Optimal. Leaf size=61 \[ -\frac{2 \sqrt{d+e x} \sqrt{f+g x}}{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]
[Out]
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Rubi [A] time = 0.252123, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.021 \[ -\frac{2 \sqrt{d+e x} \sqrt{f+g x}}{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)/(Sqrt[f + g*x]*(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 = 21.7531, size = 56, normalized size = 0.92 \[ \frac{2 \sqrt{d + e x} \sqrt{f + g x}}{\left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(3/2)/(g*x+f)**(1/2)/(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 = 0.0779741, size = 50, normalized size = 0.82 \[ -\frac{2 \sqrt{d+e x} \sqrt{f+g x}}{\sqrt{(d+e x) (a e+c d x)} (c d f-a e g)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)/(Sqrt[f + g*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
[Out]
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Maple [A] time = 0.01, size = 63, normalized size = 1. \[ 2\,{\frac{\sqrt{gx+f} \left ( cdx+ae \right ) \left ( ex+d \right ) ^{3/2}}{ \left ( aeg-cdf \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{3/2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)/(g*x+f)^(1/2)/(a*d*e+(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)^(3/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*sqrt(g*x + f)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.283882, size = 169, normalized size = 2.77 \[ -\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{g x + f}}{a c d^{2} e f - a^{2} d e^{2} g +{\left (c^{2} d^{2} e f - a c d e^{2} g\right )} x^{2} +{\left ({\left (c^{2} d^{3} + a c d e^{2}\right )} f -{\left (a c d^{2} e + a^{2} e^{3}\right )} g\right )} x} \]
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)^(3/2)*sqrt(g*x + f)),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)/(g*x+f)**(1/2)/(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 [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}} \sqrt{g x + f}}\,{d x} \]
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)^(3/2)*sqrt(g*x + f)),x, algorithm="giac")
[Out]