Optimal. Leaf size=267 \[ \frac{32 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^4}+\frac{16 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^3}+\frac{12 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 \sqrt{d+e x} (f+g x)^{7/2} (c d f-a e g)} \]
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Rubi [A] time = 1.09972, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{32 c^3 d^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^4}+\frac{16 c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^3}+\frac{12 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)^2}+\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 \sqrt{d+e x} (f+g x)^{7/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]/((f + g*x)^(9/2)*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 = 99.9555, size = 258, normalized size = 0.97 \[ \frac{32 c^{3} d^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{35 \sqrt{d + e x} \sqrt{f + g x} \left (a e g - c d f\right )^{4}} - \frac{16 c^{2} d^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{35 \sqrt{d + e x} \left (f + g x\right )^{\frac{3}{2}} \left (a e g - c d f\right )^{3}} + \frac{12 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{35 \sqrt{d + e x} \left (f + g x\right )^{\frac{5}{2}} \left (a e g - c d f\right )^{2}} - \frac{2 \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{7 \sqrt{d + e x} \left (f + g x\right )^{\frac{7}{2}} \left (a e g - c d f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(1/2)/(g*x+f)**(9/2)/(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 = 0.367312, size = 128, normalized size = 0.48 \[ \frac{2 \sqrt{(d+e x) (a e+c d x)} \left (8 c^2 d^2 (f+g x)^2 (c d f-a e g)+6 c d (f+g x) (c d f-a e g)^2+5 (c d f-a e g)^3+16 c^3 d^3 (f+g x)^3\right )}{35 \sqrt{d+e x} (f+g x)^{7/2} (c d f-a e g)^4} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]/((f + g*x)^(9/2)*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 = 0.017, size = 260, normalized size = 1. \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -16\,{c}^{3}{d}^{3}{g}^{3}{x}^{3}+8\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}-56\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}-6\,{a}^{2}cd{e}^{2}{g}^{3}x+28\,a{c}^{2}{d}^{2}ef{g}^{2}x-70\,{c}^{3}{d}^{3}{f}^{2}gx+5\,{a}^{3}{e}^{3}{g}^{3}-21\,{a}^{2}cd{e}^{2}f{g}^{2}+35\,a{c}^{2}{d}^{2}e{f}^{2}g-35\,{c}^{3}{d}^{3}{f}^{3} \right ) }{35\,{g}^{4}{e}^{4}{a}^{4}-140\,cd{g}^{3}f{e}^{3}{a}^{3}+210\,{c}^{2}{d}^{2}{g}^{2}{f}^{2}{e}^{2}{a}^{2}-140\,{c}^{3}{d}^{3}g{f}^{3}ea+35\,{c}^{4}{d}^{4}{f}^{4}}\sqrt{ex+d} \left ( gx+f \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(1/2)/(g*x+f)^(9/2)/(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. \[ \int \frac{\sqrt{e x + d}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{\frac{9}{2}}}\,{d x} \]
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)^(9/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.297505, size = 1287, normalized size = 4.82 \[ \text{result too large to display} \]
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)^(9/2)),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)**(9/2)/(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. \[ \int \frac{\sqrt{e x + d}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}^{\frac{9}{2}}}\,{d x} \]
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)^(9/2)),x, algorithm="giac")
[Out]