b \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}+{\frac{\sqrt{2}}{16\,bc}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{\sqrt{2}}{8\,bc}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{\sqrt{2}}{8\,bc}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(9/2)/(c*x^4+b*x^2)^2,x)

[Out]

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

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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^(9/2)/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284939, size = 235, normalized size = 1.08 \[ \frac{4 \,{\left (b c x^{2} + b^{2}\right )} \left (-\frac{1}{b^{5} c^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{4} c^{2} \left (-\frac{1}{b^{5} c^{3}}\right )^{\frac{3}{4}}}{\sqrt{-b^{3} c \sqrt{-\frac{1}{b^{5} c^{3}}} + x} + \sqrt{x}}\right ) +{\left (b c x^{2} + b^{2}\right )} \left (-\frac{1}{b^{5} c^{3}}\right )^{\frac{1}{4}} \log \left (b^{4} c^{2} \left (-\frac{1}{b^{5} c^{3}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) -{\left (b c x^{2} + b^{2}\right )} \left (-\frac{1}{b^{5} c^{3}}\right )^{\frac{1}{4}} \log \left (-b^{4} c^{2} \left (-\frac{1}{b^{5} c^{3}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) + 4 \, x^{\frac{3}{2}}}{8 \,{\left (b c x^{2} + b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

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

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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**(9/2)/(c*x**4+b*x**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280132, size = 269, normalized size = 1.23 \[ \frac{x^{\frac{3}{2}}}{2 \,{\left (c x^{2} + b\right )} b} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{2} c^{3}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{2} c^{3}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{2} c^{3}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

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