3.204 \(\int \frac{\sqrt{x} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=310 \[ -\frac{\sqrt [4]{c} (5 b B-9 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{13/4}}+\frac{\sqrt [4]{c} (5 b B-9 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{13/4}}+\frac{\sqrt [4]{c} (5 b B-9 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{13/4}}-\frac{\sqrt [4]{c} (5 b B-9 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{13/4}}-\frac{5 b B-9 A c}{2 b^3 \sqrt{x}}+\frac{5 b B-9 A c}{10 b^2 c x^{5/2}}-\frac{b B-A c}{2 b c x^{5/2} \left (b+c x^2\right )} \]

[Out]

(5*b*B - 9*A*c)/(10*b^2*c*x^(5/2)) - (5*b*B - 9*A*c)/(2*b^3*Sqrt[x]) - (b*B - A*
c)/(2*b*c*x^(5/2)*(b + c*x^2)) + (c^(1/4)*(5*b*B - 9*A*c)*ArcTan[1 - (Sqrt[2]*c^
(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(13/4)) - (c^(1/4)*(5*b*B - 9*A*c)*ArcTan[
1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(13/4)) - (c^(1/4)*(5*b*B -
9*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b
^(13/4)) + (c^(1/4)*(5*b*B - 9*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x
] + Sqrt[c]*x])/(8*Sqrt[2]*b^(13/4))

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Rubi [A]  time = 0.522795, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{\sqrt [4]{c} (5 b B-9 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{13/4}}+\frac{\sqrt [4]{c} (5 b B-9 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{13/4}}+\frac{\sqrt [4]{c} (5 b B-9 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{13/4}}-\frac{\sqrt [4]{c} (5 b B-9 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{13/4}}-\frac{5 b B-9 A c}{2 b^3 \sqrt{x}}+\frac{5 b B-9 A c}{10 b^2 c x^{5/2}}-\frac{b B-A c}{2 b c x^{5/2} \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(5*b*B - 9*A*c)/(10*b^2*c*x^(5/2)) - (5*b*B - 9*A*c)/(2*b^3*Sqrt[x]) - (b*B - A*
c)/(2*b*c*x^(5/2)*(b + c*x^2)) + (c^(1/4)*(5*b*B - 9*A*c)*ArcTan[1 - (Sqrt[2]*c^
(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(13/4)) - (c^(1/4)*(5*b*B - 9*A*c)*ArcTan[
1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(13/4)) - (c^(1/4)*(5*b*B -
9*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b
^(13/4)) + (c^(1/4)*(5*b*B - 9*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x
] + Sqrt[c]*x])/(8*Sqrt[2]*b^(13/4))

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Rubi in Sympy [A]  time = 83.8051, size = 289, normalized size = 0.93 \[ \frac{A c - B b}{2 b c x^{\frac{5}{2}} \left (b + c x^{2}\right )} - \frac{9 A c - 5 B b}{10 b^{2} c x^{\frac{5}{2}}} + \frac{9 A c - 5 B b}{2 b^{3} \sqrt{x}} + \frac{\sqrt{2} \sqrt [4]{c} \left (9 A c - 5 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{13}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (9 A c - 5 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{13}{4}}} - \frac{\sqrt{2} \sqrt [4]{c} \left (9 A c - 5 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{13}{4}}} + \frac{\sqrt{2} \sqrt [4]{c} \left (9 A c - 5 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)*x**(1/2)/(c*x**4+b*x**2)**2,x)

[Out]

(A*c - B*b)/(2*b*c*x**(5/2)*(b + c*x**2)) - (9*A*c - 5*B*b)/(10*b**2*c*x**(5/2))
+ (9*A*c - 5*B*b)/(2*b**3*sqrt(x)) + sqrt(2)*c**(1/4)*(9*A*c - 5*B*b)*log(-sqrt
(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(16*b**(13/4)) - sqrt(2)*c*
*(1/4)*(9*A*c - 5*B*b)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)
*x)/(16*b**(13/4)) - sqrt(2)*c**(1/4)*(9*A*c - 5*B*b)*atan(1 - sqrt(2)*c**(1/4)*
sqrt(x)/b**(1/4))/(8*b**(13/4)) + sqrt(2)*c**(1/4)*(9*A*c - 5*B*b)*atan(1 + sqrt
(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*b**(13/4))

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Mathematica [A]  time = 0.53535, size = 277, normalized size = 0.89 \[ \frac{-\frac{32 A b^{5/4}}{x^{5/2}}-\frac{40 \sqrt [4]{b} c x^{3/2} (b B-A c)}{b+c x^2}-\frac{160 \sqrt [4]{b} (b B-2 A c)}{\sqrt{x}}+5 \sqrt{2} \sqrt [4]{c} (9 A c-5 b B) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+5 \sqrt{2} \sqrt [4]{c} (5 b B-9 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-10 \sqrt{2} \sqrt [4]{c} (9 A c-5 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+10 \sqrt{2} \sqrt [4]{c} (9 A c-5 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{80 b^{13/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-32*A*b^(5/4))/x^(5/2) - (160*b^(1/4)*(b*B - 2*A*c))/Sqrt[x] - (40*b^(1/4)*c*(
b*B - A*c)*x^(3/2))/(b + c*x^2) - 10*Sqrt[2]*c^(1/4)*(-5*b*B + 9*A*c)*ArcTan[1 -
(Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 10*Sqrt[2]*c^(1/4)*(-5*b*B + 9*A*c)*ArcTan
[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 5*Sqrt[2]*c^(1/4)*(-5*b*B + 9*A*c)*Log
[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 5*Sqrt[2]*c^(1/4)*(5*b
*B - 9*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(80*b^(1
3/4))

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Maple [A]  time = 0.027, size = 339, normalized size = 1.1 \[ -{\frac{2\,A}{5\,{b}^{2}}{x}^{-{\frac{5}{2}}}}+4\,{\frac{Ac}{\sqrt{x}{b}^{3}}}-2\,{\frac{B}{\sqrt{x}{b}^{2}}}+{\frac{A{c}^{2}}{2\,{b}^{3} \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}-{\frac{Bc}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }{x}^{{\frac{3}{2}}}}+{\frac{9\,c\sqrt{2}A}{16\,{b}^{3}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{9\,c\sqrt{2}A}{8\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{9\,c\sqrt{2}A}{8\,{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{5\,\sqrt{2}B}{16\,{b}^{2}}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{5\,\sqrt{2}B}{8\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{5\,\sqrt{2}B}{8\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)*x^(1/2)/(c*x^4+b*x^2)^2,x)

[Out]

-2/5*A/b^2/x^(5/2)+4/x^(1/2)/b^3*A*c-2/x^(1/2)/b^2*B+1/2/b^3*c^2*x^(3/2)/(c*x^2+
b)*A-1/2/b^2*c*x^(3/2)/(c*x^2+b)*B+9/16/b^3*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)))+
9/8/b^3*c/(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)+9/8/b^3*c/
(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)-5/16/b^2/(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)))-5/8/b^2/(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*
x^(1/2)+1)-5/8/b^2/(b/c)^(1/4)*2^(1/2)*B*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((B*x^2 + A)*sqrt(x)/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252297, size = 1164, normalized size = 3.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/40*(20*(5*B*b*c - 9*A*c^2)*x^4 + 16*A*b^2 + 16*(5*B*b^2 - 9*A*b*c)*x^2 - 20*(
b^3*c*x^4 + b^4*x^2)*sqrt(x)*(-(625*B^4*b^4*c - 4500*A*B^3*b^3*c^2 + 12150*A^2*B
^2*b^2*c^3 - 14580*A^3*B*b*c^4 + 6561*A^4*c^5)/b^13)^(1/4)*arctan(-b^10*(-(625*B
^4*b^4*c - 4500*A*B^3*b^3*c^2 + 12150*A^2*B^2*b^2*c^3 - 14580*A^3*B*b*c^4 + 6561
*A^4*c^5)/b^13)^(3/4)/((125*B^3*b^3*c - 675*A*B^2*b^2*c^2 + 1215*A^2*B*b*c^3 - 7
29*A^3*c^4)*sqrt(x) - sqrt((15625*B^6*b^6*c^2 - 168750*A*B^5*b^5*c^3 + 759375*A^
2*B^4*b^4*c^4 - 1822500*A^3*B^3*b^3*c^5 + 2460375*A^4*B^2*b^2*c^6 - 1771470*A^5*
B*b*c^7 + 531441*A^6*c^8)*x - (625*B^4*b^11*c - 4500*A*B^3*b^10*c^2 + 12150*A^2*
B^2*b^9*c^3 - 14580*A^3*B*b^8*c^4 + 6561*A^4*b^7*c^5)*sqrt(-(625*B^4*b^4*c - 450
0*A*B^3*b^3*c^2 + 12150*A^2*B^2*b^2*c^3 - 14580*A^3*B*b*c^4 + 6561*A^4*c^5)/b^13
)))) - 5*(b^3*c*x^4 + b^4*x^2)*sqrt(x)*(-(625*B^4*b^4*c - 4500*A*B^3*b^3*c^2 + 1
2150*A^2*B^2*b^2*c^3 - 14580*A^3*B*b*c^4 + 6561*A^4*c^5)/b^13)^(1/4)*log(b^10*(-
(625*B^4*b^4*c - 4500*A*B^3*b^3*c^2 + 12150*A^2*B^2*b^2*c^3 - 14580*A^3*B*b*c^4
+ 6561*A^4*c^5)/b^13)^(3/4) - (125*B^3*b^3*c - 675*A*B^2*b^2*c^2 + 1215*A^2*B*b*
c^3 - 729*A^3*c^4)*sqrt(x)) + 5*(b^3*c*x^4 + b^4*x^2)*sqrt(x)*(-(625*B^4*b^4*c -
4500*A*B^3*b^3*c^2 + 12150*A^2*B^2*b^2*c^3 - 14580*A^3*B*b*c^4 + 6561*A^4*c^5)/
b^13)^(1/4)*log(-b^10*(-(625*B^4*b^4*c - 4500*A*B^3*b^3*c^2 + 12150*A^2*B^2*b^2*
c^3 - 14580*A^3*B*b*c^4 + 6561*A^4*c^5)/b^13)^(3/4) - (125*B^3*b^3*c - 675*A*B^2
*b^2*c^2 + 1215*A^2*B*b*c^3 - 729*A^3*c^4)*sqrt(x)))/((b^3*c*x^4 + b^4*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((B*x**2+A)*x**(1/2)/(c*x**4+b*x**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.226043, size = 409, normalized size = 1.32 \[ -\frac{B b c x^{\frac{3}{2}} - A c^{2} x^{\frac{3}{2}}}{2 \,{\left (c x^{2} + b\right )} b^{3}} - \frac{\sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 9 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{4} c^{2}} - \frac{\sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 9 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{4} c^{2}} + \frac{\sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 9 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{4} c^{2}} - \frac{\sqrt{2}{\left (5 \, \left (b c^{3}\right )^{\frac{3}{4}} B b - 9 \, \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{4} c^{2}} - \frac{2 \,{\left (5 \, B b x^{2} - 10 \, A c x^{2} + A b\right )}}{5 \, b^{3} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(B*b*c*x^(3/2) - A*c^2*x^(3/2))/((c*x^2 + b)*b^3) - 1/8*sqrt(2)*(5*(b*c^3)^
(3/4)*B*b - 9*(b*c^3)^(3/4)*A*c)*arctan(1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) + 2*sqr
t(x))/(b/c)^(1/4))/(b^4*c^2) - 1/8*sqrt(2)*(5*(b*c^3)^(3/4)*B*b - 9*(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^4*c
^2) + 1/16*sqrt(2)*(5*(b*c^3)^(3/4)*B*b - 9*(b*c^3)^(3/4)*A*c)*ln(sqrt(2)*sqrt(x
)*(b/c)^(1/4) + x + sqrt(b/c))/(b^4*c^2) - 1/16*sqrt(2)*(5*(b*c^3)^(3/4)*B*b - 9
*(b*c^3)^(3/4)*A*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^4*c^2) -
2/5*(5*B*b*x^2 - 10*A*c*x^2 + A*b)/(b^3*x^(5/2))