3.215 \(\int \frac{x^4}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=76 \[ -\frac{15 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2}}-\frac{15}{8 b^3 x}+\frac{5}{8 b^2 x \left (b+c x^2\right )}+\frac{1}{4 b x \left (b+c x^2\right )^2} \]

[Out]

-15/(8*b^3*x) + 1/(4*b*x*(b + c*x^2)^2) + 5/(8*b^2*x*(b + c*x^2)) - (15*Sqrt[c]*
ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(7/2))

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Rubi [A]  time = 0.0855407, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{15 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2}}-\frac{15}{8 b^3 x}+\frac{5}{8 b^2 x \left (b+c x^2\right )}+\frac{1}{4 b x \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(b*x^2 + c*x^4)^3,x]

[Out]

-15/(8*b^3*x) + 1/(4*b*x*(b + c*x^2)^2) + 5/(8*b^2*x*(b + c*x^2)) - (15*Sqrt[c]*
ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(7/2))

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Rubi in Sympy [A]  time = 16.8851, size = 65, normalized size = 0.86 \[ \frac{1}{4 b x \left (b + c x^{2}\right )^{2}} + \frac{5}{8 b^{2} x \left (b + c x^{2}\right )} - \frac{15}{8 b^{3} x} - \frac{15 \sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{8 b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(c*x**4+b*x**2)**3,x)

[Out]

1/(4*b*x*(b + c*x**2)**2) + 5/(8*b**2*x*(b + c*x**2)) - 15/(8*b**3*x) - 15*sqrt(
c)*atan(sqrt(c)*x/sqrt(b))/(8*b**(7/2))

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Mathematica [A]  time = 0.074645, size = 68, normalized size = 0.89 \[ -\frac{15 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2}}-\frac{8 b^2+25 b c x^2+15 c^2 x^4}{8 b^3 x \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(b*x^2 + c*x^4)^3,x]

[Out]

-(8*b^2 + 25*b*c*x^2 + 15*c^2*x^4)/(8*b^3*x*(b + c*x^2)^2) - (15*Sqrt[c]*ArcTan[
(Sqrt[c]*x)/Sqrt[b]])/(8*b^(7/2))

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Maple [A]  time = 0.016, size = 66, normalized size = 0.9 \[ -{\frac{7\,{c}^{2}{x}^{3}}{8\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{9\,cx}{8\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{15\,c}{8\,{b}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{1}{{b}^{3}x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(c*x^4+b*x^2)^3,x)

[Out]

-7/8/b^3*c^2/(c*x^2+b)^2*x^3-9/8/b^2*c/(c*x^2+b)^2*x-15/8/b^3*c/(b*c)^(1/2)*arct
an(c*x/(b*c)^(1/2))-1/b^3/x

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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^4/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264044, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \, c^{2} x^{4} + 50 \, b c x^{2} - 15 \,{\left (c^{2} x^{5} + 2 \, b c x^{3} + b^{2} x\right )} \sqrt{-\frac{c}{b}} \log \left (\frac{c x^{2} - 2 \, b x \sqrt{-\frac{c}{b}} - b}{c x^{2} + b}\right ) + 16 \, b^{2}}{16 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )}}, -\frac{15 \, c^{2} x^{4} + 25 \, b c x^{2} + 15 \,{\left (c^{2} x^{5} + 2 \, b c x^{3} + b^{2} x\right )} \sqrt{\frac{c}{b}} \arctan \left (\frac{c x}{b \sqrt{\frac{c}{b}}}\right ) + 8 \, b^{2}}{8 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

[-1/16*(30*c^2*x^4 + 50*b*c*x^2 - 15*(c^2*x^5 + 2*b*c*x^3 + b^2*x)*sqrt(-c/b)*lo
g((c*x^2 - 2*b*x*sqrt(-c/b) - b)/(c*x^2 + b)) + 16*b^2)/(b^3*c^2*x^5 + 2*b^4*c*x
^3 + b^5*x), -1/8*(15*c^2*x^4 + 25*b*c*x^2 + 15*(c^2*x^5 + 2*b*c*x^3 + b^2*x)*sq
rt(c/b)*arctan(c*x/(b*sqrt(c/b))) + 8*b^2)/(b^3*c^2*x^5 + 2*b^4*c*x^3 + b^5*x)]

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Sympy [A]  time = 2.3972, size = 114, normalized size = 1.5 \[ \frac{15 \sqrt{- \frac{c}{b^{7}}} \log{\left (- \frac{b^{4} \sqrt{- \frac{c}{b^{7}}}}{c} + x \right )}}{16} - \frac{15 \sqrt{- \frac{c}{b^{7}}} \log{\left (\frac{b^{4} \sqrt{- \frac{c}{b^{7}}}}{c} + x \right )}}{16} - \frac{8 b^{2} + 25 b c x^{2} + 15 c^{2} x^{4}}{8 b^{5} x + 16 b^{4} c x^{3} + 8 b^{3} c^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(c*x**4+b*x**2)**3,x)

[Out]

15*sqrt(-c/b**7)*log(-b**4*sqrt(-c/b**7)/c + x)/16 - 15*sqrt(-c/b**7)*log(b**4*s
qrt(-c/b**7)/c + x)/16 - (8*b**2 + 25*b*c*x**2 + 15*c**2*x**4)/(8*b**5*x + 16*b*
*4*c*x**3 + 8*b**3*c**2*x**5)

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GIAC/XCAS [A]  time = 0.271848, size = 77, normalized size = 1.01 \[ -\frac{15 \, c \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{8 \, \sqrt{b c} b^{3}} - \frac{7 \, c^{2} x^{3} + 9 \, b c x}{8 \,{\left (c x^{2} + b\right )}^{2} b^{3}} - \frac{1}{b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

-15/8*c*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*b^3) - 1/8*(7*c^2*x^3 + 9*b*c*x)/((c*x^
2 + b)^2*b^3) - 1/(b^3*x)