Optimal. Leaf size=113 \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c} e}-\frac{(B d-A e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{\sqrt{d} e \sqrt{c d-b e}} \]
[Out]
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Rubi [A] time = 0.233531, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 B \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c} e}-\frac{(B d-A e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{\sqrt{d} e \sqrt{c d-b e}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((d + e*x)*Sqrt[b*x + c*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 27.6356, size = 97, normalized size = 0.86 \[ \frac{2 B \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{\sqrt{c} e} + \frac{\left (A e - B d\right ) \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{\sqrt{d} e \sqrt{b e - c d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(e*x+d)/(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.164296, size = 144, normalized size = 1.27 \[ \frac{2 \sqrt{x} \sqrt{b+c x} \left (B \sqrt{d} \sqrt{b e-c d} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )-\sqrt{c} (B d-A e) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )\right )}{\sqrt{c} \sqrt{d} e \sqrt{x (b+c x)} \sqrt{b e-c d}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((d + e*x)*Sqrt[b*x + c*x^2]),x]
[Out]
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Maple [B] time = 0.01, size = 298, normalized size = 2.6 \[{\frac{B}{e}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}}-{\frac{A}{e}\ln \left ({1 \left ( -2\,{\frac{d \left ( be-cd \right ) }{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}}}+{\frac{Bd}{{e}^{2}}\ln \left ({1 \left ( -2\,{\frac{d \left ( be-cd \right ) }{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{-{\frac{d \left ( be-cd \right ) }{{e}^{2}}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)/(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)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.358037, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{c d^{2} - b d e} B \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) -{\left (B d - A e\right )} \sqrt{c} \log \left (\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} + \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right )}{\sqrt{c d^{2} - b d e} \sqrt{c} e}, \frac{2 \,{\left (B d - A e\right )} \sqrt{c} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + \sqrt{-c d^{2} + b d e} B \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right )}{\sqrt{-c d^{2} + b d e} \sqrt{c} e}, \frac{2 \, \sqrt{c d^{2} - b d e} B \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (B d - A e\right )} \sqrt{-c} \log \left (\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} + \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right )}{\sqrt{c d^{2} - b d e} \sqrt{-c} e}, \frac{2 \,{\left ({\left (B d - A e\right )} \sqrt{-c} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right ) + \sqrt{-c d^{2} + b d e} B \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )\right )}}{\sqrt{-c d^{2} + b d e} \sqrt{-c} e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\sqrt{x \left (b + c x\right )} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(e*x+d)/(c*x**2+b*x)**(1/2),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((B*x + A)/(sqrt(c*x^2 + b*x)*(e*x + d)),x, algorithm="giac")
[Out]