Optimal. Leaf size=176 \[ \frac{15 e^2}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^3}+\frac{5 e}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{1}{2 \sqrt{d+e x} \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac{15 \sqrt{c} \sqrt{d} e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}} \]
[Out]
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Rubi [A] time = 0.32937, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ \frac{15 e^2}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^3}+\frac{5 e}{4 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{1}{2 \sqrt{d+e x} \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac{15 \sqrt{c} \sqrt{d} e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \left (c d^2-a e^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 67.1226, size = 167, normalized size = 0.95 \[ - \frac{15 \sqrt{c} \sqrt{d} e^{2} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{4 \left (a e^{2} - c d^{2}\right )^{\frac{7}{2}}} - \frac{15 c d e \sqrt{d + e x}}{4 \left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )^{3}} - \frac{5 e}{2 \sqrt{d + e x} \left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )^{2}} + \frac{1}{2 \sqrt{d + e x} \left (a e + c d x\right )^{2} \left (a e^{2} - c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
[Out]
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Mathematica [A] time = 0.482651, size = 157, normalized size = 0.89 \[ \frac{1}{4} \left (\frac{8 a^2 e^4+a c d e^2 (9 d+25 e x)+c^2 d^2 \left (-2 d^2+5 d e x+15 e^2 x^2\right )}{\sqrt{d+e x} \left (c d^2-a e^2\right )^3 (a e+c d x)^2}-\frac{15 \sqrt{c} \sqrt{d} e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
[Out]
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Maple [A] time = 0.028, size = 226, normalized size = 1.3 \[ -2\,{\frac{{e}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}\sqrt{ex+d}}}-{\frac{7\,{d}^{2}{e}^{2}{c}^{2}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdex+a{e}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{9\,d{e}^{4}ca}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}+{\frac{9\,{d}^{3}{e}^{2}{c}^{2}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}-{\frac{15\,{e}^{2}cd}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,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((e*x + d)^(3/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.246397, size = 1, normalized size = 0.01 \[ \left [\frac{30 \, c^{2} d^{2} e^{2} x^{2} - 4 \, c^{2} d^{4} + 18 \, a c d^{2} e^{2} + 16 \, a^{2} e^{4} + 15 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \,{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right ) + 10 \,{\left (c^{2} d^{3} e + 5 \, a c d e^{3}\right )} x}{8 \,{\left (a^{2} c^{3} d^{6} e^{2} - 3 \, a^{3} c^{2} d^{4} e^{4} + 3 \, a^{4} c d^{2} e^{6} - a^{5} e^{8} +{\left (c^{5} d^{8} - 3 \, a c^{4} d^{6} e^{2} + 3 \, a^{2} c^{3} d^{4} e^{4} - a^{3} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (a c^{4} d^{7} e - 3 \, a^{2} c^{3} d^{5} e^{3} + 3 \, a^{3} c^{2} d^{3} e^{5} - a^{4} c d e^{7}\right )} x\right )} \sqrt{e x + d}}, \frac{15 \, c^{2} d^{2} e^{2} x^{2} - 2 \, c^{2} d^{4} + 9 \, a c d^{2} e^{2} + 8 \, a^{2} e^{4} - 15 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \sqrt{e x + d} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac{{\left (c d^{2} - a e^{2}\right )} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}}}{\sqrt{e x + d} c d}\right ) + 5 \,{\left (c^{2} d^{3} e + 5 \, a c d e^{3}\right )} x}{4 \,{\left (a^{2} c^{3} d^{6} e^{2} - 3 \, a^{3} c^{2} d^{4} e^{4} + 3 \, a^{4} c d^{2} e^{6} - a^{5} e^{8} +{\left (c^{5} d^{8} - 3 \, a c^{4} d^{6} e^{2} + 3 \, a^{2} c^{3} d^{4} e^{4} - a^{3} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \,{\left (a c^{4} d^{7} e - 3 \, a^{2} c^{3} d^{5} e^{3} + 3 \, a^{3} c^{2} d^{3} e^{5} - a^{4} c d e^{7}\right )} x\right )} \sqrt{e x + d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="fricas")
[Out]
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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((e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="giac")
[Out]