Optimal. Leaf size=109 \[ \frac{4 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d} \]
[Out]
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Rubi [A] time = 0.173833, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{4 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c d} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
[Out]
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Rubi in Sympy [A] time = 32.3623, size = 100, normalized size = 0.92 \[ \frac{2 \sqrt{d + e x} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 c d} - \frac{4 \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 c^{2} d^{2} \sqrt{d + e x}} \]
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)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0561004, size = 54, normalized size = 0.5 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (c d (3 d+e x)-2 a e^2\right )}{3 c^2 d^2 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]
[Out]
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Maple [A] time = 0.005, size = 69, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -cdex+2\,a{e}^{2}-3\,c{d}^{2} \right ) }{3\,{c}^{2}{d}^{2}}\sqrt{ex+d}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
[Out]
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Maxima [A] time = 0.720509, size = 88, normalized size = 0.81 \[ \frac{2 \,{\left (c^{2} d^{2} e x^{2} + 3 \, a c d^{2} e - 2 \, a^{2} e^{3} +{\left (3 \, c^{2} d^{3} - a c d e^{2}\right )} x\right )}}{3 \, \sqrt{c d x + a e} c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.263745, size = 167, normalized size = 1.53 \[ \frac{2 \,{\left (c^{2} d^{2} e^{2} x^{3} + 3 \, a c d^{3} e - 2 \, a^{2} d e^{3} +{\left (4 \, c^{2} d^{3} e - a c d e^{3}\right )} x^{2} +{\left (3 \, c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 2 \, a^{2} e^{4}\right )} x\right )}}{3 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),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)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(3/2)/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")
[Out]