3.346 \(\int \frac{x^{9/2}}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=251 \[ -\frac{45 \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{13/4}}+\frac{45 \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{13/4}}+\frac{45 \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{13/4}}-\frac{45 \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{13/4}}-\frac{45}{16 b^3 \sqrt{x}}+\frac{9}{16 b^2 \sqrt{x} \left (b+c x^2\right )}+\frac{1}{4 b \sqrt{x} \left (b+c x^2\right )^2} \]

[Out]

-45/(16*b^3*Sqrt[x]) + 1/(4*b*Sqrt[x]*(b + c*x^2)^2) + 9/(16*b^2*Sqrt[x]*(b + c*
x^2)) + (45*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b
^(13/4)) - (45*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2
]*b^(13/4)) - (45*c^(1/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c
]*x])/(64*Sqrt[2]*b^(13/4)) + (45*c^(1/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*
Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(13/4))

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Rubi [A]  time = 0.437017, antiderivative size = 251, 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 \[ -\frac{45 \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{13/4}}+\frac{45 \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{13/4}}+\frac{45 \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{13/4}}-\frac{45 \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{13/4}}-\frac{45}{16 b^3 \sqrt{x}}+\frac{9}{16 b^2 \sqrt{x} \left (b+c x^2\right )}+\frac{1}{4 b \sqrt{x} \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(9/2)/(b*x^2 + c*x^4)^3,x]

[Out]

-45/(16*b^3*Sqrt[x]) + 1/(4*b*Sqrt[x]*(b + c*x^2)^2) + 9/(16*b^2*Sqrt[x]*(b + c*
x^2)) + (45*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b
^(13/4)) - (45*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2
]*b^(13/4)) - (45*c^(1/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c
]*x])/(64*Sqrt[2]*b^(13/4)) + (45*c^(1/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*
Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(13/4))

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Rubi in Sympy [A]  time = 80.4502, size = 238, normalized size = 0.95 \[ \frac{1}{4 b \sqrt{x} \left (b + c x^{2}\right )^{2}} + \frac{9}{16 b^{2} \sqrt{x} \left (b + c x^{2}\right )} - \frac{45}{16 b^{3} \sqrt{x}} - \frac{45 \sqrt{2} \sqrt [4]{c} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{13}{4}}} + \frac{45 \sqrt{2} \sqrt [4]{c} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{13}{4}}} + \frac{45 \sqrt{2} \sqrt [4]{c} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{13}{4}}} - \frac{45 \sqrt{2} \sqrt [4]{c} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(9/2)/(c*x**4+b*x**2)**3,x)

[Out]

1/(4*b*sqrt(x)*(b + c*x**2)**2) + 9/(16*b**2*sqrt(x)*(b + c*x**2)) - 45/(16*b**3
*sqrt(x)) - 45*sqrt(2)*c**(1/4)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b)
+ sqrt(c)*x)/(128*b**(13/4)) + 45*sqrt(2)*c**(1/4)*log(sqrt(2)*b**(1/4)*c**(1/4
)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(128*b**(13/4)) + 45*sqrt(2)*c**(1/4)*atan(1 -
sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64*b**(13/4)) - 45*sqrt(2)*c**(1/4)*atan(1 +
sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(64*b**(13/4))

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Mathematica [A]  time = 0.220509, size = 234, normalized size = 0.93 \[ \frac{-\frac{32 b^{5/4} c x^{3/2}}{\left (b+c x^2\right )^2}-\frac{104 \sqrt [4]{b} c x^{3/2}}{b+c x^2}-45 \sqrt{2} \sqrt [4]{c} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+45 \sqrt{2} \sqrt [4]{c} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+90 \sqrt{2} \sqrt [4]{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-90 \sqrt{2} \sqrt [4]{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )-\frac{256 \sqrt [4]{b}}{\sqrt{x}}}{128 b^{13/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(9/2)/(b*x^2 + c*x^4)^3,x]

[Out]

((-256*b^(1/4))/Sqrt[x] - (32*b^(5/4)*c*x^(3/2))/(b + c*x^2)^2 - (104*b^(1/4)*c*
x^(3/2))/(b + c*x^2) + 90*Sqrt[2]*c^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b
^(1/4)] - 90*Sqrt[2]*c^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] - 45*
Sqrt[2]*c^(1/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 45*
Sqrt[2]*c^(1/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(128
*b^(13/4))

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Maple [A]  time = 0.025, size = 178, normalized size = 0.7 \[ -{\frac{13\,{c}^{2}}{16\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{7}{2}}}}-{\frac{17\,c}{16\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{45\,\sqrt{2}}{128\,{b}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{45\,\sqrt{2}}{64\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{45\,\sqrt{2}}{64\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-2\,{\frac{1}{{b}^{3}\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(9/2)/(c*x^4+b*x^2)^3,x)

[Out]

-13/16*c^2/b^3/(c*x^2+b)^2*x^(7/2)-17/16*c/b^2/(c*x^2+b)^2*x^(3/2)-45/128/b^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)))-45/64/b^3/(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)
^(1/4)*x^(1/2)+1)-45/64/b^3/(b/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/
2)-1)-2/b^3/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^(9/2)/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.289779, size = 333, normalized size = 1.33 \[ -\frac{180 \, c^{2} x^{4} + 324 \, b c x^{2} + 180 \,{\left (b^{3} c^{2} x^{4} + 2 \, b^{4} c x^{2} + b^{5}\right )} \sqrt{x} \left (-\frac{c}{b^{13}}\right )^{\frac{1}{4}} \arctan \left (\frac{91125 \, b^{10} \left (-\frac{c}{b^{13}}\right )^{\frac{3}{4}}}{91125 \, c \sqrt{x} + \sqrt{-8303765625 \, b^{7} c \sqrt{-\frac{c}{b^{13}}} + 8303765625 \, c^{2} x}}\right ) + 45 \,{\left (b^{3} c^{2} x^{4} + 2 \, b^{4} c x^{2} + b^{5}\right )} \sqrt{x} \left (-\frac{c}{b^{13}}\right )^{\frac{1}{4}} \log \left (91125 \, b^{10} \left (-\frac{c}{b^{13}}\right )^{\frac{3}{4}} + 91125 \, c \sqrt{x}\right ) - 45 \,{\left (b^{3} c^{2} x^{4} + 2 \, b^{4} c x^{2} + b^{5}\right )} \sqrt{x} \left (-\frac{c}{b^{13}}\right )^{\frac{1}{4}} \log \left (-91125 \, b^{10} \left (-\frac{c}{b^{13}}\right )^{\frac{3}{4}} + 91125 \, c \sqrt{x}\right ) + 128 \, b^{2}}{64 \,{\left (b^{3} c^{2} x^{4} + 2 \, b^{4} c x^{2} + b^{5}\right )} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

-1/64*(180*c^2*x^4 + 324*b*c*x^2 + 180*(b^3*c^2*x^4 + 2*b^4*c*x^2 + b^5)*sqrt(x)
*(-c/b^13)^(1/4)*arctan(91125*b^10*(-c/b^13)^(3/4)/(91125*c*sqrt(x) + sqrt(-8303
765625*b^7*c*sqrt(-c/b^13) + 8303765625*c^2*x))) + 45*(b^3*c^2*x^4 + 2*b^4*c*x^2
+ b^5)*sqrt(x)*(-c/b^13)^(1/4)*log(91125*b^10*(-c/b^13)^(3/4) + 91125*c*sqrt(x)
) - 45*(b^3*c^2*x^4 + 2*b^4*c*x^2 + b^5)*sqrt(x)*(-c/b^13)^(1/4)*log(-91125*b^10
*(-c/b^13)^(3/4) + 91125*c*sqrt(x)) + 128*b^2)/((b^3*c^2*x^4 + 2*b^4*c*x^2 + b^5
)*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**(9/2)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.280329, size = 297, normalized size = 1.18 \[ -\frac{2}{b^{3} \sqrt{x}} - \frac{45 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{4} c^{2}} - \frac{45 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{4} c^{2}} + \frac{45 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{4} c^{2}} - \frac{45 \, \sqrt{2} \left (b c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{4} c^{2}} - \frac{13 \, c^{2} x^{\frac{7}{2}} + 17 \, b c x^{\frac{3}{2}}}{16 \,{\left (c x^{2} + b\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(9/2)/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

-2/(b^3*sqrt(x)) - 45/64*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^4*c^2) - 45/64*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^4*c^2) + 45/128*s
qrt(2)*(b*c^3)^(3/4)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^4*c^2) -
45/128*sqrt(2)*(b*c^3)^(3/4)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(
b^4*c^2) - 1/16*(13*c^2*x^(7/2) + 17*b*c*x^(3/2))/((c*x^2 + b)^2*b^3)