3.328 \(\int \frac{x^{17/2}}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=230 \[ -\frac{7 b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{11/4}}+\frac{7 b^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{11/4}}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{11/4}}-\frac{7 b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{11/4}}-\frac{x^{7/2}}{2 c \left (b+c x^2\right )}+\frac{7 x^{3/2}}{6 c^2} \]

[Out]

(7*x^(3/2))/(6*c^2) - x^(7/2)/(2*c*(b + c*x^2)) + (7*b^(3/4)*ArcTan[1 - (Sqrt[2]
*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*c^(11/4)) - (7*b^(3/4)*ArcTan[1 + (Sqrt[2
]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*c^(11/4)) - (7*b^(3/4)*Log[Sqrt[b] - Sqr
t[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*c^(11/4)) + (7*b^(3/4)*Log
[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*c^(11/4))

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Rubi [A]  time = 0.393876, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526 \[ -\frac{7 b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{11/4}}+\frac{7 b^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} c^{11/4}}+\frac{7 b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} c^{11/4}}-\frac{7 b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} c^{11/4}}-\frac{x^{7/2}}{2 c \left (b+c x^2\right )}+\frac{7 x^{3/2}}{6 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(17/2)/(b*x^2 + c*x^4)^2,x]

[Out]

(7*x^(3/2))/(6*c^2) - x^(7/2)/(2*c*(b + c*x^2)) + (7*b^(3/4)*ArcTan[1 - (Sqrt[2]
*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*c^(11/4)) - (7*b^(3/4)*ArcTan[1 + (Sqrt[2
]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*c^(11/4)) - (7*b^(3/4)*Log[Sqrt[b] - Sqr
t[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*c^(11/4)) + (7*b^(3/4)*Log
[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*c^(11/4))

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Rubi in Sympy [A]  time = 72.9672, size = 216, normalized size = 0.94 \[ - \frac{7 \sqrt{2} b^{\frac{3}{4}} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 c^{\frac{11}{4}}} + \frac{7 \sqrt{2} b^{\frac{3}{4}} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 c^{\frac{11}{4}}} + \frac{7 \sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 c^{\frac{11}{4}}} - \frac{7 \sqrt{2} b^{\frac{3}{4}} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 c^{\frac{11}{4}}} - \frac{x^{\frac{7}{2}}}{2 c \left (b + c x^{2}\right )} + \frac{7 x^{\frac{3}{2}}}{6 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(17/2)/(c*x**4+b*x**2)**2,x)

[Out]

-7*sqrt(2)*b**(3/4)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x
)/(16*c**(11/4)) + 7*sqrt(2)*b**(3/4)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sq
rt(b) + sqrt(c)*x)/(16*c**(11/4)) + 7*sqrt(2)*b**(3/4)*atan(1 - sqrt(2)*c**(1/4)
*sqrt(x)/b**(1/4))/(8*c**(11/4)) - 7*sqrt(2)*b**(3/4)*atan(1 + sqrt(2)*c**(1/4)*
sqrt(x)/b**(1/4))/(8*c**(11/4)) - x**(7/2)/(2*c*(b + c*x**2)) + 7*x**(3/2)/(6*c*
*2)

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Mathematica [A]  time = 0.28352, size = 212, normalized size = 0.92 \[ \frac{-21 \sqrt{2} b^{3/4} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+21 \sqrt{2} b^{3/4} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+42 \sqrt{2} b^{3/4} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-42 \sqrt{2} b^{3/4} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )+\frac{24 b c^{3/4} x^{3/2}}{b+c x^2}+32 c^{3/4} x^{3/2}}{48 c^{11/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(17/2)/(b*x^2 + c*x^4)^2,x]

[Out]

(32*c^(3/4)*x^(3/2) + (24*b*c^(3/4)*x^(3/2))/(b + c*x^2) + 42*Sqrt[2]*b^(3/4)*Ar
cTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 42*Sqrt[2]*b^(3/4)*ArcTan[1 + (Sqr
t[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 21*Sqrt[2]*b^(3/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4
)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 21*Sqrt[2]*b^(3/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4
)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(48*c^(11/4))

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Maple [A]  time = 0.018, size = 161, normalized size = 0.7 \[{\frac{2}{3\,{c}^{2}}{x}^{{\frac{3}{2}}}}+{\frac{b}{2\,{c}^{2} \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}-{\frac{7\,b\sqrt{2}}{16\,{c}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{7\,b\sqrt{2}}{8\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{7\,b\sqrt{2}}{8\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(17/2)/(c*x^4+b*x^2)^2,x)

[Out]

2/3*x^(3/2)/c^2+1/2*b/c^2*x^(3/2)/(c*x^2+b)-7/16*b/c^3/(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)))-7/8*b/c^3/(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-7/8*b
/c^3/(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^(17/2)/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28522, size = 288, normalized size = 1.25 \[ -\frac{84 \,{\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac{b^{3}}{c^{11}}\right )^{\frac{1}{4}} \arctan \left (\frac{343 \, c^{8} \left (-\frac{b^{3}}{c^{11}}\right )^{\frac{3}{4}}}{343 \, b^{2} \sqrt{x} + \sqrt{-117649 \, b^{3} c^{5} \sqrt{-\frac{b^{3}}{c^{11}}} + 117649 \, b^{4} x}}\right ) + 21 \,{\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac{b^{3}}{c^{11}}\right )^{\frac{1}{4}} \log \left (343 \, c^{8} \left (-\frac{b^{3}}{c^{11}}\right )^{\frac{3}{4}} + 343 \, b^{2} \sqrt{x}\right ) - 21 \,{\left (c^{3} x^{2} + b c^{2}\right )} \left (-\frac{b^{3}}{c^{11}}\right )^{\frac{1}{4}} \log \left (-343 \, c^{8} \left (-\frac{b^{3}}{c^{11}}\right )^{\frac{3}{4}} + 343 \, b^{2} \sqrt{x}\right ) - 4 \,{\left (4 \, c x^{3} + 7 \, b x\right )} \sqrt{x}}{24 \,{\left (c^{3} x^{2} + b c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(17/2)/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

-1/24*(84*(c^3*x^2 + b*c^2)*(-b^3/c^11)^(1/4)*arctan(343*c^8*(-b^3/c^11)^(3/4)/(
343*b^2*sqrt(x) + sqrt(-117649*b^3*c^5*sqrt(-b^3/c^11) + 117649*b^4*x))) + 21*(c
^3*x^2 + b*c^2)*(-b^3/c^11)^(1/4)*log(343*c^8*(-b^3/c^11)^(3/4) + 343*b^2*sqrt(x
)) - 21*(c^3*x^2 + b*c^2)*(-b^3/c^11)^(1/4)*log(-343*c^8*(-b^3/c^11)^(3/4) + 343
*b^2*sqrt(x)) - 4*(4*c*x^3 + 7*b*x)*sqrt(x))/(c^3*x^2 + b*c^2)

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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**(17/2)/(c*x**4+b*x**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.279171, size = 265, normalized size = 1.15 \[ \frac{b x^{\frac{3}{2}}}{2 \,{\left (c x^{2} + b\right )} c^{2}} + \frac{2 \, x^{\frac{3}{2}}}{3 \, c^{2}} - \frac{7 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, c^{5}} - \frac{7 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, c^{5}} + \frac{7 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, c^{5}} - \frac{7 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(17/2)/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

1/2*b*x^(3/2)/((c*x^2 + b)*c^2) + 2/3*x^(3/2)/c^2 - 7/8*sqrt(2)*(b*c^3)^(3/4)*ar
ctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/c^5 - 7/8*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
))/c^5 + 7/16*sqrt(2)*(b*c^3)^(3/4)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/
c))/c^5 - 7/16*sqrt(2)*(b*c^3)^(3/4)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(
b/c))/c^5