Optimal. Leaf size=200 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^2 (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-4 c d g+2 c e f)}{e^2 (2 c d-b e)}-\frac{(b e g-4 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 \sqrt{c} e^2} \]
[Out]
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Rubi [A] time = 0.728305, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 (e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{e^2 (d+e x)^2 (2 c d-b e)}-\frac{\sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (b e g-4 c d g+2 c e f)}{e^2 (2 c d-b e)}-\frac{(b e g-4 c d g+2 c e f) \tan ^{-1}\left (\frac{e (b+2 c x)}{2 \sqrt{c} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{2 \sqrt{c} e^2} \]
Antiderivative was successfully verified.
[In] Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 68.3405, size = 189, normalized size = 0.94 \[ \frac{2 \left (\frac{b e g}{2} - 2 c d g + c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{e^{2} \left (b e - 2 c d\right )} - \frac{2 \left (d g - e f\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}}{e^{2} \left (d + e x\right )^{2} \left (b e - 2 c d\right )} - \frac{2 \left (\frac{b e g}{4} - c d g + \frac{c e f}{2}\right ) \operatorname{atan}{\left (- \frac{e \left (- b - 2 c x\right )}{2 \sqrt{c} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \right )}}{\sqrt{c} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**2,x)
[Out]
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Mathematica [C] time = 0.39099, size = 147, normalized size = 0.74 \[ \frac{\sqrt{(d+e x) (c (d-e x)-b e)} \left (-\frac{i (b e g-4 c d g+2 c e f) \log \left (2 \sqrt{d+e x} \sqrt{c (d-e x)-b e}-\frac{i e (b+2 c x)}{\sqrt{c}}\right )}{\sqrt{c} \sqrt{d+e x} \sqrt{c (d-e x)-b e}}+\frac{4 (d g-e f)}{d+e x}+2 g\right )}{2 e^2} \]
Antiderivative was successfully verified.
[In] Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^2,x]
[Out]
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Maple [B] time = 0.021, size = 880, normalized size = 4.4 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^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(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.797548, size = 1, normalized size = 0. \[ \left [\frac{4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (e g x - 2 \, e f + 3 \, d g\right )} \sqrt{-c} +{\left (2 \, c d e f -{\left (4 \, c d^{2} - b d e\right )} g +{\left (2 \, c e^{2} f -{\left (4 \, c d e - b e^{2}\right )} g\right )} x\right )} \log \left (-4 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (2 \, c^{2} e x + b c e\right )} +{\left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2}\right )} \sqrt{-c}\right )}{4 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{-c}}, \frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (e g x - 2 \, e f + 3 \, d g\right )} \sqrt{c} -{\left (2 \, c d e f -{\left (4 \, c d^{2} - b d e\right )} g +{\left (2 \, c e^{2} f -{\left (4 \, c d e - b e^{2}\right )} g\right )} x\right )} \arctan \left (\frac{2 \, c e x + b e}{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{c}}\right )}{2 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**2,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(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^2,x, algorithm="giac")
[Out]