Optimal. Leaf size=260 \[ \frac{32 c d g^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^4}+\frac{16 g^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^3}+\frac{4 g \sqrt{d+e x}}{(f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2}}{3 (f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]
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Rubi [A] time = 1.09818, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{32 c d g^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^4}+\frac{16 g^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^3}+\frac{4 g \sqrt{d+e x}}{(f+g x)^{3/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}-\frac{2 (d+e x)^{3/2}}{3 (f+g x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(5/2)/((f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 99.3271, size = 252, normalized size = 0.97 \[ \frac{32 c d g^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 \sqrt{d + e x} \sqrt{f + g x} \left (a e g - c d f\right )^{4}} - \frac{16 g^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{3 \sqrt{d + e x} \left (f + g x\right )^{\frac{3}{2}} \left (a e g - c d f\right )^{3}} + \frac{4 g \sqrt{d + e x}}{\left (f + g x\right )^{\frac{3}{2}} \left (a e g - c d f\right )^{2} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 \left (f + g x\right )^{\frac{3}{2}} \left (a e g - c d f\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(5/2)/(g*x+f)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.320296, size = 152, normalized size = 0.58 \[ \frac{2 (d+e x)^{3/2} \left (-a^3 e^3 g^3+3 a^2 c d e^2 g^2 (3 f+2 g x)+3 a c^2 d^2 e g \left (3 f^2+12 f g x+8 g^2 x^2\right )+c^3 d^3 \left (-f^3+6 f^2 g x+24 f g^2 x^2+16 g^3 x^3\right )\right )}{3 (f+g x)^{3/2} ((d+e x) (a e+c d x))^{3/2} (c d f-a e g)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(5/2)/((f + g*x)^(5/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)),x]
[Out]
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Maple [A] time = 0.015, size = 258, normalized size = 1. \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -16\,{c}^{3}{d}^{3}{g}^{3}{x}^{3}-24\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}-24\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}-6\,{a}^{2}cd{e}^{2}{g}^{3}x-36\,a{c}^{2}{d}^{2}ef{g}^{2}x-6\,{c}^{3}{d}^{3}{f}^{2}gx+{a}^{3}{e}^{3}{g}^{3}-9\,{a}^{2}cd{e}^{2}f{g}^{2}-9\,a{c}^{2}{d}^{2}e{f}^{2}g+{c}^{3}{d}^{3}{f}^{3} \right ) }{3\,{g}^{4}{e}^{4}{a}^{4}-12\,cd{g}^{3}f{e}^{3}{a}^{3}+18\,{c}^{2}{d}^{2}{g}^{2}{f}^{2}{e}^{2}{a}^{2}-12\,{c}^{3}{d}^{3}g{f}^{3}ea+3\,{c}^{4}{d}^{4}{f}^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( gx+f \right ) ^{-{\frac{3}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(5/2)/(g*x+f)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{5}{2}}{\left (g x + f\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(g*x + f)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.321584, size = 1438, normalized size = 5.53 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(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)**(5/2)/(g*x+f)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)*(g*x + f)^(5/2)),x, algorithm="giac")
[Out]