Optimal. Leaf size=122 \[ \frac{2 \left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{5/2} \sqrt{e}}+\frac{4 e (a+b x)^{3/2} \sqrt{d+e x}}{b^2}+\frac{2 \sqrt{a+b x} \sqrt{d+e x} (7 b d-5 a e)}{b^2} \]
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Rubi [A] time = 0.287447, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 \left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{b^{5/2} \sqrt{e}}+\frac{4 e (a+b x)^{3/2} \sqrt{d+e x}}{b^2}+\frac{2 \sqrt{a+b x} \sqrt{d+e x} (7 b d-5 a e)}{b^2} \]
Antiderivative was successfully verified.
[In] Int[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*Sqrt[d + e*x]),x]
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Rubi in Sympy [A] time = 43.0614, size = 184, normalized size = 1.51 \[ \frac{20 d \sqrt{a + b x} \sqrt{d + e x}}{b} + \frac{4 e x \sqrt{a + b x} \sqrt{d + e x}}{b} - \frac{6 \sqrt{a + b x} \sqrt{d + e x} \left (a e + b d\right )}{b^{2}} - \frac{10 d \left (2 a e - b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{b^{\frac{3}{2}} \sqrt{e}} - \frac{8 \left (a b d e - \frac{3 \left (a e + b d\right )^{2}}{4}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{b^{\frac{5}{2}} \sqrt{e}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((8*e**2*x**2+20*d*e*x+15*d**2)/(e*x+d)**(1/2)/(b*x+a)**(1/2),x)
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Mathematica [A] time = 0.108814, size = 115, normalized size = 0.94 \[ \frac{\left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right ) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{b^{5/2} \sqrt{e}}+\frac{2 \sqrt{a+b x} \sqrt{d+e x} (-3 a e+7 b d+2 b e x)}{b^2} \]
Antiderivative was successfully verified.
[In] Integrate[(15*d^2 + 20*d*e*x + 8*e^2*x^2)/(Sqrt[a + b*x]*Sqrt[d + e*x]),x]
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Maple [B] time = 0.035, size = 247, normalized size = 2. \[{\frac{1}{{b}^{2}} \left ( 3\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}{e}^{2}-8\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) abde+8\,\ln \left ( 1/2\,{\frac{2\,bex+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{2}{d}^{2}+4\,e\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }xb\sqrt{be}-6\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}ae+14\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}bd \right ) \sqrt{ex+d}\sqrt{bx+a}{\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((8*e^2*x^2+20*d*e*x+15*d^2)/(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((8*e^2*x^2 + 20*d*e*x + 15*d^2)/(sqrt(b*x + a)*sqrt(e*x + d)),x, algorithm="maxima")
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Fricas [A] time = 0.317592, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, b e x + 7 \, b d - 3 \, a e\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\right )} \log \left (4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{2 \, \sqrt{b e} b^{2}}, \frac{2 \,{\left (2 \, b e x + 7 \, b d - 3 \, a e\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{\sqrt{-b e} b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((8*e^2*x^2 + 20*d*e*x + 15*d^2)/(sqrt(b*x + a)*sqrt(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{15 d^{2} + 20 d e x + 8 e^{2} x^{2}}{\sqrt{a + b x} \sqrt{d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((8*e**2*x**2+20*d*e*x+15*d**2)/(e*x+d)**(1/2)/(b*x+a)**(1/2),x)
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GIAC/XCAS [A] time = 0.285379, size = 196, normalized size = 1.61 \[ \frac{2 \,{\left (\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )} e}{b^{3}} + \frac{{\left (7 \, b^{6} d e^{2} - 5 \, a b^{5} e^{3}\right )} e^{\left (-2\right )}}{b^{8}}\right )} - \frac{{\left (8 \, b^{2} d^{2} - 8 \, a b d e + 3 \, a^{2} e^{2}\right )} e^{\left (-\frac{1}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac{5}{2}}}\right )} b}{{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((8*e^2*x^2 + 20*d*e*x + 15*d^2)/(sqrt(b*x + a)*sqrt(e*x + d)),x, algorithm="giac")
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