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

Optimal. Leaf size=69 \[ \frac{\sqrt{b x^2+c x^4} (2 b B-A c)}{b c^2 x}-\frac{x^3 (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]

[Out]

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

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Rubi [A]  time = 0.220653, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\sqrt{b x^2+c x^4} (2 b B-A c)}{b c^2 x}-\frac{x^3 (b B-A c)}{b c \sqrt{b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.1368, size = 54, normalized size = 0.78 \[ \frac{x^{3} \left (A c - B b\right )}{b c \sqrt{b x^{2} + c x^{4}}} - \frac{\left (A c - 2 B b\right ) \sqrt{b x^{2} + c x^{4}}}{b c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0311094, size = 35, normalized size = 0.51 \[ \frac{x \left (-A c+2 b B+B c x^2\right )}{c^2 \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 44, normalized size = 0.6 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -Bc{x}^{2}+Ac-2\,Bb \right ){x}^{3}}{{c}^{2}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.4043, size = 53, normalized size = 0.77 \[ \frac{{\left (c x^{2} + 2 \, b\right )} B}{\sqrt{c x^{2} + b} c^{2}} - \frac{A}{\sqrt{c x^{2} + b} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x^2 + 2*b)*B/(sqrt(c*x^2 + b)*c^2) - A/(sqrt(c*x^2 + b)*c)

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Fricas [A]  time = 0.220152, size = 61, normalized size = 0.88 \[ \frac{\sqrt{c x^{4} + b x^{2}}{\left (B c x^{2} + 2 \, B b - A c\right )}}{c^{3} x^{3} + b c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.232933, size = 81, normalized size = 1.17 \[ -\frac{2 \, B \sqrt{b}}{{\left ({\left (\sqrt{c + \frac{b}{x^{2}}} - \frac{\sqrt{b}}{x}\right )}^{2} - c\right )} c} + \frac{B b - A c}{\sqrt{c + \frac{b}{x^{2}}} c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*B*sqrt(b)/(((sqrt(c + b/x^2) - sqrt(b)/x)^2 - c)*c) + (B*b - A*c)/(sqrt(c + b
/x^2)*c^2*x)