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

Optimal. Leaf size=94 \[ \frac{2 b \sqrt{b x^2+c x^4} (4 b B-5 A c)}{15 c^3 x}-\frac{x \sqrt{b x^2+c x^4} (4 b B-5 A c)}{15 c^2}+\frac{B x^3 \sqrt{b x^2+c x^4}}{5 c} \]

[Out]

(2*b*(4*b*B - 5*A*c)*Sqrt[b*x^2 + c*x^4])/(15*c^3*x) - ((4*b*B - 5*A*c)*x*Sqrt[b
*x^2 + c*x^4])/(15*c^2) + (B*x^3*Sqrt[b*x^2 + c*x^4])/(5*c)

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Rubi [A]  time = 0.28775, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2 b \sqrt{b x^2+c x^4} (4 b B-5 A c)}{15 c^3 x}-\frac{x \sqrt{b x^2+c x^4} (4 b B-5 A c)}{15 c^2}+\frac{B x^3 \sqrt{b x^2+c x^4}}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*b*(4*b*B - 5*A*c)*Sqrt[b*x^2 + c*x^4])/(15*c^3*x) - ((4*b*B - 5*A*c)*x*Sqrt[b
*x^2 + c*x^4])/(15*c^2) + (B*x^3*Sqrt[b*x^2 + c*x^4])/(5*c)

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Rubi in Sympy [A]  time = 26.1165, size = 85, normalized size = 0.9 \[ \frac{B x^{3} \sqrt{b x^{2} + c x^{4}}}{5 c} - \frac{2 b \left (5 A c - 4 B b\right ) \sqrt{b x^{2} + c x^{4}}}{15 c^{3} x} + \frac{x \left (5 A c - 4 B b\right ) \sqrt{b x^{2} + c x^{4}}}{15 c^{2}} \]

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)**(1/2),x)

[Out]

B*x**3*sqrt(b*x**2 + c*x**4)/(5*c) - 2*b*(5*A*c - 4*B*b)*sqrt(b*x**2 + c*x**4)/(
15*c**3*x) + x*(5*A*c - 4*B*b)*sqrt(b*x**2 + c*x**4)/(15*c**2)

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Mathematica [A]  time = 0.0716247, size = 63, normalized size = 0.67 \[ \frac{\sqrt{x^2 \left (b+c x^2\right )} \left (-2 b c \left (5 A+2 B x^2\right )+c^2 x^2 \left (5 A+3 B x^2\right )+8 b^2 B\right )}{15 c^3 x} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[x^2*(b + c*x^2)]*(8*b^2*B - 2*b*c*(5*A + 2*B*x^2) + c^2*x^2*(5*A + 3*B*x^2
)))/(15*c^3*x)

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Maple [A]  time = 0.008, size = 65, normalized size = 0.7 \[ -{\frac{ \left ( c{x}^{2}+b \right ) \left ( -3\,B{c}^{2}{x}^{4}-5\,A{x}^{2}{c}^{2}+4\,B{x}^{2}bc+10\,Abc-8\,{b}^{2}B \right ) x}{15\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(c*x^2+b)*(-3*B*c^2*x^4-5*A*c^2*x^2+4*B*b*c*x^2+10*A*b*c-8*B*b^2)*x/c^3/(c
*x^4+b*x^2)^(1/2)

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Maxima [A]  time = 1.39257, size = 112, normalized size = 1.19 \[ \frac{{\left (c^{2} x^{4} - b c x^{2} - 2 \, b^{2}\right )} A}{3 \, \sqrt{c x^{2} + b} c^{2}} + \frac{{\left (3 \, c^{3} x^{6} - b c^{2} x^{4} + 4 \, b^{2} c x^{2} + 8 \, b^{3}\right )} B}{15 \, \sqrt{c x^{2} + b} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(c^2*x^4 - b*c*x^2 - 2*b^2)*A/(sqrt(c*x^2 + b)*c^2) + 1/15*(3*c^3*x^6 - b*c^
2*x^4 + 4*b^2*c*x^2 + 8*b^3)*B/(sqrt(c*x^2 + b)*c^3)

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Fricas [A]  time = 0.230826, size = 80, normalized size = 0.85 \[ \frac{{\left (3 \, B c^{2} x^{4} + 8 \, B b^{2} - 10 \, A b c -{\left (4 \, B b c - 5 \, A c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15 \, c^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(3*B*c^2*x^4 + 8*B*b^2 - 10*A*b*c - (4*B*b*c - 5*A*c^2)*x^2)*sqrt(c*x^4 + b
*x^2)/(c^3*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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