Optimal. Leaf size=80 \[ \frac{8 b^2 \left (b x^2+c x^4\right )^{5/2}}{315 c^3 x^5}-\frac{4 b \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac{\left (b x^2+c x^4\right )^{5/2}}{9 c x} \]
[Out]
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Rubi [A] time = 0.173749, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{8 b^2 \left (b x^2+c x^4\right )^{5/2}}{315 c^3 x^5}-\frac{4 b \left (b x^2+c x^4\right )^{5/2}}{63 c^2 x^3}+\frac{\left (b x^2+c x^4\right )^{5/2}}{9 c x} \]
Antiderivative was successfully verified.
[In] Int[x^2*(b*x^2 + c*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 20.3437, size = 70, normalized size = 0.88 \[ \frac{8 b^{2} \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{315 c^{3} x^{5}} - \frac{4 b \left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{63 c^{2} x^{3}} + \frac{\left (b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{9 c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(c*x**4+b*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0399144, size = 53, normalized size = 0.66 \[ \frac{x \left (b+c x^2\right )^3 \left (8 b^2-20 b c x^2+35 c^2 x^4\right )}{315 c^3 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(b*x^2 + c*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.008, size = 50, normalized size = 0.6 \[{\frac{ \left ( c{x}^{2}+b \right ) \left ( 35\,{c}^{2}{x}^{4}-20\,bc{x}^{2}+8\,{b}^{2} \right ) }{315\,{c}^{3}{x}^{3}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(c*x^4+b*x^2)^(3/2),x)
[Out]
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Maxima [A] time = 0.695611, size = 77, normalized size = 0.96 \[ \frac{{\left (35 \, c^{4} x^{8} + 50 \, b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{4} - 4 \, b^{3} c x^{2} + 8 \, b^{4}\right )} \sqrt{c x^{2} + b}}{315 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2)^(3/2)*x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.264651, size = 86, normalized size = 1.08 \[ \frac{{\left (35 \, c^{4} x^{8} + 50 \, b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{4} - 4 \, b^{3} c x^{2} + 8 \, b^{4}\right )} \sqrt{c x^{4} + b x^{2}}}{315 \, c^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2)^(3/2)*x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(c*x**4+b*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.272949, size = 163, normalized size = 2.04 \[ -\frac{8 \, b^{\frac{9}{2}}{\rm sign}\left (x\right )}{315 \, c^{3}} + \frac{\frac{3 \,{\left (15 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{2}\right )} b{\rm sign}\left (x\right )}{c^{2}} + \frac{{\left (35 \,{\left (c x^{2} + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{3}\right )}{\rm sign}\left (x\right )}{c^{2}}}{315 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2)^(3/2)*x^2,x, algorithm="giac")
[Out]