Optimal. Leaf size=140 \[ -\frac{(b d-a e) (3 a B e-4 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{5/2} e^{3/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (3 a B e-4 A b e+b B d)}{4 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{3/2}}{2 b e} \]
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Rubi [A] time = 0.274312, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{(b d-a e) (3 a B e-4 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 b^{5/2} e^{3/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (3 a B e-4 A b e+b B d)}{4 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{3/2}}{2 b e} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[d + e*x])/Sqrt[a + b*x],x]
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Rubi in Sympy [A] time = 18.7734, size = 131, normalized size = 0.94 \[ \frac{B \sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}}}{2 b e} + \frac{\sqrt{a + b x} \sqrt{d + e x} \left (4 A b e - 3 B a e - B b d\right )}{4 b^{2} e} - \frac{\left (a e - b d\right ) \left (4 A b e - 3 B a e - B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{4 b^{\frac{5}{2}} e^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(1/2)/(b*x+a)**(1/2),x)
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Mathematica [A] time = 0.131295, size = 129, normalized size = 0.92 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} (-3 a B e+4 A b e+b B (d+2 e x))}{4 b^2 e}-\frac{(b d-a e) (3 a B e-4 A b e+b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{8 b^{5/2} e^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[d + e*x])/Sqrt[a + b*x],x]
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Maple [B] time = 0.025, size = 375, normalized size = 2.7 \[ -{\frac{1}{8\,{b}^{2}e}\sqrt{ex+d}\sqrt{bx+a} \left ( 4\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) aA{e}^{2}b-4\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{2}dAe-3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}{e}^{2}+2\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) aBdeb+\ln \left ({\frac{1}{2} \left ( 2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd \right ){\frac{1}{\sqrt{be}}}} \right ){b}^{2}{d}^{2}B-4\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }xeb\sqrt{be}-8\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }Aeb\sqrt{be}+6\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }ae\sqrt{be}-2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }Bdb\sqrt{be} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(1/2)/(b*x+a)^(1/2),x)
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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(e*x + d)/sqrt(b*x + a),x, algorithm="maxima")
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Fricas [A] time = 0.429015, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, B b e x + B b d -{\left (3 \, B a - 4 \, A b\right )} e\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} +{\left (B b^{2} d^{2} + 2 \,{\left (B a b - 2 \, A b^{2}\right )} d e -{\left (3 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} \log \left (-4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{16 \, \sqrt{b e} b^{2} e}, \frac{2 \,{\left (2 \, B b e x + B b d -{\left (3 \, B a - 4 \, A b\right )} e\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} -{\left (B b^{2} d^{2} + 2 \,{\left (B a b - 2 \, A b^{2}\right )} d e -{\left (3 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{8 \, \sqrt{-b e} b^{2} e}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x + d)/sqrt(b*x + a),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**(1/2)/(b*x+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.250857, size = 328, normalized size = 2.34 \[ -\frac{\frac{48 \,{\left (\frac{{\left (b^{2} d - a b e\right )} e^{\left (-\frac{1}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt{b}} - \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \sqrt{b x + a}\right )} A{\left | b \right |}}{b^{2}} - \frac{{\left (\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )} e^{\left (-2\right )}}{b^{4}} + \frac{{\left (b d e - 5 \, a e^{2}\right )} e^{\left (-4\right )}}{b^{4}}\right )} + \frac{{\left (b^{2} d^{2} + 2 \, a b d e - 3 \, a^{2} e^{2}\right )} e^{\left (-\frac{7}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac{7}{2}}}\right )} B{\left | b \right |}}{b^{3}}}{48 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(e*x + d)/sqrt(b*x + a),x, algorithm="giac")
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