Optimal. Leaf size=198 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)} \]
[Out]
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Rubi [A] time = 0.814011, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{315 (d+e x)^{5/2} (f+g x)^{5/2} (c d f-a e g)^3}+\frac{8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{63 (d+e x)^{5/2} (f+g x)^{7/2} (c d f-a e g)^2}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{9 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(11/2)),x]
[Out]
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Rubi in Sympy [A] time = 68.8613, size = 190, normalized size = 0.96 \[ - \frac{16 c^{2} d^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{315 \left (d + e x\right )^{\frac{5}{2}} \left (f + g x\right )^{\frac{5}{2}} \left (a e g - c d f\right )^{3}} + \frac{8 c d \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{63 \left (d + e x\right )^{\frac{5}{2}} \left (f + g x\right )^{\frac{7}{2}} \left (a e g - c d f\right )^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{9 \left (d + e x\right )^{\frac{5}{2}} \left (f + g x\right )^{\frac{9}{2}} \left (a e g - c d f\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2)/(g*x+f)**(11/2),x)
[Out]
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Mathematica [A] time = 0.284453, size = 105, normalized size = 0.53 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (35 a^2 e^2 g^2-10 a c d e g (9 f+2 g x)+c^2 d^2 \left (63 f^2+36 f g x+8 g^2 x^2\right )\right )}{315 (d+e x)^{5/2} (f+g x)^{9/2} (c d f-a e g)^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/((d + e*x)^(3/2)*(f + g*x)^(11/2)),x]
[Out]
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Maple [A] time = 0.014, size = 169, normalized size = 0.9 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 8\,{c}^{2}{d}^{2}{g}^{2}{x}^{2}-20\,acde{g}^{2}x+36\,{c}^{2}{d}^{2}fgx+35\,{a}^{2}{e}^{2}{g}^{2}-90\,acdefg+63\,{c}^{2}{d}^{2}{f}^{2} \right ) }{315\,{a}^{3}{e}^{3}{g}^{3}-945\,{a}^{2}cd{e}^{2}f{g}^{2}+945\,a{c}^{2}{d}^{2}e{f}^{2}g-315\,{c}^{3}{d}^{3}{f}^{3}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( gx+f \right ) ^{-{\frac{9}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2)/(g*x+f)^(11/2),x)
[Out]
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Maxima [A] time = 1.40244, size = 1245, normalized size = 6.29 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)^(3/2)*(g*x + f)^(11/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.306683, size = 1239, normalized size = 6.26 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)^(3/2)*(g*x + f)^(11/2)),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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2)/(g*x+f)**(11/2),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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)^(3/2)*(g*x + f)^(11/2)),x, algorithm="giac")
[Out]