3.327 \(\int \frac{x^{19/2}}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=243 \[ -\frac{9 b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}-\frac{9 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{13/4}}-\frac{9 b \sqrt{x}}{2 c^3}-\frac{x^{9/2}}{2 c \left (b+c x^2\right )}+\frac{9 x^{5/2}}{10 c^2} \]

[Out]

(-9*b*Sqrt[x])/(2*c^3) + (9*x^(5/2))/(10*c^2) - x^(9/2)/(2*c*(b + c*x^2)) - (9*b
^(5/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*c^(13/4)) + (9*
b^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*c^(13/4)) - (9
*b^(5/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*
c^(13/4)) + (9*b^(5/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x
])/(8*Sqrt[2]*c^(13/4))

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Rubi [A]  time = 0.440643, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ -\frac{9 b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{13/4}}-\frac{9 b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{13/4}}+\frac{9 b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{13/4}}-\frac{9 b \sqrt{x}}{2 c^3}-\frac{x^{9/2}}{2 c \left (b+c x^2\right )}+\frac{9 x^{5/2}}{10 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(19/2)/(b*x^2 + c*x^4)^2,x]

[Out]

(-9*b*Sqrt[x])/(2*c^3) + (9*x^(5/2))/(10*c^2) - x^(9/2)/(2*c*(b + c*x^2)) - (9*b
^(5/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*c^(13/4)) + (9*
b^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*c^(13/4)) - (9
*b^(5/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*
c^(13/4)) + (9*b^(5/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x
])/(8*Sqrt[2]*c^(13/4))

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Rubi in Sympy [A]  time = 79.4076, size = 230, normalized size = 0.95 \[ - \frac{9 \sqrt{2} b^{\frac{5}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 c^{\frac{13}{4}}} + \frac{9 \sqrt{2} b^{\frac{5}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 c^{\frac{13}{4}}} - \frac{9 \sqrt{2} b^{\frac{5}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 c^{\frac{13}{4}}} + \frac{9 \sqrt{2} b^{\frac{5}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 c^{\frac{13}{4}}} - \frac{9 b \sqrt{x}}{2 c^{3}} - \frac{x^{\frac{9}{2}}}{2 c \left (b + c x^{2}\right )} + \frac{9 x^{\frac{5}{2}}}{10 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(19/2)/(c*x**4+b*x**2)**2,x)

[Out]

-9*sqrt(2)*b**(5/4)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x
)/(16*c**(13/4)) + 9*sqrt(2)*b**(5/4)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sq
rt(b) + sqrt(c)*x)/(16*c**(13/4)) - 9*sqrt(2)*b**(5/4)*atan(1 - sqrt(2)*c**(1/4)
*sqrt(x)/b**(1/4))/(8*c**(13/4)) + 9*sqrt(2)*b**(5/4)*atan(1 + sqrt(2)*c**(1/4)*
sqrt(x)/b**(1/4))/(8*c**(13/4)) - 9*b*sqrt(x)/(2*c**3) - x**(9/2)/(2*c*(b + c*x*
*2)) + 9*x**(5/2)/(10*c**2)

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Mathematica [A]  time = 0.376528, size = 227, normalized size = 0.93 \[ \frac{-45 \sqrt{2} b^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+45 \sqrt{2} b^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-90 \sqrt{2} b^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+90 \sqrt{2} b^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )-\frac{40 b^2 \sqrt [4]{c} \sqrt{x}}{b+c x^2}-320 b \sqrt [4]{c} \sqrt{x}+32 c^{5/4} x^{5/2}}{80 c^{13/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(19/2)/(b*x^2 + c*x^4)^2,x]

[Out]

(-320*b*c^(1/4)*Sqrt[x] + 32*c^(5/4)*x^(5/2) - (40*b^2*c^(1/4)*Sqrt[x])/(b + c*x
^2) - 90*Sqrt[2]*b^(5/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 90*Sqrt
[2]*b^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 45*Sqrt[2]*b^(5/4)*L
og[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 45*Sqrt[2]*b^(5/4)*L
og[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(80*c^(13/4))

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Maple [A]  time = 0.018, size = 172, normalized size = 0.7 \[{\frac{2}{5\,{c}^{2}}{x}^{{\frac{5}{2}}}}-4\,{\frac{b\sqrt{x}}{{c}^{3}}}-{\frac{{b}^{2}}{2\,{c}^{3} \left ( c{x}^{2}+b \right ) }\sqrt{x}}+{\frac{9\,b\sqrt{2}}{16\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{9\,b\sqrt{2}}{8\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }+{\frac{9\,b\sqrt{2}}{8\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(19/2)/(c*x^4+b*x^2)^2,x)

[Out]

2/5*x^(5/2)/c^2-4*b*x^(1/2)/c^3-1/2/c^3*b^2*x^(1/2)/(c*x^2+b)+9/16/c^3*b*(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)))+9/8/c^3*b*(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4
)*x^(1/2)+1)+9/8/c^3*b*(b/c)^(1/4)*2^(1/2)*arctan(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^(19/2)/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285359, size = 284, normalized size = 1.17 \[ -\frac{180 \,{\left (c^{4} x^{2} + b c^{3}\right )} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} \arctan \left (\frac{c^{3} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}}}{b \sqrt{x} + \sqrt{c^{6} \sqrt{-\frac{b^{5}}{c^{13}}} + b^{2} x}}\right ) - 45 \,{\left (c^{4} x^{2} + b c^{3}\right )} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} \log \left (9 \, c^{3} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} + 9 \, b \sqrt{x}\right ) + 45 \,{\left (c^{4} x^{2} + b c^{3}\right )} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} \log \left (-9 \, c^{3} \left (-\frac{b^{5}}{c^{13}}\right )^{\frac{1}{4}} + 9 \, b \sqrt{x}\right ) - 4 \,{\left (4 \, c^{2} x^{4} - 36 \, b c x^{2} - 45 \, b^{2}\right )} \sqrt{x}}{40 \,{\left (c^{4} x^{2} + b c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(19/2)/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

-1/40*(180*(c^4*x^2 + b*c^3)*(-b^5/c^13)^(1/4)*arctan(c^3*(-b^5/c^13)^(1/4)/(b*s
qrt(x) + sqrt(c^6*sqrt(-b^5/c^13) + b^2*x))) - 45*(c^4*x^2 + b*c^3)*(-b^5/c^13)^
(1/4)*log(9*c^3*(-b^5/c^13)^(1/4) + 9*b*sqrt(x)) + 45*(c^4*x^2 + b*c^3)*(-b^5/c^
13)^(1/4)*log(-9*c^3*(-b^5/c^13)^(1/4) + 9*b*sqrt(x)) - 4*(4*c^2*x^4 - 36*b*c*x^
2 - 45*b^2)*sqrt(x))/(c^4*x^2 + b*c^3)

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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**(19/2)/(c*x**4+b*x**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278981, size = 292, normalized size = 1.2 \[ \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, c^{4}} + \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, c^{4}} + \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, c^{4}} - \frac{9 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} b{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, c^{4}} - \frac{b^{2} \sqrt{x}}{2 \,{\left (c x^{2} + b\right )} c^{3}} + \frac{2 \,{\left (c^{8} x^{\frac{5}{2}} - 10 \, b c^{7} \sqrt{x}\right )}}{5 \, c^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(19/2)/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

9/8*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^4 + 9/8*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^4 + 9/16*sqrt(2)*(b*c^3)^(1/4)*b*ln(sqrt(2)
*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/c^4 - 9/16*sqrt(2)*(b*c^3)^(1/4)*b*ln(-sqr
t(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/c^4 - 1/2*b^2*sqrt(x)/((c*x^2 + b)*c^3
) + 2/5*(c^8*x^(5/2) - 10*b*c^7*sqrt(x))/c^10