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

Optimal. Leaf size=235 \[ \frac{(b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{5/4} c^{3/4}}-\frac{(b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{5/4} c^{3/4}}-\frac{(b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{5/4} c^{3/4}}+\frac{(b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{5/4} c^{3/4}}-\frac{2 A}{b \sqrt{x}} \]

[Out]

(-2*A)/(b*Sqrt[x]) - ((b*B - A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])
/(Sqrt[2]*b^(5/4)*c^(3/4)) + ((b*B - A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b
^(1/4)])/(Sqrt[2]*b^(5/4)*c^(3/4)) + ((b*B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*
c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(5/4)*c^(3/4)) - ((b*B - A*c)*Log[Sqr
t[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(5/4)*c^(3/4))

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Rubi [A]  time = 0.395755, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346 \[ \frac{(b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{5/4} c^{3/4}}-\frac{(b B-A c) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{2 \sqrt{2} b^{5/4} c^{3/4}}-\frac{(b B-A c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{\sqrt{2} b^{5/4} c^{3/4}}+\frac{(b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{\sqrt{2} b^{5/4} c^{3/4}}-\frac{2 A}{b \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*A)/(b*Sqrt[x]) - ((b*B - A*c)*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/b^(1/4)])
/(Sqrt[2]*b^(5/4)*c^(3/4)) + ((b*B - A*c)*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/b
^(1/4)])/(Sqrt[2]*b^(5/4)*c^(3/4)) + ((b*B - A*c)*Log[Sqrt[b] - Sqrt[2]*b^(1/4)*
c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(5/4)*c^(3/4)) - ((b*B - A*c)*Log[Sqr
t[b] + Sqrt[2]*b^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(2*Sqrt[2]*b^(5/4)*c^(3/4))

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Rubi in Sympy [A]  time = 65.5439, size = 219, normalized size = 0.93 \[ - \frac{2 A}{b \sqrt{x}} - \frac{\sqrt{2} \left (A c - B b\right ) \log{\left (- \sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{5}{4}} c^{\frac{3}{4}}} + \frac{\sqrt{2} \left (A c - B b\right ) \log{\left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x} + \sqrt{b} + \sqrt{c} x \right )}}{4 b^{\frac{5}{4}} c^{\frac{3}{4}}} + \frac{\sqrt{2} \left (A c - B b\right ) \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{5}{4}} c^{\frac{3}{4}}} - \frac{\sqrt{2} \left (A c - B b\right ) \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}}{2 b^{\frac{5}{4}} c^{\frac{3}{4}}} \]

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),x)

[Out]

-2*A/(b*sqrt(x)) - sqrt(2)*(A*c - B*b)*log(-sqrt(2)*b**(1/4)*c**(1/4)*sqrt(x) +
sqrt(b) + sqrt(c)*x)/(4*b**(5/4)*c**(3/4)) + sqrt(2)*(A*c - B*b)*log(sqrt(2)*b**
(1/4)*c**(1/4)*sqrt(x) + sqrt(b) + sqrt(c)*x)/(4*b**(5/4)*c**(3/4)) + sqrt(2)*(A
*c - B*b)*atan(1 - sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(2*b**(5/4)*c**(3/4)) - sq
rt(2)*(A*c - B*b)*atan(1 + sqrt(2)*c**(1/4)*sqrt(x)/b**(1/4))/(2*b**(5/4)*c**(3/
4))

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Mathematica [A]  time = 0.363427, size = 221, normalized size = 0.94 \[ \frac{\frac{\sqrt{2} (b B-A c) \log \left (-\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{c^{3/4}}+\frac{\sqrt{2} (A c-b B) \log \left (\sqrt{2} \sqrt [4]{b} \sqrt [4]{c} \sqrt{x}+\sqrt{b}+\sqrt{c} x\right )}{c^{3/4}}+\frac{2 \sqrt{2} (A c-b B) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )}{c^{3/4}}+\frac{2 \sqrt{2} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}+1\right )}{c^{3/4}}-\frac{8 A \sqrt [4]{b}}{\sqrt{x}}}{4 b^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 277, normalized size = 1.2 \[ -{\frac{\sqrt{2}A}{2\,b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{\sqrt{2}A}{4\,b}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-{\frac{\sqrt{2}A}{2\,b}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{\sqrt{2}B}{2\,c}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{\sqrt{2}B}{4\,c}\ln \left ({1 \left ( x-\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{b}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{b}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+{\frac{\sqrt{2}B}{2\,c}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{b}{c}}}}}}-2\,{\frac{A}{b\sqrt{x}}} \]

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),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245064, size = 996, normalized size = 4.24 \[ -\frac{4 \, b \sqrt{x} \left (-\frac{B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{3}}\right )^{\frac{1}{4}} \arctan \left (-\frac{b^{4} c^{2} \left (-\frac{B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{3}}\right )^{\frac{3}{4}}}{{\left (B^{3} b^{3} - 3 \, A B^{2} b^{2} c + 3 \, A^{2} B b c^{2} - A^{3} c^{3}\right )} \sqrt{x} - \sqrt{{\left (B^{6} b^{6} - 6 \, A B^{5} b^{5} c + 15 \, A^{2} B^{4} b^{4} c^{2} - 20 \, A^{3} B^{3} b^{3} c^{3} + 15 \, A^{4} B^{2} b^{2} c^{4} - 6 \, A^{5} B b c^{5} + A^{6} c^{6}\right )} x -{\left (B^{4} b^{7} c - 4 \, A B^{3} b^{6} c^{2} + 6 \, A^{2} B^{2} b^{5} c^{3} - 4 \, A^{3} B b^{4} c^{4} + A^{4} b^{3} c^{5}\right )} \sqrt{-\frac{B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{3}}}}}\right ) + b \sqrt{x} \left (-\frac{B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{3}}\right )^{\frac{1}{4}} \log \left (b^{4} c^{2} \left (-\frac{B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{3}}\right )^{\frac{3}{4}} -{\left (B^{3} b^{3} - 3 \, A B^{2} b^{2} c + 3 \, A^{2} B b c^{2} - A^{3} c^{3}\right )} \sqrt{x}\right ) - b \sqrt{x} \left (-\frac{B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{3}}\right )^{\frac{1}{4}} \log \left (-b^{4} c^{2} \left (-\frac{B^{4} b^{4} - 4 \, A B^{3} b^{3} c + 6 \, A^{2} B^{2} b^{2} c^{2} - 4 \, A^{3} B b c^{3} + A^{4} c^{4}}{b^{5} c^{3}}\right )^{\frac{3}{4}} -{\left (B^{3} b^{3} - 3 \, A B^{2} b^{2} c + 3 \, A^{2} B b c^{2} - A^{3} c^{3}\right )} \sqrt{x}\right ) + 4 \, A}{2 \, b \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(4*b*sqrt(x)*(-(B^4*b^4 - 4*A*B^3*b^3*c + 6*A^2*B^2*b^2*c^2 - 4*A^3*B*b*c^3
+ A^4*c^4)/(b^5*c^3))^(1/4)*arctan(-b^4*c^2*(-(B^4*b^4 - 4*A*B^3*b^3*c + 6*A^2*
B^2*b^2*c^2 - 4*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^3))^(3/4)/((B^3*b^3 - 3*A*B^2*b^2*
c + 3*A^2*B*b*c^2 - A^3*c^3)*sqrt(x) - sqrt((B^6*b^6 - 6*A*B^5*b^5*c + 15*A^2*B^
4*b^4*c^2 - 20*A^3*B^3*b^3*c^3 + 15*A^4*B^2*b^2*c^4 - 6*A^5*B*b*c^5 + A^6*c^6)*x
- (B^4*b^7*c - 4*A*B^3*b^6*c^2 + 6*A^2*B^2*b^5*c^3 - 4*A^3*B*b^4*c^4 + A^4*b^3*
c^5)*sqrt(-(B^4*b^4 - 4*A*B^3*b^3*c + 6*A^2*B^2*b^2*c^2 - 4*A^3*B*b*c^3 + A^4*c^
4)/(b^5*c^3))))) + b*sqrt(x)*(-(B^4*b^4 - 4*A*B^3*b^3*c + 6*A^2*B^2*b^2*c^2 - 4*
A^3*B*b*c^3 + A^4*c^4)/(b^5*c^3))^(1/4)*log(b^4*c^2*(-(B^4*b^4 - 4*A*B^3*b^3*c +
6*A^2*B^2*b^2*c^2 - 4*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^3))^(3/4) - (B^3*b^3 - 3*A*
B^2*b^2*c + 3*A^2*B*b*c^2 - A^3*c^3)*sqrt(x)) - b*sqrt(x)*(-(B^4*b^4 - 4*A*B^3*b
^3*c + 6*A^2*B^2*b^2*c^2 - 4*A^3*B*b*c^3 + A^4*c^4)/(b^5*c^3))^(1/4)*log(-b^4*c^
2*(-(B^4*b^4 - 4*A*B^3*b^3*c + 6*A^2*B^2*b^2*c^2 - 4*A^3*B*b*c^3 + A^4*c^4)/(b^5
*c^3))^(3/4) - (B^3*b^3 - 3*A*B^2*b^2*c + 3*A^2*B*b*c^2 - A^3*c^3)*sqrt(x)) + 4*
A)/(b*sqrt(x))

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Sympy [A]  time = 51.3887, size = 374, normalized size = 1.59 \[ \begin{cases} \tilde{\infty } \left (- \frac{2 A}{5 x^{\frac{5}{2}}} - \frac{2 B}{\sqrt{x}}\right ) & \text{for}\: b = 0 \wedge c = 0 \\\frac{- \frac{2 A}{5 x^{\frac{5}{2}}} - \frac{2 B}{\sqrt{x}}}{c} & \text{for}\: b = 0 \\\frac{- \frac{2 A}{\sqrt{x}} + \frac{2 B x^{\frac{3}{2}}}{3}}{b} & \text{for}\: c = 0 \\- \frac{2 A}{b \sqrt{x}} + \frac{\left (-1\right )^{\frac{3}{4}} A \log{\left (- \sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{5}{4}} c^{6} \left (\frac{1}{c}\right )^{\frac{25}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} A \log{\left (\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 b^{\frac{5}{4}} c^{6} \left (\frac{1}{c}\right )^{\frac{25}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} A \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{b} \sqrt [4]{\frac{1}{c}}} \right )}}{b^{\frac{5}{4}} c^{6} \left (\frac{1}{c}\right )^{\frac{25}{4}}} - \frac{\left (-1\right )^{\frac{3}{4}} B \log{\left (- \sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 \sqrt [4]{b} c^{7} \left (\frac{1}{c}\right )^{\frac{25}{4}}} + \frac{\left (-1\right )^{\frac{3}{4}} B \log{\left (\sqrt [4]{-1} \sqrt [4]{b} \sqrt [4]{\frac{1}{c}} + \sqrt{x} \right )}}{2 \sqrt [4]{b} c^{7} \left (\frac{1}{c}\right )^{\frac{25}{4}}} + \frac{\left (-1\right )^{\frac{3}{4}} B \operatorname{atan}{\left (\frac{\left (-1\right )^{\frac{3}{4}} \sqrt{x}}{\sqrt [4]{b} \sqrt [4]{\frac{1}{c}}} \right )}}{\sqrt [4]{b} c^{7} \left (\frac{1}{c}\right )^{\frac{25}{4}}} & \text{otherwise} \end{cases} \]

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),x)

[Out]

Piecewise((zoo*(-2*A/(5*x**(5/2)) - 2*B/sqrt(x)), Eq(b, 0) & Eq(c, 0)), ((-2*A/(
5*x**(5/2)) - 2*B/sqrt(x))/c, Eq(b, 0)), ((-2*A/sqrt(x) + 2*B*x**(3/2)/3)/b, Eq(
c, 0)), (-2*A/(b*sqrt(x)) + (-1)**(3/4)*A*log(-(-1)**(1/4)*b**(1/4)*(1/c)**(1/4)
+ sqrt(x))/(2*b**(5/4)*c**6*(1/c)**(25/4)) - (-1)**(3/4)*A*log((-1)**(1/4)*b**(
1/4)*(1/c)**(1/4) + sqrt(x))/(2*b**(5/4)*c**6*(1/c)**(25/4)) - (-1)**(3/4)*A*ata
n((-1)**(3/4)*sqrt(x)/(b**(1/4)*(1/c)**(1/4)))/(b**(5/4)*c**6*(1/c)**(25/4)) - (
-1)**(3/4)*B*log(-(-1)**(1/4)*b**(1/4)*(1/c)**(1/4) + sqrt(x))/(2*b**(1/4)*c**7*
(1/c)**(25/4)) + (-1)**(3/4)*B*log((-1)**(1/4)*b**(1/4)*(1/c)**(1/4) + sqrt(x))/
(2*b**(1/4)*c**7*(1/c)**(25/4)) + (-1)**(3/4)*B*atan((-1)**(3/4)*sqrt(x)/(b**(1/
4)*(1/c)**(1/4)))/(b**(1/4)*c**7*(1/c)**(25/4)), True))

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GIAC/XCAS [A]  time = 0.222468, size = 339, normalized size = 1.44 \[ -\frac{2 \, A}{b \sqrt{x}} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{3}{4}} B b - \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{2} c^{3}} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{3}{4}} B b - \left (b c^{3}\right )^{\frac{3}{4}} A c\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{b}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{b}{c}\right )^{\frac{1}{4}}}\right )}{2 \, b^{2} c^{3}} - \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{3}{4}} B b - \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{2} c^{3}} + \frac{\sqrt{2}{\left (\left (b c^{3}\right )^{\frac{3}{4}} B b - \left (b c^{3}\right )^{\frac{3}{4}} A c\right )}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{b}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{b}{c}}\right )}{4 \, b^{2} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/(b*sqrt(x)) + 1/2*sqrt(2)*((b*c^3)^(3/4)*B*b - (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^2*c^3) + 1/2*sqrt(2)
*((b*c^3)^(3/4)*B*b - (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^2*c^3) - 1/4*sqrt(2)*((b*c^3)^(3/4)*B*b - (b*c^3)
^(3/4)*A*c)*ln(sqrt(2)*sqrt(x)*(b/c)^(1/4) + x + sqrt(b/c))/(b^2*c^3) + 1/4*sqrt
(2)*((b*c^3)^(3/4)*B*b - (b*c^3)^(3/4)*A*c)*ln(-sqrt(2)*sqrt(x)*(b/c)^(1/4) + x
+ sqrt(b/c))/(b^2*c^3)