Optimal. Leaf size=180 \[ \frac{3 \sqrt{e} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{5/2} d^{5/2}}+\frac{3 e \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^2 d^2}-\frac{2 (d+e x)^2}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
[Out]
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Rubi [A] time = 0.292731, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ \frac{3 \sqrt{e} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{5/2} d^{5/2}}+\frac{3 e \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^2 d^2}-\frac{2 (d+e x)^2}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 42.7609, size = 173, normalized size = 0.96 \[ - \frac{2 \left (d + e x\right )^{2}}{c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{3 e \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{c^{2} d^{2}} - \frac{3 \sqrt{e} \left (a e^{2} - c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{2 c^{\frac{5}{2}} d^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.325259, size = 162, normalized size = 0.9 \[ \frac{3 \sqrt{e} \sqrt{d+e x} \left (c d^2-a e^2\right ) \sqrt{a e+c d x} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )-2 \sqrt{c} \sqrt{d} (d+e x) \left (c d (2 d-e x)-3 a e^2\right )}{2 c^{5/2} d^{5/2} \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2),x]
[Out]
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Maple [B] time = 0.015, size = 1047, normalized size = 5.8 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),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/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.349505, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a c d^{2} e - a^{2} e^{3} +{\left (c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt{\frac{e}{c d}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \,{\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{e}{c d}}\right ) + 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d e x - 2 \, c d^{2} + 3 \, a e^{2}\right )}}{4 \,{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )}}, \frac{3 \,{\left (a c d^{2} e - a^{2} e^{3} +{\left (c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt{-\frac{e}{c d}} \arctan \left (\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d \sqrt{-\frac{e}{c d}}}\right ) + 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d e x - 2 \, c d^{2} + 3 \, a e^{2}\right )}}{2 \,{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.264442, size = 478, normalized size = 2.66 \[ \frac{{\left (\frac{{\left (c^{3} d^{5} e^{3} - 2 \, a c^{2} d^{3} e^{5} + a^{2} c d e^{7}\right )} x}{c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}} - \frac{c^{3} d^{6} e^{2} - 5 \, a c^{2} d^{4} e^{4} + 7 \, a^{2} c d^{2} e^{6} - 3 \, a^{3} e^{8}}{c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}}\right )} x - \frac{2 \, c^{3} d^{7} e - 7 \, a c^{2} d^{5} e^{3} + 8 \, a^{2} c d^{3} e^{5} - 3 \, a^{3} d e^{7}}{c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}}}{\sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}} - \frac{3 \,{\left (c d^{2} e - a e^{3}\right )} \sqrt{c d} e^{\left (-\frac{1}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{2 \, c^{3} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2),x, algorithm="giac")
[Out]