Optimal. Leaf size=117 \[ -\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+c d g+3 c e f)}{3 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2} \]
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Rubi [A] time = 0.455701, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{2 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-2 b e g+c d g+3 c e f)}{3 c^2 e^2 \sqrt{d+e x}}-\frac{2 g \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}{3 c e^2} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[d + e*x]*(f + g*x))/Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 46.5025, size = 109, normalized size = 0.93 \[ - \frac{2 g \sqrt{d + e x} \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{3 c e^{2}} + \frac{2 \left (2 b e g - c d g - 3 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{3 c^{2} e^{2} \sqrt{d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(1/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0989573, size = 63, normalized size = 0.54 \[ -\frac{2 \sqrt{(d+e x) (c (d-e x)-b e)} (c (2 d g+3 e f+e g x)-2 b e g)}{3 c^2 e^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[d + e*x]*(f + g*x))/Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2],x]
[Out]
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Maple [A] time = 0.008, size = 79, normalized size = 0.7 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -cegx+2\,beg-2\,cdg-3\,fce \right ) }{3\,{c}^{2}{e}^{2}}\sqrt{ex+d}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(1/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.730278, size = 149, normalized size = 1.27 \[ \frac{2 \,{\left (c e x - c d + b e\right )} f}{\sqrt{-c e x + c d - b e} c e} + \frac{2 \,{\left (c^{2} e^{2} x^{2} - 2 \, c^{2} d^{2} + 4 \, b c d e - 2 \, b^{2} e^{2} +{\left (c^{2} d e - b c e^{2}\right )} x\right )} g}{3 \, \sqrt{-c e x + c d - b e} c^{2} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)*(g*x + f)/sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272447, size = 236, normalized size = 2.02 \[ \frac{2 \,{\left (c^{2} e^{3} g x^{3} +{\left (3 \, c^{2} e^{3} f +{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g\right )} x^{2} - 3 \,{\left (c^{2} d^{2} e - b c d e^{2}\right )} f - 2 \,{\left (c^{2} d^{3} - 2 \, b c d^{2} e + b^{2} d e^{2}\right )} g +{\left (3 \, b c e^{3} f -{\left (c^{2} d^{2} e - 3 \, b c d e^{2} + 2 \, b^{2} e^{3}\right )} g\right )} x\right )}}{3 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c^{2} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)*(g*x + f)/sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d + e x} \left (f + g x\right )}{\sqrt{- \left (d + e x\right ) \left (b e - c d + c e x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(1/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 22.8779, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)*(g*x + f)/sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e),x, algorithm="giac")
[Out]