3.339 \(\int \frac{x^{23/2}}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=251 \[ \frac{45 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{13/4}}-\frac{45 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{13/4}}+\frac{45 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} c^{13/4}}-\frac{45 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} c^{13/4}}-\frac{9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}-\frac{x^{9/2}}{4 c \left (b+c x^2\right )^2}+\frac{45 \sqrt{x}}{16 c^3} \]

[Out]

(45*Sqrt[x])/(16*c^3) - x^(9/2)/(4*c*(b + c*x^2)^2) - (9*x^(5/2))/(16*c^2*(b + c
*x^2)) + (45*b^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*
c^(13/4)) - (45*b^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[
2]*c^(13/4)) + (45*b^(1/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[
c]*x])/(64*Sqrt[2]*c^(13/4)) - (45*b^(1/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)
*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*c^(13/4))

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Rubi [A]  time = 0.438835, 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 \[ \frac{45 \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{13/4}}-\frac{45 \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} c^{13/4}}+\frac{45 \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} c^{13/4}}-\frac{45 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} c^{13/4}}-\frac{9 x^{5/2}}{16 c^2 \left (b+c x^2\right )}-\frac{x^{9/2}}{4 c \left (b+c x^2\right )^2}+\frac{45 \sqrt{x}}{16 c^3} \]

Antiderivative was successfully verified.

[In]  Int[x^(23/2)/(b*x^2 + c*x^4)^3,x]

[Out]

(45*Sqrt[x])/(16*c^3) - x^(9/2)/(4*c*(b + c*x^2)^2) - (9*x^(5/2))/(16*c^2*(b + c
*x^2)) + (45*b^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*
c^(13/4)) - (45*b^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[
2]*c^(13/4)) + (45*b^(1/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[
c]*x])/(64*Sqrt[2]*c^(13/4)) - (45*b^(1/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)
*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*c^(13/4))

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Rubi in Sympy [A]  time = 79.5493, size = 236, normalized size = 0.94 \[ \frac{45 \sqrt{2} \sqrt [4]{b} \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 c^{\frac{13}{4}}} - \frac{45 \sqrt{2} \sqrt [4]{b} \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 c^{\frac{13}{4}}} + \frac{45 \sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 c^{\frac{13}{4}}} - \frac{45 \sqrt{2} \sqrt [4]{b} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 c^{\frac{13}{4}}} - \frac{x^{\frac{9}{2}}}{4 c \left (b + c x^{2}\right )^{2}} - \frac{9 x^{\frac{5}{2}}}{16 c^{2} \left (b + c x^{2}\right )} + \frac{45 \sqrt{x}}{16 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(23/2)/(c*x**4+b*x**2)**3,x)

[Out]

45*sqrt(2)*b**(1/4)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x
)/(128*c**(13/4)) - 45*sqrt(2)*b**(1/4)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) +
sqrt(b) + sqrt(c)*x)/(128*c**(13/4)) + 45*sqrt(2)*b**(1/4)*atan(1 - sqrt(2)*c**(
1/4)*sqrt(x)/b**(1/4))/(64*c**(13/4)) - 45*sqrt(2)*b**(1/4)*atan(1 + sqrt(2)*c**
(1/4)*sqrt(x)/b**(1/4))/(64*c**(13/4)) - x**(9/2)/(4*c*(b + c*x**2)**2) - 9*x**(
5/2)/(16*c**2*(b + c*x**2)) + 45*sqrt(x)/(16*c**3)

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Mathematica [A]  time = 0.19042, size = 236, normalized size = 0.94 \[ \frac{-\frac{32 b^2 \sqrt [4]{c} \sqrt{x}}{\left (b+c x^2\right )^2}+\frac{136 b \sqrt [4]{c} \sqrt{x}}{b+c x^2}+45 \sqrt{2} \sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-45 \sqrt{2} \sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+90 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )-90 \sqrt{2} \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )+256 \sqrt [4]{c} \sqrt{x}}{128 c^{13/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(23/2)/(b*x^2 + c*x^4)^3,x]

[Out]

(256*c^(1/4)*Sqrt[x] - (32*b^2*c^(1/4)*Sqrt[x])/(b + c*x^2)^2 + (136*b*c^(1/4)*S
qrt[x])/(b + c*x^2) + 90*Sqrt[2]*b^(1/4)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^
(1/4)] - 90*Sqrt[2]*b^(1/4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 45*S
qrt[2]*b^(1/4)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] - 45*S
qrt[2]*b^(1/4)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(128*
c^(13/4))

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Maple [A]  time = 0.024, size = 178, normalized size = 0.7 \[ 2\,{\frac{\sqrt{x}}{{c}^{3}}}+{\frac{17\,b}{16\,{c}^{2} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{5}{2}}}}+{\frac{13\,{b}^{2}}{16\,{c}^{3} \left ( c{x}^{2}+b \right ) ^{2}}\sqrt{x}}-{\frac{45\,\sqrt{2}}{128\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }-{\frac{45\,\sqrt{2}}{64\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{45\,\sqrt{2}}{64\,{c}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(23/2)/(c*x^4+b*x^2)^3,x)

[Out]

2*x^(1/2)/c^3+17/16/c^2*b/(c*x^2+b)^2*x^(5/2)+13/16/c^3*b^2/(c*x^2+b)^2*x^(1/2)-
45/128/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)))-45/64/c^3*(b/c)^(1/4)*2^(1/2)*arctan(
2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-45/64/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^(23/2)/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.285824, size = 313, normalized size = 1.25 \[ \frac{180 \,{\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )} \left (-\frac{b}{c^{13}}\right )^{\frac{1}{4}} \arctan \left (\frac{c^{3} \left (-\frac{b}{c^{13}}\right )^{\frac{1}{4}}}{\sqrt{c^{6} \sqrt{-\frac{b}{c^{13}}} + x} + \sqrt{x}}\right ) - 45 \,{\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )} \left (-\frac{b}{c^{13}}\right )^{\frac{1}{4}} \log \left (45 \, c^{3} \left (-\frac{b}{c^{13}}\right )^{\frac{1}{4}} + 45 \, \sqrt{x}\right ) + 45 \,{\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )} \left (-\frac{b}{c^{13}}\right )^{\frac{1}{4}} \log \left (-45 \, c^{3} \left (-\frac{b}{c^{13}}\right )^{\frac{1}{4}} + 45 \, \sqrt{x}\right ) + 4 \,{\left (32 \, c^{2} x^{4} + 81 \, b c x^{2} + 45 \, b^{2}\right )} \sqrt{x}}{64 \,{\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(23/2)/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

1/64*(180*(c^5*x^4 + 2*b*c^4*x^2 + b^2*c^3)*(-b/c^13)^(1/4)*arctan(c^3*(-b/c^13)
^(1/4)/(sqrt(c^6*sqrt(-b/c^13) + x) + sqrt(x))) - 45*(c^5*x^4 + 2*b*c^4*x^2 + b^
2*c^3)*(-b/c^13)^(1/4)*log(45*c^3*(-b/c^13)^(1/4) + 45*sqrt(x)) + 45*(c^5*x^4 +
2*b*c^4*x^2 + b^2*c^3)*(-b/c^13)^(1/4)*log(-45*c^3*(-b/c^13)^(1/4) + 45*sqrt(x))
+ 4*(32*c^2*x^4 + 81*b*c*x^2 + 45*b^2)*sqrt(x))/(c^5*x^4 + 2*b*c^4*x^2 + b^2*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**(23/2)/(c*x**4+b*x**2)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.279218, size = 281, normalized size = 1.12 \[ -\frac{45 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, c^{4}} - \frac{45 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, c^{4}} - \frac{45 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, c^{4}} + \frac{45 \, \sqrt{2} \left (b c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, c^{4}} + \frac{2 \, \sqrt{x}}{c^{3}} + \frac{17 \, b c x^{\frac{5}{2}} + 13 \, b^{2} \sqrt{x}}{16 \,{\left (c x^{2} + b\right )}^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(23/2)/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

-45/64*sqrt(2)*(b*c^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqrt(x)
)/(b/c)^(1/4))/c^4 - 45/64*sqrt(2)*(b*c^3)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b
/c)^(1/4) - 2*sqrt(x))/(b/c)^(1/4))/c^4 - 45/128*sqrt(2)*(b*c^3)^(1/4)*ln(sqrt(2
)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/c^4 + 45/128*sqrt(2)*(b*c^3)^(1/4)*ln(-sq
rt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/c^4 + 2*sqrt(x)/c^3 + 1/16*(17*b*c*x^
(5/2) + 13*b^2*sqrt(x))/((c*x^2 + b)^2*c^3)