Optimal. Leaf size=207 \[ \frac{\left (c d^2-a e^2\right ) \left (a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{5/2}}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 c d e (d+e x)}-\frac{1}{4} \left (\frac{a}{c d}+\frac{3 d}{e^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \]
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Rubi [A] time = 0.485473, antiderivative size = 207, 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 \[ \frac{\left (c d^2-a e^2\right ) \left (a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{5/2}}+\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 c d e (d+e x)}-\frac{1}{4} \left (\frac{a}{c d}+\frac{3 d}{e^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} \]
Antiderivative was successfully verified.
[In] Int[(x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d + e*x),x]
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Rubi in Sympy [A] time = 44.3452, size = 189, normalized size = 0.91 \[ - \left (\frac{a}{4 c d} + \frac{3 d}{4 e^{2}}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} + \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{2 c d e \left (d + e x\right )} - \frac{\left (a e^{2} - c d^{2}\right ) \left (a e^{2} + 3 c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{8 c^{\frac{3}{2}} d^{\frac{3}{2}} e^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d),x)
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Mathematica [A] time = 0.382126, size = 159, normalized size = 0.77 \[ \frac{1}{8} \sqrt{(d+e x) (a e+c d x)} \left (\frac{\left (c d^2-a e^2\right ) \left (a e^2+3 c d^2\right ) \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{3/2} d^{3/2} e^{5/2} \sqrt{d+e x} \sqrt{a e+c d x}}+\frac{2 a}{c d}-\frac{6 d}{e^2}+\frac{4 x}{e}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(x*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d + e*x),x]
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Maple [B] time = 0.013, size = 516, normalized size = 2.5 \[{\frac{x}{2\,e}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{a}{4\,cd}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{d}{4\,{e}^{2}}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}-{\frac{{e}^{2}{a}^{2}}{8\,cd}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{cde}}}}+\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{cde}}}}+{\frac{ad}{4}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{cde}}}}+\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{cde}}}}-{\frac{c{d}^{3}}{8\,{e}^{2}}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{cde}}}}+\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{cde}}}}-{\frac{d}{{e}^{2}}\sqrt{cde \left ( x+{\frac{d}{e}} \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( x+{\frac{d}{e}} \right ) }}-{\frac{ad}{2}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}-{\frac{c{d}^{2}}{2}}+ \left ( x+{\frac{d}{e}} \right ) cde \right ){\frac{1}{\sqrt{cde}}}}+\sqrt{cde \left ( x+{\frac{d}{e}} \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( x+{\frac{d}{e}} \right ) } \right ){\frac{1}{\sqrt{cde}}}}+{\frac{c{d}^{3}}{2\,{e}^{2}}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}-{\frac{c{d}^{2}}{2}}+ \left ( x+{\frac{d}{e}} \right ) cde \right ){\frac{1}{\sqrt{cde}}}}+\sqrt{cde \left ( x+{\frac{d}{e}} \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( x+{\frac{d}{e}} \right ) } \right ){\frac{1}{\sqrt{cde}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d),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(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*x/(e*x + d),x, algorithm="maxima")
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Fricas [A] time = 0.307921, size = 1, normalized size = 0. \[ \left [\frac{4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d e x - 3 \, c d^{2} + a e^{2}\right )} \sqrt{c d e} -{\left (3 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \log \left (-4 \,{\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e + a c d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} +{\left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{c d e}\right )}{16 \, \sqrt{c d e} c d e^{2}}, \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d e x - 3 \, c d^{2} + a e^{2}\right )} \sqrt{-c d e} +{\left (3 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - a^{2} e^{4}\right )} \arctan \left (\frac{{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-c d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d e}\right )}{8 \, \sqrt{-c d e} c d e^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*x/(e*x + d),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \sqrt{\left (d + e x\right ) \left (a e + c d x\right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d),x)
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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(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*x/(e*x + d),x, algorithm="giac")
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