Optimal. Leaf size=552 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (-a b e-a c d+b^2 d\right )}{a^3 \sqrt{d}}+\frac{\sqrt{c} \left (b^2 \left (d \sqrt{b^2-4 a c}-a e\right )-a b \left (e \sqrt{b^2-4 a c}+3 c d\right )-a c \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\sqrt{c} \left (-b^2 \left (d \sqrt{b^2-4 a c}+a e\right )-a b \left (3 c d-e \sqrt{b^2-4 a c}\right )+a c \left (d \sqrt{b^2-4 a c}+2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{e (b d-a e) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 a^2 d^{3/2}}+\frac{\sqrt{d+e x^2} (b d-a e)}{2 a^2 d x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{8 a d^{3/2}}+\frac{3 e \sqrt{d+e x^2}}{8 a d x^2}-\frac{\sqrt{d+e x^2}}{4 a x^4} \]
[Out]
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Rubi [A] time = 9.29025, antiderivative size = 552, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right ) \left (-a b e-a c d+b^2 d\right )}{a^3 \sqrt{d}}+\frac{\sqrt{c} \left (b^2 \left (d \sqrt{b^2-4 a c}-a e\right )-a b \left (e \sqrt{b^2-4 a c}+3 c d\right )-a c \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}\right )}{\sqrt{2} a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{\sqrt{c} \left (-b^2 \left (d \sqrt{b^2-4 a c}+a e\right )-a b \left (3 c d-e \sqrt{b^2-4 a c}\right )+a c \left (d \sqrt{b^2-4 a c}+2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2} a^3 \sqrt{b^2-4 a c} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{e (b d-a e) \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{2 a^2 d^{3/2}}+\frac{\sqrt{d+e x^2} (b d-a e)}{2 a^2 d x^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d}}\right )}{8 a d^{3/2}}+\frac{3 e \sqrt{d+e x^2}}{8 a d x^2}-\frac{\sqrt{d+e x^2}}{4 a x^4} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x^2]/(x^5*(a + b*x^2 + c*x^4)),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x**2+d)**(1/2)/x**5/(c*x**4+b*x**2+a),x)
[Out]
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Mathematica [A] time = 2.95803, size = 468, normalized size = 0.85 \[ \frac{\frac{\log \left (\sqrt{d} \sqrt{d+e x^2}+d\right ) \left (4 a b d e+a \left (a e^2+8 c d^2\right )-8 b^2 d^2\right )}{d^{3/2}}-\frac{\log (x) \left (4 a b d e+a \left (a e^2+8 c d^2\right )-8 b^2 d^2\right )}{d^{3/2}}-\frac{4 \sqrt{2} \left (\frac{c \left (b^2 \left (a e-d \sqrt{b^2-4 a c}\right )+a b \left (e \sqrt{b^2-4 a c}+3 c d\right )+a c \left (d \sqrt{b^2-4 a c}-2 a e\right )+b^3 (-d)\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{e \sqrt{b^2-4 a c}-b e+2 c d}}\right )}{\sqrt{e \left (\sqrt{b^2-4 a c}-b\right )+2 c d}}+\frac{c \left (-b^2 \left (d \sqrt{b^2-4 a c}+a e\right )+a b \left (e \sqrt{b^2-4 a c}-3 c d\right )+a c \left (d \sqrt{b^2-4 a c}+2 a e\right )+b^3 d\right ) \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{d+e x^2}}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}\right )}{\sqrt{c} \sqrt{b^2-4 a c}}+\frac{a \sqrt{d+e x^2} \left (4 b d x^2-a \left (2 d+e x^2\right )\right )}{d x^4}}{8 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x^2]/(x^5*(a + b*x^2 + c*x^4)),x]
[Out]
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Maple [C] time = 0.047, size = 655, normalized size = 1.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x^2+d)^(1/2)/x^5/(c*x^4+b*x^2+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x^{2} + d}}{{\left (c x^{4} + b x^{2} + a\right )} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x^2 + d)/((c*x^4 + b*x^2 + a)*x^5),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(sqrt(e*x^2 + d)/((c*x^4 + b*x^2 + a)*x^5),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((e*x**2+d)**(1/2)/x**5/(c*x**4+b*x**2+a),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x^2 + d)/((c*x^4 + b*x^2 + a)*x^5),x, algorithm="giac")
[Out]