Optimal. Leaf size=85 \[ \frac{2 B e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}-\frac{2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt{b x+c x^2}} \]
[Out]
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Rubi [A] time = 0.127252, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{2 B e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}}-\frac{2 \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x))/(b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 12.8314, size = 83, normalized size = 0.98 \[ \frac{2 B e \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{3}{2}}} - \frac{2 \left (A b c d + x \left (2 A c^{2} d + B b^{2} e - b c \left (A e + B d\right )\right )\right )}{b^{2} c \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)/(c*x**2+b*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.149608, size = 102, normalized size = 1.2 \[ \frac{2 \left (\sqrt{c} (A c (-b d+b e x-2 c d x)+b B x (c d-b e))+b^2 B e \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )\right )}{b^2 c^{3/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x))/(b*x + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.01, size = 113, normalized size = 1.3 \[ -2\,{\frac{Ad \left ( 2\,cx+b \right ) }{{b}^{2}\sqrt{c{x}^{2}+bx}}}+2\,{\frac{xAe}{b\sqrt{c{x}^{2}+bx}}}+2\,{\frac{Bxd}{b\sqrt{c{x}^{2}+bx}}}-2\,{\frac{Bex}{c\sqrt{c{x}^{2}+bx}}}+{Be\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)/(c*x^2+b*x)^(3/2),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)*(e*x + d)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.303426, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{c x^{2} + b x} B b^{2} e \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) - 2 \,{\left (A b c d -{\left ({\left (B b c - 2 \, A c^{2}\right )} d -{\left (B b^{2} - A b c\right )} e\right )} x\right )} \sqrt{c}}{\sqrt{c x^{2} + b x} b^{2} c^{\frac{3}{2}}}, \frac{2 \,{\left (\sqrt{c x^{2} + b x} B b^{2} e \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (A b c d -{\left ({\left (B b c - 2 \, A c^{2}\right )} d -{\left (B b^{2} - A b c\right )} e\right )} x\right )} \sqrt{-c}\right )}}{\sqrt{c x^{2} + b x} b^{2} \sqrt{-c} c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (d + e x\right )}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)/(c*x**2+b*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.318023, size = 128, normalized size = 1.51 \[ -\frac{B e{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{3}{2}}} - \frac{2 \,{\left (\frac{A d}{b} - \frac{{\left (B b c d - 2 \, A c^{2} d - B b^{2} e + A b c e\right )} x}{b^{2} c}\right )}}{\sqrt{c x^{2} + b x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)/(c*x^2 + b*x)^(3/2),x, algorithm="giac")
[Out]