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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]