Optimal. Leaf size=219 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c e (a e g-b d g+b e f)+b^2 e^2 g+8 c^2 d (e f-d g)\right )}{8 c^{3/2} e^3}+\frac{(e f-d g) \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^3}+\frac{\sqrt{a+b x+c x^2} (b e g-4 c d g+4 c e f+2 c e g x)}{4 c e^2} \]
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Rubi [A] time = 0.712034, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c e (a e g-b d g+b e f)+b^2 e^2 g+8 c^2 d (e f-d g)\right )}{8 c^{3/2} e^3}+\frac{(e f-d g) \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^3}+\frac{\sqrt{a+b x+c x^2} (b e g-4 c d g+4 c e f+2 c e g x)}{4 c e^2} \]
Antiderivative was successfully verified.
[In] Int[((f + g*x)*Sqrt[a + b*x + c*x^2])/(d + e*x),x]
[Out]
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Rubi in Sympy [A] time = 101.168, size = 211, normalized size = 0.96 \[ \frac{\left (d g - e f\right ) \sqrt{a e^{2} - b d e + c d^{2}} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{e^{3}} + \frac{\sqrt{a + b x + c x^{2}} \left (2 c e f + c e g x + \frac{g \left (b e - 4 c d\right )}{2}\right )}{2 c e^{2}} - \frac{\left (- 4 c e f \left (b e - 2 c d\right ) + g \left (b^{2} e^{2} - 8 c^{2} d^{2} - 4 c e \left (a e - b d\right )\right )\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 c^{\frac{3}{2}} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)*(c*x**2+b*x+a)**(1/2)/(e*x+d),x)
[Out]
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Mathematica [A] time = 0.849658, size = 253, normalized size = 1.16 \[ \frac{\log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (4 c e (a e g-b d g+b e f)-b^2 e^2 g+8 c^2 d (d g-e f)\right )+8 c^{3/2} (e f-d g) \log (d+e x) \sqrt{e (a e-b d)+c d^2}+2 \sqrt{c} \left (4 c (d g-e f) \sqrt{e (a e-b d)+c d^2} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )+e \sqrt{a+x (b+c x)} (b e g+2 c (-2 d g+2 e f+e g x))\right )}{8 c^{3/2} e^3} \]
Antiderivative was successfully verified.
[In] Integrate[((f + g*x)*Sqrt[a + b*x + c*x^2])/(d + e*x),x]
[Out]
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Maple [B] time = 0.012, size = 1559, normalized size = 7.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)*(c*x^2+b*x+a)^(1/2)/(e*x+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(sqrt(c*x^2 + b*x + a)*(g*x + f)/(e*x + 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(sqrt(c*x^2 + b*x + a)*(g*x + f)/(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (f + g x\right ) \sqrt{a + b x + c x^{2}}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x+f)*(c*x**2+b*x+a)**(1/2)/(e*x+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(sqrt(c*x^2 + b*x + a)*(g*x + f)/(e*x + d),x, algorithm="giac")
[Out]