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

Optimal. Leaf size=310 \[ \frac{c^{3/4} (7 b B-11 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{15/4}}-\frac{c^{3/4} (7 b B-11 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{15/4}}+\frac{c^{3/4} (7 b B-11 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{15/4}}-\frac{c^{3/4} (7 b B-11 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{15/4}}-\frac{7 b B-11 A c}{6 b^3 x^{3/2}}+\frac{7 b B-11 A c}{14 b^2 c x^{7/2}}-\frac{b B-A c}{2 b c x^{7/2} \left (b+c x^2\right )} \]

[Out]

(7*b*B - 11*A*c)/(14*b^2*c*x^(7/2)) - (7*b*B - 11*A*c)/(6*b^3*x^(3/2)) - (b*B -
A*c)/(2*b*c*x^(7/2)*(b + c*x^2)) + (c^(3/4)*(7*b*B - 11*A*c)*ArcTan[1 - (Sqrt[2]
*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(15/4)) - (c^(3/4)*(7*b*B - 11*A*c)*Arc
Tan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(15/4)) + (c^(3/4)*(7*b
*B - 11*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt
[2]*b^(15/4)) - (c^(3/4)*(7*b*B - 11*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*
Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(15/4))

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Rubi [A]  time = 0.524771, 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 \[ \frac{c^{3/4} (7 b B-11 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{15/4}}-\frac{c^{3/4} (7 b B-11 A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{8 \sqrt{2} b^{15/4}}+\frac{c^{3/4} (7 b B-11 A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{4 \sqrt{2} b^{15/4}}-\frac{c^{3/4} (7 b B-11 A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{4 \sqrt{2} b^{15/4}}-\frac{7 b B-11 A c}{6 b^3 x^{3/2}}+\frac{7 b B-11 A c}{14 b^2 c x^{7/2}}-\frac{b B-A c}{2 b c x^{7/2} \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(7*b*B - 11*A*c)/(14*b^2*c*x^(7/2)) - (7*b*B - 11*A*c)/(6*b^3*x^(3/2)) - (b*B -
A*c)/(2*b*c*x^(7/2)*(b + c*x^2)) + (c^(3/4)*(7*b*B - 11*A*c)*ArcTan[1 - (Sqrt[2]
*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(15/4)) - (c^(3/4)*(7*b*B - 11*A*c)*Arc
Tan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])/(4*Sqrt[2]*b^(15/4)) + (c^(3/4)*(7*b
*B - 11*A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(8*Sqrt
[2]*b^(15/4)) - (c^(3/4)*(7*b*B - 11*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*
Sqrt[x] + Sqrt[c]*x])/(8*Sqrt[2]*b^(15/4))

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Rubi in Sympy [A]  time = 82.6396, size = 289, normalized size = 0.93 \[ \frac{A c - B b}{2 b c x^{\frac{7}{2}} \left (b + c x^{2}\right )} - \frac{11 A c - 7 B b}{14 b^{2} c x^{\frac{7}{2}}} + \frac{11 A c - 7 B b}{6 b^{3} x^{\frac{3}{2}}} - \frac{\sqrt{2} c^{\frac{3}{4}} \left (11 A c - 7 B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{15}{4}}} + \frac{\sqrt{2} c^{\frac{3}{4}} \left (11 A c - 7 B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{16 b^{\frac{15}{4}}} - \frac{\sqrt{2} c^{\frac{3}{4}} \left (11 A c - 7 B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{15}{4}}} + \frac{\sqrt{2} c^{\frac{3}{4}} \left (11 A c - 7 B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{8 b^{\frac{15}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*c - B*b)/(2*b*c*x**(7/2)*(b + c*x**2)) - (11*A*c - 7*B*b)/(14*b**2*c*x**(7/2)
) + (11*A*c - 7*B*b)/(6*b**3*x**(3/2)) - sqrt(2)*c**(3/4)*(11*A*c - 7*B*b)*log(-
sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(16*b**(15/4)) + sqrt(2
)*c**(3/4)*(11*A*c - 7*B*b)*log(sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sq
rt(c)*x)/(16*b**(15/4)) - sqrt(2)*c**(3/4)*(11*A*c - 7*B*b)*atan(1 - sqrt(2)*c**
(1/4)*sqrt(x)/b**(1/4))/(8*b**(15/4)) + sqrt(2)*c**(3/4)*(11*A*c - 7*B*b)*atan(1
+ sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(8*b**(15/4))

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Mathematica [A]  time = 0.487227, size = 277, normalized size = 0.89 \[ \frac{-\frac{224 b^{3/4} (b B-2 A c)}{x^{3/2}}-\frac{168 b^{3/4} c \sqrt{x} (b B-A c)}{b+c x^2}-\frac{96 A b^{7/4}}{x^{7/2}}+21 \sqrt{2} c^{3/4} (7 b B-11 A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )+21 \sqrt{2} c^{3/4} (11 A c-7 b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )-42 \sqrt{2} c^{3/4} (11 A c-7 b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )+42 \sqrt{2} c^{3/4} (11 A c-7 b B) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{336 b^{15/4}} \]

Antiderivative was successfully verified.

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

[Out]

((-96*A*b^(7/4))/x^(7/2) - (224*b^(3/4)*(b*B - 2*A*c))/x^(3/2) - (168*b^(3/4)*c*
(b*B - A*c)*Sqrt[x])/(b + c*x^2) - 42*Sqrt[2]*c^(3/4)*(-7*b*B + 11*A*c)*ArcTan[1
- (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 42*Sqrt[2]*c^(3/4)*(-7*b*B + 11*A*c)*Arc
Tan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)] + 21*Sqrt[2]*c^(3/4)*(7*b*B - 11*A*c)
*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x] + 21*Sqrt[2]*c^(3/4)
*(-7*b*B + 11*A*c)*Log[Sqrt[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(
336*b^(15/4))

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Maple [A]  time = 0.025, size = 348, normalized size = 1.1 \[ -{\frac{2\,A}{7\,{b}^{2}}{x}^{-{\frac{7}{2}}}}+{\frac{4\,Ac}{3\,{b}^{3}}{x}^{-{\frac{3}{2}}}}-{\frac{2\,B}{3\,{b}^{2}}{x}^{-{\frac{3}{2}}}}+{\frac{A{c}^{2}}{2\,{b}^{3} \left ( c{x}^{2}+b \right ) }\sqrt{x}}-{\frac{Bc}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }\sqrt{x}}+{\frac{11\,{c}^{2}\sqrt{2}A}{8\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }+{\frac{11\,{c}^{2}\sqrt{2}A}{16\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }+{\frac{11\,{c}^{2}\sqrt{2}A}{8\,{b}^{4}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) }-{\frac{7\,c\sqrt{2}B}{8\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ) }-{\frac{7\,c\sqrt{2}B}{16\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ) }-{\frac{7\,c\sqrt{2}B}{8\,{b}^{3}}\sqrt [4]{{\frac{b}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/7*A/b^2/x^(7/2)+4/3/x^(3/2)/b^3*A*c-2/3/x^(3/2)/b^2*B+1/2/b^3*c^2*x^(1/2)/(c*
x^2+b)*A-1/2/b^2*c*x^(1/2)/(c*x^2+b)*B+11/8/b^4*c^2*(b/c)^(1/4)*2^(1/2)*A*arctan
(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)+11/16/b^4*c^2*(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)))+
11/8/b^4*c^2*(b/c)^(1/4)*2^(1/2)*A*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)+1)-7/8/b^3
*c*(b/c)^(1/4)*2^(1/2)*B*arctan(2^(1/2)/(b/c)^(1/4)*x^(1/2)-1)-7/16/b^3*c*(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)))-7/8/b^3*c*(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)/((c*x^4 + b*x^2)^2*sqrt(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.249994, size = 917, normalized size = 2.96 \[ -\frac{28 \,{\left (7 \, B b c - 11 \, A c^{2}\right )} x^{4} + 48 \, A b^{2} + 16 \,{\left (7 \, B b^{2} - 11 \, A b c\right )} x^{2} + 84 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )} \sqrt{x} \left (-\frac{2401 \, B^{4} b^{4} c^{3} - 15092 \, A B^{3} b^{3} c^{4} + 35574 \, A^{2} B^{2} b^{2} c^{5} - 37268 \, A^{3} B b c^{6} + 14641 \, A^{4} c^{7}}{b^{15}}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{4} \left (-\frac{2401 \, B^{4} b^{4} c^{3} - 15092 \, A B^{3} b^{3} c^{4} + 35574 \, A^{2} B^{2} b^{2} c^{5} - 37268 \, A^{3} B b c^{6} + 14641 \, A^{4} c^{7}}{b^{15}}\right )^{\frac{1}{4}}}{{\left (7 \, B b c - 11 \, A c^{2}\right )} \sqrt{x} - \sqrt{b^{8} \sqrt{-\frac{2401 \, B^{4} b^{4} c^{3} - 15092 \, A B^{3} b^{3} c^{4} + 35574 \, A^{2} B^{2} b^{2} c^{5} - 37268 \, A^{3} B b c^{6} + 14641 \, A^{4} c^{7}}{b^{15}}} +{\left (49 \, B^{2} b^{2} c^{2} - 154 \, A B b c^{3} + 121 \, A^{2} c^{4}\right )} x}}\right ) - 21 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )} \sqrt{x} \left (-\frac{2401 \, B^{4} b^{4} c^{3} - 15092 \, A B^{3} b^{3} c^{4} + 35574 \, A^{2} B^{2} b^{2} c^{5} - 37268 \, A^{3} B b c^{6} + 14641 \, A^{4} c^{7}}{b^{15}}\right )^{\frac{1}{4}} \log \left (b^{4} \left (-\frac{2401 \, B^{4} b^{4} c^{3} - 15092 \, A B^{3} b^{3} c^{4} + 35574 \, A^{2} B^{2} b^{2} c^{5} - 37268 \, A^{3} B b c^{6} + 14641 \, A^{4} c^{7}}{b^{15}}\right )^{\frac{1}{4}} -{\left (7 \, B b c - 11 \, A c^{2}\right )} \sqrt{x}\right ) + 21 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )} \sqrt{x} \left (-\frac{2401 \, B^{4} b^{4} c^{3} - 15092 \, A B^{3} b^{3} c^{4} + 35574 \, A^{2} B^{2} b^{2} c^{5} - 37268 \, A^{3} B b c^{6} + 14641 \, A^{4} c^{7}}{b^{15}}\right )^{\frac{1}{4}} \log \left (-b^{4} \left (-\frac{2401 \, B^{4} b^{4} c^{3} - 15092 \, A B^{3} b^{3} c^{4} + 35574 \, A^{2} B^{2} b^{2} c^{5} - 37268 \, A^{3} B b c^{6} + 14641 \, A^{4} c^{7}}{b^{15}}\right )^{\frac{1}{4}} -{\left (7 \, B b c - 11 \, A c^{2}\right )} \sqrt{x}\right )}{168 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )} \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/168*(28*(7*B*b*c - 11*A*c^2)*x^4 + 48*A*b^2 + 16*(7*B*b^2 - 11*A*b*c)*x^2 + 8
4*(b^3*c*x^5 + b^4*x^3)*sqrt(x)*(-(2401*B^4*b^4*c^3 - 15092*A*B^3*b^3*c^4 + 3557
4*A^2*B^2*b^2*c^5 - 37268*A^3*B*b*c^6 + 14641*A^4*c^7)/b^15)^(1/4)*arctan(-b^4*(
-(2401*B^4*b^4*c^3 - 15092*A*B^3*b^3*c^4 + 35574*A^2*B^2*b^2*c^5 - 37268*A^3*B*b
*c^6 + 14641*A^4*c^7)/b^15)^(1/4)/((7*B*b*c - 11*A*c^2)*sqrt(x) - sqrt(b^8*sqrt(
-(2401*B^4*b^4*c^3 - 15092*A*B^3*b^3*c^4 + 35574*A^2*B^2*b^2*c^5 - 37268*A^3*B*b
*c^6 + 14641*A^4*c^7)/b^15) + (49*B^2*b^2*c^2 - 154*A*B*b*c^3 + 121*A^2*c^4)*x))
) - 21*(b^3*c*x^5 + b^4*x^3)*sqrt(x)*(-(2401*B^4*b^4*c^3 - 15092*A*B^3*b^3*c^4 +
35574*A^2*B^2*b^2*c^5 - 37268*A^3*B*b*c^6 + 14641*A^4*c^7)/b^15)^(1/4)*log(b^4*
(-(2401*B^4*b^4*c^3 - 15092*A*B^3*b^3*c^4 + 35574*A^2*B^2*b^2*c^5 - 37268*A^3*B*
b*c^6 + 14641*A^4*c^7)/b^15)^(1/4) - (7*B*b*c - 11*A*c^2)*sqrt(x)) + 21*(b^3*c*x
^5 + b^4*x^3)*sqrt(x)*(-(2401*B^4*b^4*c^3 - 15092*A*B^3*b^3*c^4 + 35574*A^2*B^2*
b^2*c^5 - 37268*A^3*B*b*c^6 + 14641*A^4*c^7)/b^15)^(1/4)*log(-b^4*(-(2401*B^4*b^
4*c^3 - 15092*A*B^3*b^3*c^4 + 35574*A^2*B^2*b^2*c^5 - 37268*A^3*B*b*c^6 + 14641*
A^4*c^7)/b^15)^(1/4) - (7*B*b*c - 11*A*c^2)*sqrt(x)))/((b^3*c*x^5 + b^4*x^3)*sqr
t(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)/(c*x**4+b*x**2)**2/x**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221011, size = 394, normalized size = 1.27 \[ -\frac{\sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 11 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{4}} - \frac{\sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 11 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{8 \, b^{4}} - \frac{\sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 11 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{4}} + \frac{\sqrt{2}{\left (7 \, \left (b c^{3}\right )^{\frac{1}{4}} B b - 11 \, \left (b c^{3}\right )^{\frac{1}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{16 \, b^{4}} - \frac{B b c \sqrt{x} - A c^{2} \sqrt{x}}{2 \,{\left (c x^{2} + b\right )} b^{3}} - \frac{2 \,{\left (7 \, B b x^{2} - 14 \, A c x^{2} + 3 \, A b\right )}}{21 \, b^{3} x^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*sqrt(2)*(7*(b*c^3)^(1/4)*B*b - 11*(b*c^3)^(1/4)*A*c)*arctan(1/2*sqrt(2)*(sq
rt(2)*(b/c)^(1/4) + 2*sqrt(x))/(b/c)^(1/4))/b^4 - 1/8*sqrt(2)*(7*(b*c^3)^(1/4)*B
*b - 11*(b*c^3)^(1/4)*A*c)*arctan(-1/2*sqrt(2)*(sqrt(2)*(b/c)^(1/4) - 2*sqrt(x))
/(b/c)^(1/4))/b^4 - 1/16*sqrt(2)*(7*(b*c^3)^(1/4)*B*b - 11*(b*c^3)^(1/4)*A*c)*ln
(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b^4 + 1/16*sqrt(2)*(7*(b*c^3)^(1/4
)*B*b - 11*(b*c^3)^(1/4)*A*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/b
^4 - 1/2*(B*b*c*sqrt(x) - A*c^2*sqrt(x))/((c*x^2 + b)*b^3) - 2/21*(7*B*b*x^2 - 1
4*A*c*x^2 + 3*A*b)/(b^3*x^(7/2))