Optimal. Leaf size=360 \[ \frac{\left (8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{5/2} e^5}-\frac{\sqrt{a+b x^2+c x^4} \left (-2 c e x^2 \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+3 b^3 e^3+64 c^3 d^3\right )}{128 c^2 e^4}-\frac{d \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^5}-\frac{\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b e+8 c d-6 c e x^2\right )}{48 c e^2} \]
[Out]
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Rubi [A] time = 1.49269, antiderivative size = 360, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{\left (8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{256 c^{5/2} e^5}-\frac{\sqrt{a+b x^2+c x^4} \left (-2 c e x^2 \left (-4 c e (2 b d-3 a e)-3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-4 a e)+4 b c e^2 (2 b d-3 a e)+3 b^3 e^3+64 c^3 d^3\right )}{128 c^2 e^4}-\frac{d \left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^5}-\frac{\left (a+b x^2+c x^4\right )^{3/2} \left (-3 b e+8 c d-6 c e x^2\right )}{48 c e^2} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2),x]
[Out]
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Rubi in Sympy [A] time = 127.517, size = 388, normalized size = 1.08 \[ \frac{d \left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 e^{5}} + \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}} \left (\frac{3 b e}{2} - 4 c d + 3 c e x^{2}\right )}{24 c e^{2}} - \frac{\sqrt{a + b x^{2} + c x^{4}} \left (- 3 a b c e^{3} + 16 a c^{2} d e^{2} + \frac{3 b^{3} e^{3}}{4} + 2 b^{2} c d e^{2} - 20 b c^{2} d^{2} e + 16 c^{3} d^{3} + \frac{c e x^{2} \left (3 b^{2} e^{2} - 16 c^{2} d^{2} - 4 c e \left (3 a e - 2 b d\right )\right )}{2}\right )}{32 c^{2} e^{4}} + \frac{\left (- c d e \left (b e - 2 c d\right ) \left (4 a c e + b \left (3 b e - 8 c d\right )\right ) + \left (\frac{b^{2} e^{2}}{4} - 2 c^{2} d^{2} - c e \left (a e - b d\right )\right ) \left (3 b^{2} e^{2} - 16 c^{2} d^{2} - 4 c e \left (3 a e - 2 b d\right )\right )\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{64 c^{\frac{5}{2}} e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(c*x**4+b*x**2+a)**(3/2)/(e*x**2+d),x)
[Out]
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Mathematica [A] time = 0.473627, size = 376, normalized size = 1.04 \[ \frac{-2 \sqrt{c} e \sqrt{a+b x^2+c x^4} \left (-8 c^2 e \left (a e \left (15 e x^2-32 d\right )+b \left (30 d^2-14 d e x^2+9 e^2 x^4\right )\right )-6 b c e^2 \left (10 a e-4 b d+b e x^2\right )+9 b^3 e^3+16 c^3 \left (12 d^3-6 d^2 e x^2+4 d e^2 x^4-3 e^3 x^6\right )\right )+3 \left (8 b^2 c e^3 (b d-3 a e)-192 c^3 d^2 e (b d-a e)+48 c^2 e^2 (b d-a e)^2+3 b^4 e^4+128 c^4 d^4\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )+384 c^{5/2} d \left (e (a e-b d)+c d^2\right )^{3/2} \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )-384 c^{5/2} d \log \left (d+e x^2\right ) \left (e (a e-b d)+c d^2\right )^{3/2}}{768 c^{5/2} e^5} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(a + b*x^2 + c*x^4)^(3/2))/(d + e*x^2),x]
[Out]
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Maple [B] time = 0.017, size = 1696, normalized size = 4.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(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^3/(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^3/(e*x^2 + d),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(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^3/(e*x^2 + d),x, algorithm="giac")
[Out]