Optimal. Leaf size=269 \[ \frac{\sqrt{a+b x^2+c x^4} \left (-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x^2 (2 c d-b e)+8 c^2 d^2\right )}{16 c e^3}-\frac{(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{3/2} e^4}+\frac{\left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^4}+\frac{\left (a+b x^2+c x^4\right )^{3/2}}{6 e} \]
[Out]
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Rubi [A] time = 1.00379, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259 \[ \frac{\sqrt{a+b x^2+c x^4} \left (-2 c e (5 b d-4 a e)+b^2 e^2-2 c e x^2 (2 c d-b e)+8 c^2 d^2\right )}{16 c e^3}-\frac{(2 c d-b e) \left (-4 c e (2 b d-3 a e)-b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{32 c^{3/2} e^4}+\frac{\left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^4}+\frac{\left (a+b x^2+c x^4\right )^{3/2}}{6 e} \]
Antiderivative was successfully verified.
[In] Int[(x*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2),x]
[Out]
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Rubi in Sympy [A] time = 109.583, size = 253, normalized size = 0.94 \[ \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{6 e} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 e^{4}} + \frac{\sqrt{a + b x^{2} + c x^{4}} \left (4 a c e^{2} + \frac{b^{2} e^{2}}{2} - 5 b c d e + 4 c^{2} d^{2} + c e x^{2} \left (b e - 2 c d\right )\right )}{8 c e^{3}} - \frac{\left (b e - 2 c d\right ) \left (- 12 a c e^{2} + b^{2} e^{2} + 8 b c d e - 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{32 c^{\frac{3}{2}} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(c*x**4+b*x**2+a)**(3/2)/(e*x**2+d),x)
[Out]
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Mathematica [A] time = 0.416168, size = 276, normalized size = 1.03 \[ \frac{\frac{2 e \sqrt{a+b x^2+c x^4} \left (2 c e \left (16 a e-15 b d+7 b e x^2\right )+3 b^2 e^2+4 c^2 \left (6 d^2-3 d e x^2+2 e^2 x^4\right )\right )}{c}-\frac{3 (2 c d-b e) \left (4 c e (3 a e-2 b d)-b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{c^{3/2}}-48 \left (e (a e-b d)+c d^2\right )^{3/2} \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )+48 \log \left (d+e x^2\right ) \left (e (a e-b d)+c d^2\right )^{3/2}}{96 e^4} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2),x]
[Out]
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Maple [B] time = 0.011, size = 1411, normalized size = 5.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(c*x^4+b*x^2+a)^(3/2)/(e*x^2+d),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^(3/2)*x/(e*x^2 + d),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^(3/2)*x/(e*x^2 + d),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{d + e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(c*x**4+b*x**2+a)**(3/2)/(e*x**2+d),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^(3/2)*x/(e*x^2 + d),x, algorithm="giac")
[Out]