Optimal. Leaf size=128 \[ \frac{\sqrt{b x+c x^2} (B d-A e)}{d (d+e x) (c d-b e)}-\frac{(A b e-2 A c d+b B d) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}} \]
[Out]
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Rubi [A] time = 0.241654, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{\sqrt{b x+c x^2} (B d-A e)}{d (d+e x) (c d-b e)}-\frac{(A b e-2 A c d+b B d) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((d + e*x)^2*Sqrt[b*x + c*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 27.1757, size = 105, normalized size = 0.82 \[ \frac{\left (A e - B d\right ) \sqrt{b x + c x^{2}}}{d \left (d + e x\right ) \left (b e - c d\right )} + \frac{\left (- A c d + \frac{b \left (A e + B d\right )}{2}\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{d^{\frac{3}{2}} \left (b e - c d\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(e*x+d)**2/(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.254098, size = 134, normalized size = 1.05 \[ \frac{\sqrt{x} \left (\frac{\sqrt{d} \sqrt{x} (b+c x) (B d-A e)}{d+e x}-\frac{\sqrt{b+c x} (A b e-2 A c d+b B d) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b e-c d}}\right )}{d^{3/2} \sqrt{x (b+c x)} (c d-b e)} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((d + e*x)^2*Sqrt[b*x + c*x^2]),x]
[Out]
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Maple [B] time = 0.015, size = 849, normalized size = 6.6 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)^2/(c*x^2+b*x)^(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((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.319963, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{c d^{2} - b d e} \sqrt{c x^{2} + b x}{\left (B d - A e\right )} -{\left (A b d e +{\left (B b - 2 \, A c\right )} d^{2} +{\left (A b e^{2} +{\left (B b - 2 \, A c\right )} d e\right )} x\right )} \log \left (\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} + \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right )}{2 \,{\left (c d^{3} - b d^{2} e +{\left (c d^{2} e - b d e^{2}\right )} x\right )} \sqrt{c d^{2} - b d e}}, \frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}{\left (B d - A e\right )} +{\left (A b d e +{\left (B b - 2 \, A c\right )} d^{2} +{\left (A b e^{2} +{\left (B b - 2 \, A c\right )} d e\right )} x\right )} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right )}{{\left (c d^{3} - b d^{2} e +{\left (c d^{2} e - b d e^{2}\right )} x\right )} \sqrt{-c d^{2} + b d e}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\sqrt{x \left (b + c x\right )} \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(e*x+d)**2/(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{\sqrt{c x^{2} + b x}{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)^2),x, algorithm="giac")
[Out]