Optimal. Leaf size=141 \[ \frac{\sqrt{a+b x+c x^2} (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{e \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]
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Rubi [A] time = 0.291157, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{\sqrt{a+b x+c x^2} (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{e \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(b + 2*c*x)/((d + e*x)^2*Sqrt[a + b*x + c*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 35.7617, size = 126, normalized size = 0.89 \[ \frac{e \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 \left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}}} - \frac{\left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{\left (d + e x\right ) \left (a e^{2} - b d e + c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*x+b)/(e*x+d)**2/(c*x**2+b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.506476, size = 176, normalized size = 1.25 \[ \frac{\frac{e \left (b^2-4 a c\right ) \log (d+e x)}{\sqrt{e (a e-b d)+c d^2}}+\frac{e \left (4 a c-b^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{e (a e-b d)+c d^2}}+\frac{2 \sqrt{a+x (b+c x)} (b e-2 c d)}{d+e x}}{2 \left (e (b d-a e)-c d^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(b + 2*c*x)/((d + e*x)^2*Sqrt[a + b*x + c*x^2]),x]
[Out]
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Maple [B] time = 0.015, size = 860, normalized size = 6.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)/(e*x+d)^2/(c*x^2+b*x+a)^(1/2),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((2*c*x + b)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.546653, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c d^{2} - b d e + a e^{2}} \sqrt{c x^{2} + b x + a}{\left (2 \, c d - b e\right )} -{\left ({\left (b^{2} - 4 \, a c\right )} e^{2} x +{\left (b^{2} - 4 \, a c\right )} d e\right )} \log \left (\frac{{\left (8 \, a b d e - 8 \, a^{2} e^{2} -{\left (b^{2} + 4 \, a c\right )} d^{2} -{\left (8 \, c^{2} d^{2} - 8 \, b c d e +{\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{2} - 2 \,{\left (4 \, b c d^{2} + 4 \, a b e^{2} -{\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x\right )} \sqrt{c d^{2} - b d e + a e^{2}} - 4 \,{\left (b c d^{3} + 3 \, a b d e^{2} - 2 \, a^{2} e^{3} -{\left (b^{2} + 2 \, a c\right )} d^{2} e +{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{4 \,{\left (c d^{3} - b d^{2} e + a d e^{2} +{\left (c d^{2} e - b d e^{2} + a e^{3}\right )} x\right )} \sqrt{c d^{2} - b d e + a e^{2}}}, \frac{2 \, \sqrt{-c d^{2} + b d e - a e^{2}} \sqrt{c x^{2} + b x + a}{\left (2 \, c d - b e\right )} +{\left ({\left (b^{2} - 4 \, a c\right )} e^{2} x +{\left (b^{2} - 4 \, a c\right )} d e\right )} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e - a e^{2}}{\left (b d - 2 \, a e +{\left (2 \, c d - b e\right )} x\right )}}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{c x^{2} + b x + a}}\right )}{2 \,{\left (c d^{3} - b d^{2} e + a d e^{2} +{\left (c d^{2} e - b d e^{2} + a e^{3}\right )} x\right )} \sqrt{-c d^{2} + b d e - a e^{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x + b)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{b + 2 c x}{\left (d + e x\right )^{2} \sqrt{a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x+b)/(e*x+d)**2/(c*x**2+b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, c x + b}{\sqrt{c x^{2} + b x + a}{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x + b)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^2),x, algorithm="giac")
[Out]