Optimal. Leaf size=350 \[ -\frac{a^{3/2} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 d}+\frac{\left (-12 c d e (b d-a e)+b e^2 (3 b d-4 a e)+8 c^2 d^3\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{16 \sqrt{c} d e^3}-\frac{\sqrt{a+b x^2+c x^4} \left (-e (5 b d-4 a e)+4 c d^2-2 c d e x^2\right )}{8 d e^2}-\frac{\left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d e^3}+\frac{a \sqrt{a+b x^2+c x^4}}{2 d}+\frac{a b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 \sqrt{c} d} \]
[Out]
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Rubi [A] time = 1.27014, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276 \[ -\frac{a^{3/2} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 d}+\frac{\left (-12 c d e (b d-a e)+b e^2 (3 b d-4 a e)+8 c^2 d^3\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{16 \sqrt{c} d e^3}-\frac{\sqrt{a+b x^2+c x^4} \left (-e (5 b d-4 a e)+4 c d^2-2 c d e x^2\right )}{8 d e^2}-\frac{\left (a e^2-b d e+c d^2\right )^{3/2} \tanh ^{-1}\left (\frac{-2 a e+x^2 (2 c d-b e)+b d}{2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}}\right )}{2 d e^3}+\frac{a \sqrt{a+b x^2+c x^4}}{2 d}+\frac{a b \tanh ^{-1}\left (\frac{b+2 c x^2}{2 \sqrt{c} \sqrt{a+b x^2+c x^4}}\right )}{4 \sqrt{c} d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2 + c*x^4)^(3/2)/(x*(d + e*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 130.837, size = 332, normalized size = 0.95 \[ - \frac{a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{2 d} + \frac{a b \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{4 \sqrt{c} d} + \frac{a \sqrt{a + b x^{2} + c x^{4}}}{2 d} - \frac{\sqrt{a + b x^{2} + c x^{4}} \left (2 a e^{2} - \frac{5 b d e}{2} + 2 c d^{2} - c d e x^{2}\right )}{4 d e^{2}} + \frac{\left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{2 a e - b d + x^{2} \left (b e - 2 c d\right )}{2 \sqrt{a + b x^{2} + c x^{4}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 d e^{3}} + \frac{\left (- 4 a b e^{3} + 12 a c d e^{2} + 3 b^{2} d e^{2} - 12 b c d^{2} e + 8 c^{2} d^{3}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{2 \sqrt{c} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{16 \sqrt{c} d e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+b*x**2+a)**(3/2)/x/(e*x**2+d),x)
[Out]
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Mathematica [A] time = 2.25722, size = 292, normalized size = 0.83 \[ \frac{-\frac{8 a^{3/2} e^3 \log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )}{d}+\frac{8 a^{3/2} e^3 \log \left (x^2\right )}{d}+\frac{\left (12 c e (a e-b d)+3 b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^2+c x^4}+b+2 c x^2\right )}{\sqrt{c}}+\frac{8 \left (e (a e-b d)+c d^2\right )^{3/2} \log \left (2 \sqrt{a+b x^2+c x^4} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x^2-2 c d x^2\right )}{d}-\frac{8 \log \left (d+e x^2\right ) \left (e (a e-b d)+c d^2\right )^{3/2}}{d}+2 e \sqrt{a+b x^2+c x^4} \left (5 b e-4 c d+2 c e x^2\right )}{16 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2 + c*x^4)^(3/2)/(x*(d + e*x^2)),x]
[Out]
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Maple [B] time = 0.017, size = 1270, normalized size = 3.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+b*x^2+a)^(3/2)/x/(e*x^2+d),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{{\left (e x^{2} + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^(3/2)/((e*x^2 + d)*x),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)/((e*x^2 + d)*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{x \left (d + e x^{2}\right )}\, dx \]
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)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{{\left (e x^{2} + d\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2 + a)^(3/2)/((e*x^2 + d)*x),x, algorithm="giac")
[Out]