Optimal. Leaf size=112 \[ -\frac{2 \left (a e^2+c d^2\right )}{e^2 \sqrt{d+e x} (e f-d g)}-\frac{2 \left (a g^2+c f^2\right ) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right )}{g^{3/2} (e f-d g)^{3/2}}+\frac{2 c \sqrt{d+e x}}{e^2 g} \]
[Out]
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Rubi [A] time = 0.456118, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2 \left (a e^2+c d^2\right )}{e^2 \sqrt{d+e x} (e f-d g)}-\frac{2 \left (a g^2+c f^2\right ) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e f-d g}}\right )}{g^{3/2} (e f-d g)^{3/2}}+\frac{2 c \sqrt{d+e x}}{e^2 g} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)/((d + e*x)^(3/2)*(f + g*x)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{2 \left (a g^{2} + c f^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{g} \sqrt{d + e x}}{\sqrt{d g - e f}} \right )}}{g^{\frac{3}{2}} \left (d g - e f\right )^{\frac{3}{2}}} + \frac{2 \left (a e^{2} + c d^{2}\right )}{e^{2} \sqrt{d + e x} \left (d g - e f\right )} + \frac{2 \int ^{\sqrt{d + e x}} c\, dx}{e^{2} g} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)/(e*x+d)**(3/2)/(g*x+f),x)
[Out]
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Mathematica [A] time = 0.333416, size = 108, normalized size = 0.96 \[ \frac{2 \sqrt{d+e x} \left (\frac{a e^2+c d^2}{(d+e x) (d g-e f)}+\frac{c}{g}\right )}{e^2}-\frac{2 \left (a g^2+c f^2\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{d g-e f}}\right )}{g^{3/2} (d g-e f)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)/((d + e*x)^(3/2)*(f + g*x)),x]
[Out]
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Maple [A] time = 0.022, size = 114, normalized size = 1. \[ 2\,{\frac{1}{{e}^{2}} \left ({\frac{c\sqrt{ex+d}}{g}}-{\frac{{e}^{2} \left ( a{g}^{2}+c{f}^{2} \right ) }{ \left ( dg-ef \right ) g\sqrt{ \left ( dg-ef \right ) g}}{\it Artanh} \left ({\frac{g\sqrt{ex+d}}{\sqrt{ \left ( dg-ef \right ) g}}} \right ) }-{\frac{-a{e}^{2}-c{d}^{2}}{ \left ( dg-ef \right ) \sqrt{ex+d}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)/(e*x+d)^(3/2)/(g*x+f),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((c*x^2 + a)/((e*x + d)^(3/2)*(g*x + f)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.290642, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (c e^{2} f^{2} + a e^{2} g^{2}\right )} \sqrt{e x + d} \log \left (\frac{\sqrt{-e f g + d g^{2}}{\left (e g x - e f + 2 \, d g\right )} + 2 \,{\left (e f g - d g^{2}\right )} \sqrt{e x + d}}{g x + f}\right ) - 2 \,{\left (c d e f -{\left (2 \, c d^{2} + a e^{2}\right )} g +{\left (c e^{2} f - c d e g\right )} x\right )} \sqrt{-e f g + d g^{2}}}{{\left (e^{3} f g - d e^{2} g^{2}\right )} \sqrt{-e f g + d g^{2}} \sqrt{e x + d}}, -\frac{2 \,{\left ({\left (c e^{2} f^{2} + a e^{2} g^{2}\right )} \sqrt{e x + d} \arctan \left (-\frac{e f - d g}{\sqrt{e f g - d g^{2}} \sqrt{e x + d}}\right ) -{\left (c d e f -{\left (2 \, c d^{2} + a e^{2}\right )} g +{\left (c e^{2} f - c d e g\right )} x\right )} \sqrt{e f g - d g^{2}}\right )}}{{\left (e^{3} f g - d e^{2} g^{2}\right )} \sqrt{e f g - d g^{2}} \sqrt{e x + d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/((e*x + d)^(3/2)*(g*x + f)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{a + c x^{2}}{\left (d + e x\right )^{\frac{3}{2}} \left (f + g x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)/(e*x+d)**(3/2)/(g*x+f),x)
[Out]
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GIAC/XCAS [A] time = 0.288194, size = 157, normalized size = 1.4 \[ \frac{2 \, \sqrt{x e + d} c e^{\left (-2\right )}}{g} + \frac{2 \,{\left (c f^{2} + a g^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} g}{\sqrt{-d g^{2} + f g e}}\right )}{{\left (d g^{2} - f g e\right )} \sqrt{-d g^{2} + f g e}} + \frac{2 \,{\left (c d^{2} + a e^{2}\right )}}{{\left (d g e^{2} - f e^{3}\right )} \sqrt{x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/((e*x + d)^(3/2)*(g*x + f)),x, algorithm="giac")
[Out]