Optimal. Leaf size=200 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (-b^2 e^2-4 b c d e+8 c^2 d^2\right )\right )}{4 c^{3/2} e^3}-\frac{\sqrt{d} (B d-A e) \sqrt{c d-b e} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^3}-\frac{\sqrt{b x+c x^2} (-4 A c e-b B e+4 B c d-2 B c e x)}{4 c e^2} \]
[Out]
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Rubi [A] time = 0.628787, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) \left (4 A c e (2 c d-b e)-B \left (-b^2 e^2-4 b c d e+8 c^2 d^2\right )\right )}{4 c^{3/2} e^3}-\frac{\sqrt{d} (B d-A e) \sqrt{c d-b e} \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{e^3}-\frac{\sqrt{b x+c x^2} (-4 A c e-b B e+4 B c d-2 B c e x)}{4 c e^2} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x),x]
[Out]
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Rubi in Sympy [A] time = 67.9328, size = 184, normalized size = 0.92 \[ - \frac{\sqrt{d} \left (A e - B d\right ) \sqrt{b e - c d} \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{e^{3}} + \frac{\sqrt{b x + c x^{2}} \left (2 A c e + B c e x + \frac{B \left (b e - 4 c d\right )}{2}\right )}{2 c e^{2}} - \frac{\left (- 4 A c e \left (b e - 2 c d\right ) + B \left (b^{2} e^{2} + 4 b c d e - 8 c^{2} d^{2}\right )\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4 c^{\frac{3}{2}} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x)**(1/2)/(e*x+d),x)
[Out]
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Mathematica [A] time = 0.684815, size = 207, normalized size = 1.03 \[ \frac{\sqrt{x (b+c x)} \left (\frac{\log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) \left (4 A c e (b e-2 c d)+B \left (-b^2 e^2-4 b c d e+8 c^2 d^2\right )\right )}{c^{3/2} \sqrt{b+c x}}+\frac{e \sqrt{x} (4 A c e+b B e-4 B c d)}{c}+\frac{8 \sqrt{d} (B d-A e) \sqrt{b e-c d} \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{b+c x}}+2 B e^2 x^{3/2}\right )}{4 e^3 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/(d + e*x),x]
[Out]
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Maple [B] time = 0.012, size = 1069, normalized size = 5.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x)^(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)*(B*x + A)/(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.05029, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x)*(B*x + A)/(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )} \left (A + B x\right )}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x)**(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)*(B*x + A)/(e*x + d),x, algorithm="giac")
[Out]