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

Optimal. Leaf size=142 \[ -\frac{(2 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 b^{5/2}}+\frac{\sqrt{b x^2+c x^4} (2 b B-3 A c)}{2 b^2 c x^3}-\frac{2 b B-3 A c}{3 b c x \sqrt{b x^2+c x^4}}-\frac{B}{3 c x \sqrt{b x^2+c x^4}} \]

[Out]

-B/(3*c*x*Sqrt[b*x^2 + c*x^4]) - (2*b*B - 3*A*c)/(3*b*c*x*Sqrt[b*x^2 + c*x^4]) +
((2*b*B - 3*A*c)*Sqrt[b*x^2 + c*x^4])/(2*b^2*c*x^3) - ((2*b*B - 3*A*c)*ArcTanh[
(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])/(2*b^(5/2))

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Rubi [A]  time = 0.192562, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{(2 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 b^{5/2}}+\frac{\sqrt{b x^2+c x^4} (2 b B-3 A c)}{2 b^2 c x^3}-\frac{2 b B-3 A c}{3 b c x \sqrt{b x^2+c x^4}}-\frac{B}{3 c x \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

-B/(3*c*x*Sqrt[b*x^2 + c*x^4]) - (2*b*B - 3*A*c)/(3*b*c*x*Sqrt[b*x^2 + c*x^4]) +
((2*b*B - 3*A*c)*Sqrt[b*x^2 + c*x^4])/(2*b^2*c*x^3) - ((2*b*B - 3*A*c)*ArcTanh[
(Sqrt[b]*x)/Sqrt[b*x^2 + c*x^4]])/(2*b^(5/2))

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Rubi in Sympy [A]  time = 19.6348, size = 122, normalized size = 0.86 \[ - \frac{B}{3 c x \sqrt{b x^{2} + c x^{4}}} + \frac{3 A c - 2 B b}{3 b c x \sqrt{b x^{2} + c x^{4}}} - \frac{\left (3 A c - 2 B b\right ) \sqrt{b x^{2} + c x^{4}}}{2 b^{2} c x^{3}} + \frac{\left (3 A c - 2 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{2 b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-B/(3*c*x*sqrt(b*x**2 + c*x**4)) + (3*A*c - 2*B*b)/(3*b*c*x*sqrt(b*x**2 + c*x**4
)) - (3*A*c - 2*B*b)*sqrt(b*x**2 + c*x**4)/(2*b**2*c*x**3) + (3*A*c - 2*B*b)*ata
nh(sqrt(b)*x/sqrt(b*x**2 + c*x**4))/(2*b**(5/2))

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Mathematica [A]  time = 0.177967, size = 123, normalized size = 0.87 \[ \frac{\sqrt{b} \left (2 b B x^2-A \left (b+3 c x^2\right )\right )+x^2 \log (x) \sqrt{b+c x^2} (2 b B-3 A c)+x^2 \sqrt{b+c x^2} (3 A c-2 b B) \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )}{2 b^{5/2} x \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b]*(2*b*B*x^2 - A*(b + 3*c*x^2)) + (2*b*B - 3*A*c)*x^2*Sqrt[b + c*x^2]*Log
[x] + (-2*b*B + 3*A*c)*x^2*Sqrt[b + c*x^2]*Log[b + Sqrt[b]*Sqrt[b + c*x^2]])/(2*
b^(5/2)*x*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.019, size = 131, normalized size = 0.9 \[ -{\frac{ \left ( c{x}^{2}+b \right ) x}{2} \left ( 3\,Ac{x}^{2}{b}^{5/2}-2\,B{b}^{7/2}{x}^{2}+2\,B\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){b}^{3}{x}^{2}\sqrt{c{x}^{2}+b}-3\,Ac\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){b}^{2}{x}^{2}\sqrt{c{x}^{2}+b}+A{b}^{{\frac{7}{2}}} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x*(c*x^2+b)*(3*A*c*x^2*b^(5/2)-2*B*b^(7/2)*x^2+2*B*ln(2*(b^(1/2)*(c*x^2+b)^
(1/2)+b)/x)*b^3*x^2*(c*x^2+b)^(1/2)-3*A*c*ln(2*(b^(1/2)*(c*x^2+b)^(1/2)+b)/x)*b^
2*x^2*(c*x^2+b)^(1/2)+A*b^(7/2))/(c*x^4+b*x^2)^(3/2)/b^(9/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{{\left (c x^{4} + b x^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.242721, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (2 \, B b c - 3 \, A c^{2}\right )} x^{5} +{\left (2 \, B b^{2} - 3 \, A b c\right )} x^{3}\right )} \sqrt{b} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} + 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2}}{\left (A b^{2} -{\left (2 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )}}{4 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )}}, \frac{{\left ({\left (2 \, B b c - 3 \, A c^{2}\right )} x^{5} +{\left (2 \, B b^{2} - 3 \, A b c\right )} x^{3}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) - \sqrt{c x^{4} + b x^{2}}{\left (A b^{2} -{\left (2 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )}}{2 \,{\left (b^{3} c x^{5} + b^{4} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/4*(((2*B*b*c - 3*A*c^2)*x^5 + (2*B*b^2 - 3*A*b*c)*x^3)*sqrt(b)*log(-((c*x^3
+ 2*b*x)*sqrt(b) + 2*sqrt(c*x^4 + b*x^2)*b)/x^3) + 2*sqrt(c*x^4 + b*x^2)*(A*b^2
- (2*B*b^2 - 3*A*b*c)*x^2))/(b^3*c*x^5 + b^4*x^3), 1/2*(((2*B*b*c - 3*A*c^2)*x^5
+ (2*B*b^2 - 3*A*b*c)*x^3)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(c*x^4 + b*x^2)) - sq
rt(c*x^4 + b*x^2)*(A*b^2 - (2*B*b^2 - 3*A*b*c)*x^2))/(b^3*c*x^5 + b^4*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x^{2}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.565225, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x