3.333 \(\int \frac{x^{7/2}}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=218 \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{7/4} \sqrt [4]{c}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{7/4} \sqrt [4]{c}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{7/4} \sqrt [4]{c}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{7/4} \sqrt [4]{c}}+\frac{\sqrt{x}}{2 b \left (b+c x^2\right )} \]

[Out]

Sqrt[x]/(2*b*(b + c*x^2)) - (3*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4
*Sqrt[2]*b^(7/4)*c^(1/4)) + (3*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4
*Sqrt[2]*b^(7/4)*c^(1/4)) - (3*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + S
qrt[c]*x])/(8*Sqrt[2]*b^(7/4)*c^(1/4)) + (3*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4
)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(7/4)*c^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.330145, 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 \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{7/4} \sqrt [4]{c}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{7/4} \sqrt [4]{c}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{7/4} \sqrt [4]{c}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{7/4} \sqrt [4]{c}}+\frac{\sqrt{x}}{2 b \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)/(b*x^2 + c*x^4)^2,x]

[Out]

Sqrt[x]/(2*b*(b + c*x^2)) - (3*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4
*Sqrt[2]*b^(7/4)*c^(1/4)) + (3*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4
*Sqrt[2]*b^(7/4)*c^(1/4)) - (3*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + S
qrt[c]*x])/(8*Sqrt[2]*b^(7/4)*c^(1/4)) + (3*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4
)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(7/4)*c^(1/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 63.7103, size = 204, normalized size = 0.94 \[ \frac{\sqrt{x}}{2 b \left (b + c x^{2}\right )} - \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{7}{4}} \sqrt [4]{c}} + \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{7}{4}} \sqrt [4]{c}} - \frac{3 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{7}{4}} \sqrt [4]{c}} + \frac{3 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{7}{4}} \sqrt [4]{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)/(c*x**4+b*x**2)**2,x)

[Out]

sqrt(x)/(2*b*(b + c*x**2)) - 3*sqrt(2)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) +
sqrt(b) + sqrt(c)*x)/(16*b**(7/4)*c**(1/4)) + 3*sqrt(2)*log(sqrt(2)*b**(1/4)*c**
(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(16*b**(7/4)*c**(1/4)) - 3*sqrt(2)*atan(1 -
sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*b**(7/4)*c**(1/4)) + 3*sqrt(2)*atan(1 + s
qrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*b**(7/4)*c**(1/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.225975, size = 199, normalized size = 0.91 \[ \frac{\frac{8 b^{3/4} \sqrt{x}}{b+c x^2}-\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{\sqrt [4]{c}}+\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{\sqrt [4]{c}}-\frac{6 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt [4]{c}}+\frac{6 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt [4]{c}}}{16 b^{7/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)/(b*x^2 + c*x^4)^2,x]

[Out]

((8*b^(3/4)*Sqrt[x])/(b + c*x^2) - (6*Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x
])/b^(1/4)])/c^(1/4) + (6*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])
/c^(1/4) - (3*Sqrt[2]*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]
)/c^(1/4) + (3*Sqrt[2]*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x
])/c^(1/4))/(16*b^(7/4))

_______________________________________________________________________________________

Maple [A]  time = 0.012, size = 149, normalized size = 0.7 \[{\frac{1}{2\,b \left ( c{x}^{2}+b \right ) }\sqrt{x}}+{\frac{3\,\sqrt{2}}{16\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}}{8\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}}{8\,{b}^{2}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)/(c*x^4+b*x^2)^2,x)

[Out]

1/2*x^(1/2)/b/(c*x^2+b)+3/16/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)))+3/8/b^2*(b/c)^(
1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+3/8/b^2*(b/c)^(1/4)*2^(1/2)*a
rctan(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^(7/2)/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.283612, size = 221, normalized size = 1.01 \[ -\frac{12 \,{\left (b c x^{2} + b^{2}\right )} \left (-\frac{1}{b^{7} c}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{2} \left (-\frac{1}{b^{7} c}\right )^{\frac{1}{4}}}{\sqrt{b^{4} \sqrt{-\frac{1}{b^{7} c}} + x} + \sqrt{x}}\right ) - 3 \,{\left (b c x^{2} + b^{2}\right )} \left (-\frac{1}{b^{7} c}\right )^{\frac{1}{4}} \log \left (b^{2} \left (-\frac{1}{b^{7} c}\right )^{\frac{1}{4}} + \sqrt{x}\right ) + 3 \,{\left (b c x^{2} + b^{2}\right )} \left (-\frac{1}{b^{7} c}\right )^{\frac{1}{4}} \log \left (-b^{2} \left (-\frac{1}{b^{7} c}\right )^{\frac{1}{4}} + \sqrt{x}\right ) - 4 \, \sqrt{x}}{8 \,{\left (b c x^{2} + b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

-1/8*(12*(b*c*x^2 + b^2)*(-1/(b^7*c))^(1/4)*arctan(b^2*(-1/(b^7*c))^(1/4)/(sqrt(
b^4*sqrt(-1/(b^7*c)) + x) + sqrt(x))) - 3*(b*c*x^2 + b^2)*(-1/(b^7*c))^(1/4)*log
(b^2*(-1/(b^7*c))^(1/4) + sqrt(x)) + 3*(b*c*x^2 + b^2)*(-1/(b^7*c))^(1/4)*log(-b
^2*(-1/(b^7*c))^(1/4) + sqrt(x)) - 4*sqrt(x))/(b*c*x^2 + b^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**(7/2)/(c*x**4+b*x**2)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.278, size = 269, normalized size = 1.23 \[ \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{2} c} + \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{2} c} + \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{2} c} - \frac{3 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{2} c} + \frac{\sqrt{x}}{2 \,{\left (c x^{2} + b\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(7/2)/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

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