3.348 \(\int \frac{x^{5/2}}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=264 \[ \frac{117 c^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{17/4}}-\frac{117 c^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{17/4}}-\frac{117 c^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{17/4}}+\frac{117 c^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{17/4}}+\frac{117 c}{16 b^4 \sqrt{x}}-\frac{117}{80 b^3 x^{5/2}}+\frac{13}{16 b^2 x^{5/2} \left (b+c x^2\right )}+\frac{1}{4 b x^{5/2} \left (b+c x^2\right )^2} \]

[Out]

-117/(80*b^3*x^(5/2)) + (117*c)/(16*b^4*Sqrt[x]) + 1/(4*b*x^(5/2)*(b + c*x^2)^2)
+ 13/(16*b^2*x^(5/2)*(b + c*x^2)) - (117*c^(5/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sq
rt[x])/b^(1/4)])/(32*Sqrt[2]*b^(17/4)) + (117*c^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/4
)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(17/4)) + (117*c^(5/4)*Log[Sqrt[b] - Sqrt[2]*
b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(17/4)) - (117*c^(5/4)*Log[S
qrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(17/4))

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Rubi [A]  time = 0.498042, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ \frac{117 c^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{17/4}}-\frac{117 c^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{17/4}}-\frac{117 c^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{17/4}}+\frac{117 c^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{17/4}}+\frac{117 c}{16 b^4 \sqrt{x}}-\frac{117}{80 b^3 x^{5/2}}+\frac{13}{16 b^2 x^{5/2} \left (b+c x^2\right )}+\frac{1}{4 b x^{5/2} \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)/(b*x^2 + c*x^4)^3,x]

[Out]

-117/(80*b^3*x^(5/2)) + (117*c)/(16*b^4*Sqrt[x]) + 1/(4*b*x^(5/2)*(b + c*x^2)^2)
+ 13/(16*b^2*x^(5/2)*(b + c*x^2)) - (117*c^(5/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sq
rt[x])/b^(1/4)])/(32*Sqrt[2]*b^(17/4)) + (117*c^(5/4)*ArcTan[1 + (Sqrt[2]*c^(1/4
)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(17/4)) + (117*c^(5/4)*Log[Sqrt[b] - Sqrt[2]*
b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(17/4)) - (117*c^(5/4)*Log[S
qrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(17/4))

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Rubi in Sympy [A]  time = 97.4506, size = 252, normalized size = 0.95 \[ \frac{1}{4 b x^{\frac{5}{2}} \left (b + c x^{2}\right )^{2}} + \frac{13}{16 b^{2} x^{\frac{5}{2}} \left (b + c x^{2}\right )} - \frac{117}{80 b^{3} x^{\frac{5}{2}}} + \frac{117 c}{16 b^{4} \sqrt{x}} + \frac{117 \sqrt{2} c^{\frac{5}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{17}{4}}} - \frac{117 \sqrt{2} c^{\frac{5}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{17}{4}}} - \frac{117 \sqrt{2} c^{\frac{5}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{17}{4}}} + \frac{117 \sqrt{2} c^{\frac{5}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{17}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)/(c*x**4+b*x**2)**3,x)

[Out]

1/(4*b*x**(5/2)*(b + c*x**2)**2) + 13/(16*b**2*x**(5/2)*(b + c*x**2)) - 117/(80*
b**3*x**(5/2)) + 117*c/(16*b**4*sqrt(x)) + 117*sqrt(2)*c**(5/4)*log(-sqrt(2)*b**
(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(128*b**(17/4)) - 117*sqrt(2)*c**(
5/4)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(128*b**(17/4)
) - 117*sqrt(2)*c**(5/4)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64*b**(17/
4)) + 117*sqrt(2)*c**(5/4)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64*b**(1
7/4))

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Mathematica [A]  time = 0.238205, size = 251, normalized size = 0.95 \[ \frac{\frac{160 b^{5/4} c^2 x^{3/2}}{\left (b+c x^2\right )^2}-\frac{256 b^{5/4}}{x^{5/2}}+585 \sqrt{2} c^{5/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-585 \sqrt{2} c^{5/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-1170 \sqrt{2} c^{5/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+1170 \sqrt{2} c^{5/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )+\frac{840 \sqrt [4]{b} c^2 x^{3/2}}{b+c x^2}+\frac{3840 \sqrt [4]{b} c}{\sqrt{x}}}{640 b^{17/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)/(b*x^2 + c*x^4)^3,x]

[Out]

((-256*b^(5/4))/x^(5/2) + (3840*b^(1/4)*c)/Sqrt[x] + (160*b^(5/4)*c^2*x^(3/2))/(
b + c*x^2)^2 + (840*b^(1/4)*c^2*x^(3/2))/(b + c*x^2) - 1170*Sqrt[2]*c^(5/4)*ArcT
an[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 1170*Sqrt[2]*c^(5/4)*ArcTan[1 + (Sqr
t[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 585*Sqrt[2]*c^(5/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/
4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 585*Sqrt[2]*c^(5/4)*Log[Sqrt[b] + Sqrt[2]*b^(1
/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(640*b^(17/4))

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Maple [A]  time = 0.027, size = 192, normalized size = 0.7 \[{\frac{21\,{c}^{3}}{16\,{b}^{4} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{7}{2}}}}+{\frac{25\,{c}^{2}}{16\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{117\,c\sqrt{2}}{128\,{b}^{4}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{117\,c\sqrt{2}}{64\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{117\,c\sqrt{2}}{64\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{2}{5\,{b}^{3}}{x}^{-{\frac{5}{2}}}}+6\,{\frac{c}{{b}^{4}\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)/(c*x^4+b*x^2)^3,x)

[Out]

21/16*c^3/b^4/(c*x^2+b)^2*x^(7/2)+25/16*c^2/b^3/(c*x^2+b)^2*x^(3/2)+117/128*c/b^
4/(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)))+117/64*c/b^4/(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2
)/(b/c)^(1/4)*x^(1/2)+1)+117/64*c/b^4/(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(
1/4)*x^(1/2)-1)-2/5/b^3/x^(5/2)+6*c/b^4/x^(1/2)

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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^(5/2)/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289045, size = 400, normalized size = 1.52 \[ \frac{2340 \, c^{3} x^{6} + 4212 \, b c^{2} x^{4} + 1664 \, b^{2} c x^{2} - 128 \, b^{3} + 2340 \,{\left (b^{4} c^{2} x^{6} + 2 \, b^{5} c x^{4} + b^{6} x^{2}\right )} \sqrt{x} \left (-\frac{c^{5}}{b^{17}}\right )^{\frac{1}{4}} \arctan \left (\frac{1601613 \, b^{13} \left (-\frac{c^{5}}{b^{17}}\right )^{\frac{3}{4}}}{1601613 \, c^{4} \sqrt{x} + \sqrt{-2565164201769 \, b^{9} c^{5} \sqrt{-\frac{c^{5}}{b^{17}}} + 2565164201769 \, c^{8} x}}\right ) + 585 \,{\left (b^{4} c^{2} x^{6} + 2 \, b^{5} c x^{4} + b^{6} x^{2}\right )} \sqrt{x} \left (-\frac{c^{5}}{b^{17}}\right )^{\frac{1}{4}} \log \left (1601613 \, b^{13} \left (-\frac{c^{5}}{b^{17}}\right )^{\frac{3}{4}} + 1601613 \, c^{4} \sqrt{x}\right ) - 585 \,{\left (b^{4} c^{2} x^{6} + 2 \, b^{5} c x^{4} + b^{6} x^{2}\right )} \sqrt{x} \left (-\frac{c^{5}}{b^{17}}\right )^{\frac{1}{4}} \log \left (-1601613 \, b^{13} \left (-\frac{c^{5}}{b^{17}}\right )^{\frac{3}{4}} + 1601613 \, c^{4} \sqrt{x}\right )}{320 \,{\left (b^{4} c^{2} x^{6} + 2 \, b^{5} c x^{4} + b^{6} x^{2}\right )} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

1/320*(2340*c^3*x^6 + 4212*b*c^2*x^4 + 1664*b^2*c*x^2 - 128*b^3 + 2340*(b^4*c^2*
x^6 + 2*b^5*c*x^4 + b^6*x^2)*sqrt(x)*(-c^5/b^17)^(1/4)*arctan(1601613*b^13*(-c^5
/b^17)^(3/4)/(1601613*c^4*sqrt(x) + sqrt(-2565164201769*b^9*c^5*sqrt(-c^5/b^17)
+ 2565164201769*c^8*x))) + 585*(b^4*c^2*x^6 + 2*b^5*c*x^4 + b^6*x^2)*sqrt(x)*(-c
^5/b^17)^(1/4)*log(1601613*b^13*(-c^5/b^17)^(3/4) + 1601613*c^4*sqrt(x)) - 585*(
b^4*c^2*x^6 + 2*b^5*c*x^4 + b^6*x^2)*sqrt(x)*(-c^5/b^17)^(1/4)*log(-1601613*b^13
*(-c^5/b^17)^(3/4) + 1601613*c^4*sqrt(x)))/((b^4*c^2*x^6 + 2*b^5*c*x^4 + b^6*x^2
)*sqrt(x))

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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**(5/2)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.282072, size = 313, normalized size = 1.19 \[ \frac{117 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{5} c} + \frac{117 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{5} c} - \frac{117 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{5} c} + \frac{117 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{5} c} + \frac{21 \, c^{3} x^{\frac{7}{2}} + 25 \, b c^{2} x^{\frac{3}{2}}}{16 \,{\left (c x^{2} + b\right )}^{2} b^{4}} + \frac{2 \,{\left (15 \, c x^{2} - b\right )}}{5 \, b^{4} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(5/2)/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

117/64*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^5*c) + 117/64*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^5*c) - 117/128*sqrt(2)*(b*c^3)^(3/4)
*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^5*c) + 117/128*sqrt(2)*(b*c^
3)^(3/4)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^5*c) + 1/16*(21*c^3
*x^(7/2) + 25*b*c^2*x^(3/2))/((c*x^2 + b)^2*b^4) + 2/5*(15*c*x^2 - b)/(b^4*x^(5/
2))