Optimal. Leaf size=120 \[ -\frac{2 c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}}+\frac{2 c d}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}+\frac{2}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.232387, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ -\frac{2 c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}}+\frac{2 c d}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}+\frac{2}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
[In] Int[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]
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Rubi in Sympy [A] time = 49.5063, size = 105, normalized size = 0.88 \[ \frac{2 c^{\frac{3}{2}} d^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{\left (a e^{2} - c d^{2}\right )^{\frac{5}{2}}} + \frac{2 c d}{\sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2}{3 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
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Mathematica [A] time = 0.339688, size = 111, normalized size = 0.92 \[ \frac{2 \left (c d (4 d+3 e x)-a e^2\right )}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}-\frac{2 c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]
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Maple [A] time = 0.017, size = 117, normalized size = 1. \[ -{\frac{2}{3\,a{e}^{2}-3\,c{d}^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{cd}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}\sqrt{ex+d}}}+2\,{\frac{{c}^{2}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^(3/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^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(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(3/2)),x, algorithm="maxima")
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Fricas [A] time = 0.221308, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, c d e x + 8 \, c d^{2} - 2 \, a e^{2} + 3 \,{\left (c d e x + c d^{2}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \,{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right )}{3 \,{\left (c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} +{\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}, \frac{2 \,{\left (3 \, c d e x + 4 \, c d^{2} - a e^{2} - 3 \,{\left (c d e x + c d^{2}\right )} \sqrt{e x + d} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac{{\left (c d^{2} - a e^{2}\right )} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}}}{\sqrt{e x + d} c d}\right )\right )}}{3 \,{\left (c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} +{\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(3/2)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right )^{\frac{5}{2}} \left (a e + c d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),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(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(3/2)),x, algorithm="giac")
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