Optimal. Leaf size=262 \[ -\frac{32 c^2 d^2 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^4}-\frac{16 c d g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^3}-\frac{12 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]
[Out]
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Rubi [A] time = 1.09274, antiderivative size = 262, 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 \[ -\frac{32 c^2 d^2 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt{d+e x} \sqrt{f+g x} (c d f-a e g)^4}-\frac{16 c d g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt{d+e x} (f+g x)^{3/2} (c d f-a e g)^3}-\frac{12 g \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt{d+e x} (f+g x)^{5/2} (c d f-a e g)^2}-\frac{2 \sqrt{d+e x}}{(f+g x)^{5/2} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(3/2)/((f + g*x)^(7/2)*(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 = 98.6059, size = 255, normalized size = 0.97 \[ - \frac{32 c^{2} d^{2} g \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{5 \sqrt{d + e x} \sqrt{f + g x} \left (a e g - c d f\right )^{4}} + \frac{16 c d g \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{5 \sqrt{d + e x} \left (f + g x\right )^{\frac{3}{2}} \left (a e g - c d f\right )^{3}} - \frac{12 g \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{5 \sqrt{d + e x} \left (f + g x\right )^{\frac{5}{2}} \left (a e g - c d f\right )^{2}} + \frac{2 \sqrt{d + e x}}{\left (f + g x\right )^{\frac{5}{2}} \left (a e g - c d f\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(3/2)/(g*x+f)**(7/2)/(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 = 0.339974, size = 150, normalized size = 0.57 \[ -\frac{2 \sqrt{d+e x} \left (a^3 e^3 g^3-a^2 c d e^2 g^2 (5 f+2 g x)+a c^2 d^2 e g \left (15 f^2+20 f g x+8 g^2 x^2\right )+c^3 d^3 \left (5 f^3+30 f^2 g x+40 f g^2 x^2+16 g^3 x^3\right )\right )}{5 (f+g x)^{5/2} \sqrt{(d+e x) (a e+c d x)} (c d f-a e g)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(3/2)/((f + g*x)^(7/2)*(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 = 0.017, size = 259, normalized size = 1. \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 16\,{c}^{3}{d}^{3}{g}^{3}{x}^{3}+8\,a{c}^{2}{d}^{2}e{g}^{3}{x}^{2}+40\,{c}^{3}{d}^{3}f{g}^{2}{x}^{2}-2\,{a}^{2}cd{e}^{2}{g}^{3}x+20\,a{c}^{2}{d}^{2}ef{g}^{2}x+30\,{c}^{3}{d}^{3}{f}^{2}gx+{a}^{3}{e}^{3}{g}^{3}-5\,{a}^{2}cd{e}^{2}f{g}^{2}+15\,a{c}^{2}{d}^{2}e{f}^{2}g+5\,{c}^{3}{d}^{3}{f}^{3} \right ) }{5\,{g}^{4}{e}^{4}{a}^{4}-20\,cd{g}^{3}f{e}^{3}{a}^{3}+30\,{c}^{2}{d}^{2}{g}^{2}{f}^{2}{e}^{2}{a}^{2}-20\,{c}^{3}{d}^{3}g{f}^{3}ea+5\,{c}^{4}{d}^{4}{f}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( gx+f \right ) ^{-{\frac{5}{2}}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(3/2)/(g*x+f)^(7/2)/(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. \[ \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{\frac{7}{2}}}\,{d x} \]
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)*(g*x + f)^(7/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.318316, size = 1434, normalized size = 5.47 \[ \text{result too large to display} \]
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)*(g*x + f)^(7/2)),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)**(3/2)/(g*x+f)**(7/2)/(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(-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)^(3/2)/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(g*x + f)^(7/2)),x, algorithm="giac")
[Out]