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

Optimal. Leaf size=61 \[ \frac{B \left (b x^2+c x^4\right )^{3/2}}{5 c x}-\frac{\left (b x^2+c x^4\right )^{3/2} (2 b B-5 A c)}{15 c^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0609238, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{B \left (b x^2+c x^4\right )^{3/2}}{5 c x}-\frac{\left (b x^2+c x^4\right )^{3/2} (2 b B-5 A c)}{15 c^2 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.84231, size = 51, normalized size = 0.84 \[ \frac{B \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{5 c x} + \frac{\left (5 A c - 2 B b\right ) \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{15 c^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.047161, size = 41, normalized size = 0.67 \[ \frac{\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (5 A c-2 b B+3 B c x^2\right )}{15 c^2 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 45, normalized size = 0.7 \[{\frac{ \left ( c{x}^{2}+b \right ) \left ( 3\,Bc{x}^{2}+5\,Ac-2\,Bb \right ) }{15\,{c}^{2}x}\sqrt{c{x}^{4}+b{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.39787, size = 69, normalized size = 1.13 \[ \frac{{\left (c x^{2} + b\right )}^{\frac{3}{2}} A}{3 \, c} + \frac{{\left (3 \, c^{2} x^{4} + b c x^{2} - 2 \, b^{2}\right )} \sqrt{c x^{2} + b} B}{15 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.220092, size = 77, normalized size = 1.26 \[ \frac{{\left (3 \, B c^{2} x^{4} - 2 \, B b^{2} + 5 \, A b c +{\left (B b c + 5 \, A c^{2}\right )} x^{2}\right )} \sqrt{c x^{4} + b x^{2}}}{15 \, c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213487, size = 99, normalized size = 1.62 \[ \frac{5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} A{\rm sign}\left (x\right ) + \frac{{\left (3 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b\right )} B{\rm sign}\left (x\right )}{c}}{15 \, c} + \frac{{\left (2 \, B b^{\frac{5}{2}} - 5 \, A b^{\frac{3}{2}} c\right )}{\rm sign}\left (x\right )}{15 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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