Optimal. Leaf size=129 \[ \frac{4 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)} \]
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Rubi [A] time = 0.515515, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{4 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]/((f + g*x)^(5/2)*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 = 43.6975, size = 122, normalized size = 0.95 \[ \frac{4 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 \sqrt{d + e x} \sqrt{f + g x} \left (a e g - c d f\right )^{2}} - \frac{2 \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 \sqrt{d + e x} \left (f + g x\right )^{\frac{3}{2}} \left (a e g - c d f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(1/2)/(g*x+f)**(5/2)/(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 = 0.145734, size = 69, normalized size = 0.53 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} (c d (3 f+2 g x)-a e g)}{3 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^2} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]/((f + g*x)^(5/2)*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 = 0.011, size = 98, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -2\,xcdg+aeg-3\,cdf \right ) }{3\,{a}^{2}{e}^{2}{g}^{2}-6\,acdefg+3\,{c}^{2}{d}^{2}{f}^{2}}\sqrt{ex+d} \left ( gx+f \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(1/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e x + d}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.284762, size = 389, normalized size = 3.02 \[ \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d g x + 3 \, c d f - a e g\right )} \sqrt{e x + d} \sqrt{g x + f}}{3 \,{\left (c^{2} d^{3} f^{4} - 2 \, a c d^{2} e f^{3} g + a^{2} d e^{2} f^{2} g^{2} +{\left (c^{2} d^{2} e f^{2} g^{2} - 2 \, a c d e^{2} f g^{3} + a^{2} e^{3} g^{4}\right )} x^{3} +{\left (2 \, c^{2} d^{2} e f^{3} g + a^{2} d e^{2} g^{4} +{\left (c^{2} d^{3} - 4 \, a c d e^{2}\right )} f^{2} g^{2} - 2 \,{\left (a c d^{2} e - a^{2} e^{3}\right )} f g^{3}\right )} x^{2} +{\left (c^{2} d^{2} e f^{4} + 2 \, a^{2} d e^{2} f g^{3} + 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} f^{3} g -{\left (4 \, a c d^{2} e - a^{2} e^{3}\right )} f^{2} g^{2}\right )} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^(5/2)),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)**(1/2)/(g*x+f)**(5/2)/(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. \[ \int \frac{\sqrt{e x + d}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(g*x + f)^(5/2)),x, algorithm="giac")
[Out]