Optimal. Leaf size=110 \[ \frac{\left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{5/2}}+\frac{3 e \sqrt{b x+c x^2} (2 c d-b e)}{4 c^2}+\frac{e \sqrt{b x+c x^2} (d+e x)}{2 c} \]
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Rubi [A] time = 0.206391, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{\left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{5/2}}+\frac{3 e \sqrt{b x+c x^2} (2 c d-b e)}{4 c^2}+\frac{e \sqrt{b x+c x^2} (d+e x)}{2 c} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2/Sqrt[b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 22.9823, size = 102, normalized size = 0.93 \[ \frac{e \left (d + e x\right ) \sqrt{b x + c x^{2}}}{2 c} - \frac{3 e \left (b e - 2 c d\right ) \sqrt{b x + c x^{2}}}{4 c^{2}} + \frac{\left (3 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4 c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2/(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.147584, size = 110, normalized size = 1. \[ \frac{\sqrt{x} \sqrt{b+c x} \left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )+\sqrt{c} e x (b+c x) (-3 b e+8 c d+2 c e x)}{4 c^{5/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2/Sqrt[b*x + c*x^2],x]
[Out]
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Maple [A] time = 0.011, size = 158, normalized size = 1.4 \[{{d}^{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{2}x}{2\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,{e}^{2}b}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{2}{e}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+2\,{\frac{de\sqrt{c{x}^{2}+bx}}{c}}-{bde\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((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((e*x + d)^2/sqrt(c*x^2 + b*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.249755, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, c e^{2} x + 8 \, c d e - 3 \, b e^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{c} +{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{8 \, c^{\frac{5}{2}}}, \frac{{\left (2 \, c e^{2} x + 8 \, c d e - 3 \, b e^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{-c} +{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{4 \, \sqrt{-c} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/sqrt(c*x^2 + b*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{2}}{\sqrt{x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2/(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.234101, size = 131, normalized size = 1.19 \[ \frac{1}{4} \, \sqrt{c x^{2} + b x}{\left (\frac{2 \, x e^{2}}{c} + \frac{8 \, c d e - 3 \, b e^{2}}{c^{2}}\right )} - \frac{{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/sqrt(c*x^2 + b*x),x, algorithm="giac")
[Out]