3.317 \(\int \frac{x^{11/2}}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=215 \[ -\frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}+\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}-\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{9/4}}+\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} c^{9/4}}-\frac{2 b \sqrt{x}}{c^2}+\frac{2 x^{5/2}}{5 c} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.40537, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474 \[ -\frac{b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}+\frac{b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} c^{9/4}}-\frac{b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} c^{9/4}}+\frac{b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} c^{9/4}}-\frac{2 b \sqrt{x}}{c^2}+\frac{2 x^{5/2}}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[x^(11/2)/(b*x^2 + c*x^4),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 72.0698, size = 204, normalized size = 0.95 \[ - \frac{\sqrt{2} b^{\frac{5}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 c^{\frac{9}{4}}} + \frac{\sqrt{2} b^{\frac{5}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 c^{\frac{9}{4}}} - \frac{\sqrt{2} b^{\frac{5}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 c^{\frac{9}{4}}} + \frac{\sqrt{2} b^{\frac{5}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 c^{\frac{9}{4}}} - \frac{2 b \sqrt{x}}{c^{2}} + \frac{2 x^{\frac{5}{2}}}{5 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(11/2)/(c*x**4+b*x**2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0731065, size = 203, normalized size = 0.94 \[ \frac{-5 \sqrt{2} b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+5 \sqrt{2} b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-10 \sqrt{2} b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+10 \sqrt{2} b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )-40 b \sqrt [4]{c} \sqrt{x}+8 c^{5/4} x^{5/2}}{20 c^{9/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(11/2)/(b*x^2 + c*x^4),x]

[Out]

(-40*b*c^(1/4)*Sqrt[x] + 8*c^(5/4)*x^(5/2) - 10*Sqrt[2]*b^(5/4)*ArcTan[1 - (Sqrt
[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 10*Sqrt[2]*b^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*S
qrt[x])/b^(1/4)] - 5*Sqrt[2]*b^(5/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[
x] + Sqrt[c]*x] + 5*Sqrt[2]*b^(5/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x
] + Sqrt[c]*x])/(20*c^(9/4))

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 152, normalized size = 0.7 \[{\frac{2}{5\,c}{x}^{{\frac{5}{2}}}}-2\,{\frac{b\sqrt{x}}{{c}^{2}}}+{\frac{b\sqrt{2}}{4\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{b\sqrt{2}}{2\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{b\sqrt{2}}{2\,{c}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(11/2)/(c*x^4+b*x^2),x)

[Out]

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

_______________________________________________________________________________________

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^(11/2)/(c*x^4 + b*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.280556, size = 207, normalized size = 0.96 \[ -\frac{20 \, c^{2} \left (-\frac{b^{5}}{c^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{c^{2} \left (-\frac{b^{5}}{c^{9}}\right )^{\frac{1}{4}}}{b \sqrt{x} + \sqrt{c^{4} \sqrt{-\frac{b^{5}}{c^{9}}} + b^{2} x}}\right ) - 5 \, c^{2} \left (-\frac{b^{5}}{c^{9}}\right )^{\frac{1}{4}} \log \left (c^{2} \left (-\frac{b^{5}}{c^{9}}\right )^{\frac{1}{4}} + b \sqrt{x}\right ) + 5 \, c^{2} \left (-\frac{b^{5}}{c^{9}}\right )^{\frac{1}{4}} \log \left (-c^{2} \left (-\frac{b^{5}}{c^{9}}\right )^{\frac{1}{4}} + b \sqrt{x}\right ) - 4 \,{\left (c x^{2} - 5 \, b\right )} \sqrt{x}}{10 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(11/2)/(c*x^4 + b*x^2),x, algorithm="fricas")

[Out]

-1/10*(20*c^2*(-b^5/c^9)^(1/4)*arctan(c^2*(-b^5/c^9)^(1/4)/(b*sqrt(x) + sqrt(c^4
*sqrt(-b^5/c^9) + b^2*x))) - 5*c^2*(-b^5/c^9)^(1/4)*log(c^2*(-b^5/c^9)^(1/4) + b
*sqrt(x)) + 5*c^2*(-b^5/c^9)^(1/4)*log(-c^2*(-b^5/c^9)^(1/4) + b*sqrt(x)) - 4*(c
*x^2 - 5*b)*sqrt(x))/c^2

_______________________________________________________________________________________

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**(11/2)/(c*x**4+b*x**2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.276417, size = 265, normalized size = 1.23 \[ \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, c^{3}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, c^{3}} + \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, c^{3}} - \frac{\sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, c^{3}} + \frac{2 \,{\left (c^{4} x^{\frac{5}{2}} - 5 \, b c^{3} \sqrt{x}\right )}}{5 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(11/2)/(c*x^4 + b*x^2),x, algorithm="giac")

[Out]

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