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

Optimal. Leaf size=343 \[ -\frac{9 \sqrt [4]{c} (5 b B-13 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{17/4}}+\frac{9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{17/4}}+\frac{9 \sqrt [4]{c} (5 b B-13 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{17/4}}-\frac{9 \sqrt [4]{c} (5 b B-13 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{17/4}}-\frac{9 (5 b B-13 A c)}{16 b^4 \sqrt{x}}+\frac{9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac{5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}-\frac{b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2} \]

[Out]

(9*(5*b*B - 13*A*c))/(80*b^3*c*x^(5/2)) - (9*(5*b*B - 13*A*c))/(16*b^4*Sqrt[x])
- (b*B - A*c)/(4*b*c*x^(5/2)*(b + c*x^2)^2) - (5*b*B - 13*A*c)/(16*b^2*c*x^(5/2)
*(b + c*x^2)) + (9*c^(1/4)*(5*b*B - 13*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])
/b^(1/4)])/(32*Sqrt[2]*b^(17/4)) - (9*c^(1/4)*(5*b*B - 13*A*c)*ArcTan[1 + (Sqrt[
2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(17/4)) - (9*c^(1/4)*(5*b*B - 13*A*c
)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(17/
4)) + (9*c^(1/4)*(5*b*B - 13*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x]
+ Sqrt[c]*x])/(64*Sqrt[2]*b^(17/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.600914, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423 \[ -\frac{9 \sqrt [4]{c} (5 b B-13 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{17/4}}+\frac{9 \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{64 \sqrt{2} b^{17/4}}+\frac{9 \sqrt [4]{c} (5 b B-13 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{32 \sqrt{2} b^{17/4}}-\frac{9 \sqrt [4]{c} (5 b B-13 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{32 \sqrt{2} b^{17/4}}-\frac{9 (5 b B-13 A c)}{16 b^4 \sqrt{x}}+\frac{9 (5 b B-13 A c)}{80 b^3 c x^{5/2}}-\frac{5 b B-13 A c}{16 b^2 c x^{5/2} \left (b+c x^2\right )}-\frac{b B-A c}{4 b c x^{5/2} \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(9*(5*b*B - 13*A*c))/(80*b^3*c*x^(5/2)) - (9*(5*b*B - 13*A*c))/(16*b^4*Sqrt[x])
- (b*B - A*c)/(4*b*c*x^(5/2)*(b + c*x^2)^2) - (5*b*B - 13*A*c)/(16*b^2*c*x^(5/2)
*(b + c*x^2)) + (9*c^(1/4)*(5*b*B - 13*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])
/b^(1/4)])/(32*Sqrt[2]*b^(17/4)) - (9*c^(1/4)*(5*b*B - 13*A*c)*ArcTan[1 + (Sqrt[
2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(32*Sqrt[2]*b^(17/4)) - (9*c^(1/4)*(5*b*B - 13*A*c
)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*b^(17/
4)) + (9*c^(1/4)*(5*b*B - 13*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x]
+ Sqrt[c]*x])/(64*Sqrt[2]*b^(17/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 92.932, size = 326, normalized size = 0.95 \[ \frac{A c - B b}{4 b c x^{\frac{5}{2}} \left (b + c x^{2}\right )^{2}} + \frac{13 A c - 5 B b}{16 b^{2} c x^{\frac{5}{2}} \left (b + c x^{2}\right )} - \frac{9 \left (13 A c - 5 B b\right )}{80 b^{3} c x^{\frac{5}{2}}} + \frac{9 \left (13 A c - 5 B b\right )}{16 b^{4} \sqrt{x}} + \frac{9 \sqrt{2} \sqrt [4]{c} \left (13 A c - 5 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{17}{4}}} - \frac{9 \sqrt{2} \sqrt [4]{c} \left (13 A c - 5 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{128 b^{\frac{17}{4}}} - \frac{9 \sqrt{2} \sqrt [4]{c} \left (13 A c - 5 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{17}{4}}} + \frac{9 \sqrt{2} \sqrt [4]{c} \left (13 A c - 5 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{64 b^{\frac{17}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*c - B*b)/(4*b*c*x**(5/2)*(b + c*x**2)**2) + (13*A*c - 5*B*b)/(16*b**2*c*x**(5
/2)*(b + c*x**2)) - 9*(13*A*c - 5*B*b)/(80*b**3*c*x**(5/2)) + 9*(13*A*c - 5*B*b)
/(16*b**4*sqrt(x)) + 9*sqrt(2)*c**(1/4)*(13*A*c - 5*B*b)*log(-sqrt(2)*b**(1/4)*c
**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(128*b**(17/4)) - 9*sqrt(2)*c**(1/4)*(13*
A*c - 5*B*b)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(128*b
**(17/4)) - 9*sqrt(2)*c**(1/4)*(13*A*c - 5*B*b)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x
)/b**(1/4))/(64*b**(17/4)) + 9*sqrt(2)*c**(1/4)*(13*A*c - 5*B*b)*atan(1 + sqrt(2
)*c**(1/4)*sqrt(x)/b**(1/4))/(64*b**(17/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.541549, size = 308, normalized size = 0.9 \[ \frac{-\frac{160 b^{5/4} c x^{3/2} (b B-A c)}{\left (b+c x^2\right )^2}-\frac{256 A b^{5/4}}{x^{5/2}}-\frac{40 \sqrt [4]{b} c x^{3/2} (13 b B-21 A c)}{b+c x^2}-\frac{1280 \sqrt [4]{b} (b B-3 A c)}{\sqrt{x}}+45 \sqrt{2} \sqrt [4]{c} (13 A c-5 b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+45 \sqrt{2} \sqrt [4]{c} (5 b B-13 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-90 \sqrt{2} \sqrt [4]{c} (13 A c-5 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+90 \sqrt{2} \sqrt [4]{c} (13 A c-5 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{640 b^{17/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-256*A*b^(5/4))/x^(5/2) - (1280*b^(1/4)*(b*B - 3*A*c))/Sqrt[x] - (160*b^(5/4)*
c*(b*B - A*c)*x^(3/2))/(b + c*x^2)^2 - (40*b^(1/4)*c*(13*b*B - 21*A*c)*x^(3/2))/
(b + c*x^2) - 90*Sqrt[2]*c^(1/4)*(-5*b*B + 13*A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*S
qrt[x])/b^(1/4)] + 90*Sqrt[2]*c^(1/4)*(-5*b*B + 13*A*c)*ArcTan[1 + (Sqrt[2]*c^(1
/4)*Sqrt[x])/b^(1/4)] + 45*Sqrt[2]*c^(1/4)*(-5*b*B + 13*A*c)*Log[Sqrt[b] - Sqrt[
2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 45*Sqrt[2]*c^(1/4)*(5*b*B - 13*A*c)*Lo
g[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(640*b^(17/4))

_______________________________________________________________________________________

Maple [A]  time = 0.034, size = 381, normalized size = 1.1 \[ -{\frac{2\,A}{5\,{b}^{3}}{x}^{-{\frac{5}{2}}}}+6\,{\frac{Ac}{\sqrt{x}{b}^{4}}}-2\,{\frac{B}{\sqrt{x}{b}^{3}}}+{\frac{21\,A{c}^{3}}{16\,{b}^{4} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{7}{2}}}}-{\frac{13\,B{c}^{2}}{16\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{7}{2}}}}+{\frac{25\,A{c}^{2}}{16\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{17\,Bc}{16\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{117\,c\sqrt{2}A}{128\,{b}^{4}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{117\,c\sqrt{2}A}{64\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{117\,c\sqrt{2}A}{64\,{b}^{4}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{45\,\sqrt{2}B}{128\,{b}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{45\,\sqrt{2}B}{64\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{45\,\sqrt{2}B}{64\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/5*A/b^3/x^(5/2)+6/x^(1/2)/b^4*A*c-2/x^(1/2)/b^3*B+21/16/b^4*c^3/(c*x^2+b)^2*x
^(7/2)*A-13/16/b^3*c^2/(c*x^2+b)^2*x^(7/2)*B+25/16/b^3*c^2/(c*x^2+b)^2*A*x^(3/2)
-17/16/b^2*c/(c*x^2+b)^2*B*x^(3/2)+117/128/b^4*c/(b/c)^(1/4)*2^(1/2)*A*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/b^4*c/(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+117/
64/b^4*c/(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)-45/128/b^3/
(b/c)^(1/4)*2^(1/2)*B*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)*B*arctan(2^(1/2)
/(b/c)^(1/4)*x^(1/2)+1)-45/64/b^3/(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/
4)*x^(1/2)-1)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.250985, size = 1257, normalized size = 3.66 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/320*(180*(5*B*b*c^2 - 13*A*c^3)*x^6 + 324*(5*B*b^2*c - 13*A*b*c^2)*x^4 + 128*
A*b^3 + 128*(5*B*b^3 - 13*A*b^2*c)*x^2 - 180*(b^4*c^2*x^6 + 2*b^5*c*x^4 + b^6*x^
2)*sqrt(x)*(-(625*B^4*b^4*c - 6500*A*B^3*b^3*c^2 + 25350*A^2*B^2*b^2*c^3 - 43940
*A^3*B*b*c^4 + 28561*A^4*c^5)/b^17)^(1/4)*arctan(-b^13*(-(625*B^4*b^4*c - 6500*A
*B^3*b^3*c^2 + 25350*A^2*B^2*b^2*c^3 - 43940*A^3*B*b*c^4 + 28561*A^4*c^5)/b^17)^
(3/4)/((125*B^3*b^3*c - 975*A*B^2*b^2*c^2 + 2535*A^2*B*b*c^3 - 2197*A^3*c^4)*sqr
t(x) - sqrt((15625*B^6*b^6*c^2 - 243750*A*B^5*b^5*c^3 + 1584375*A^2*B^4*b^4*c^4
- 5492500*A^3*B^3*b^3*c^5 + 10710375*A^4*B^2*b^2*c^6 - 11138790*A^5*B*b*c^7 + 48
26809*A^6*c^8)*x - (625*B^4*b^13*c - 6500*A*B^3*b^12*c^2 + 25350*A^2*B^2*b^11*c^
3 - 43940*A^3*B*b^10*c^4 + 28561*A^4*b^9*c^5)*sqrt(-(625*B^4*b^4*c - 6500*A*B^3*
b^3*c^2 + 25350*A^2*B^2*b^2*c^3 - 43940*A^3*B*b*c^4 + 28561*A^4*c^5)/b^17)))) -
45*(b^4*c^2*x^6 + 2*b^5*c*x^4 + b^6*x^2)*sqrt(x)*(-(625*B^4*b^4*c - 6500*A*B^3*b
^3*c^2 + 25350*A^2*B^2*b^2*c^3 - 43940*A^3*B*b*c^4 + 28561*A^4*c^5)/b^17)^(1/4)*
log(729*b^13*(-(625*B^4*b^4*c - 6500*A*B^3*b^3*c^2 + 25350*A^2*B^2*b^2*c^3 - 439
40*A^3*B*b*c^4 + 28561*A^4*c^5)/b^17)^(3/4) - 729*(125*B^3*b^3*c - 975*A*B^2*b^2
*c^2 + 2535*A^2*B*b*c^3 - 2197*A^3*c^4)*sqrt(x)) + 45*(b^4*c^2*x^6 + 2*b^5*c*x^4
+ b^6*x^2)*sqrt(x)*(-(625*B^4*b^4*c - 6500*A*B^3*b^3*c^2 + 25350*A^2*B^2*b^2*c^
3 - 43940*A^3*B*b*c^4 + 28561*A^4*c^5)/b^17)^(1/4)*log(-729*b^13*(-(625*B^4*b^4*
c - 6500*A*B^3*b^3*c^2 + 25350*A^2*B^2*b^2*c^3 - 43940*A^3*B*b*c^4 + 28561*A^4*c
^5)/b^17)^(3/4) - 729*(125*B^3*b^3*c - 975*A*B^2*b^2*c^2 + 2535*A^2*B*b*c^3 - 21
97*A^3*c^4)*sqrt(x)))/((b^4*c^2*x^6 + 2*b^5*c*x^4 + b^6*x^2)*sqrt(x))

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.227888, size = 440, normalized size = 1.28 \[ -\frac{9 \, \sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{5} c^{2}} - \frac{9 \, \sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{64 \, b^{5} c^{2}} + \frac{9 \, \sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{5} c^{2}} - \frac{9 \, \sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 13 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{128 \, b^{5} c^{2}} - \frac{13 \, B b c^{2} x^{\frac{7}{2}} - 21 \, A c^{3} x^{\frac{7}{2}} + 17 \, B b^{2} c x^{\frac{3}{2}} - 25 \, A b c^{2} x^{\frac{3}{2}}}{16 \,{\left (c x^{2} + b\right )}^{2} b^{4}} - \frac{2 \,{\left (5 \, B b x^{2} - 15 \, A c x^{2} + A b\right )}}{5 \, b^{4} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-9/64*sqrt(2)*(5*(b*c^3)^(3/4)*B*b - 13*(b*c^3)^(3/4)*A*c)*arctan(1/2*sqrt(2)*(s
qrt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/(b^5*c^2) - 9/64*sqrt(2)*(5*(b*c^3)
^(3/4)*B*b - 13*(b*c^3)^(3/4)*A*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*
sqrt(x))/(b/c)^(1/4))/(b^5*c^2) + 9/128*sqrt(2)*(5*(b*c^3)^(3/4)*B*b - 13*(b*c^3
)^(3/4)*A*c)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^5*c^2) - 9/128*s
qrt(2)*(5*(b*c^3)^(3/4)*B*b - 13*(b*c^3)^(3/4)*A*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1
/4) + x + sqrt(b/c))/(b^5*c^2) - 1/16*(13*B*b*c^2*x^(7/2) - 21*A*c^3*x^(7/2) + 1
7*B*b^2*c*x^(3/2) - 25*A*b*c^2*x^(3/2))/((c*x^2 + b)^2*b^4) - 2/5*(5*B*b*x^2 - 1
5*A*c*x^2 + A*b)/(b^4*x^(5/2))