Optimal. Leaf size=125 \[ \frac{2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d e (d+e x)^{3/2}}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{35 c^2 d^2 e (d+e x)^{5/2}} \]
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Rubi [A] time = 0.327352, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{2 g \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d e (d+e x)^{3/2}}-\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2} \left (2 a e^2 g-c d (7 e f-5 d g)\right )}{35 c^2 d^2 e (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[((f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 34.9547, size = 121, normalized size = 0.97 \[ \frac{2 g \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}}}{7 c d e \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{5}{2}} \left (2 a e^{2} g + 5 c d^{2} g - 7 c d e f\right )}{35 c^{2} d^{2} e \left (d + e x\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0991003, size = 54, normalized size = 0.43 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} (c d (7 f+5 g x)-2 a e g)}{35 c^2 d^2 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((f + g*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(d + e*x)^(3/2),x]
[Out]
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Maple [A] time = 0.007, size = 67, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -5\,xcdg+2\,aeg-7\,cdf \right ) }{35\,{c}^{2}{d}^{2}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(e*x+d)^(3/2),x)
[Out]
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Maxima [A] time = 0.72322, size = 144, normalized size = 1.15 \[ \frac{2 \,{\left (c^{2} d^{2} x^{2} + 2 \, a c d e x + a^{2} e^{2}\right )} \sqrt{c d x + a e} f}{5 \, c d} + \frac{2 \,{\left (5 \, c^{3} d^{3} x^{3} + 8 \, a c^{2} d^{2} e x^{2} + a^{2} c d e^{2} x - 2 \, a^{3} e^{3}\right )} \sqrt{c d x + a e} g}{35 \, c^{2} d^{2}} \]
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)*(g*x + f)/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.267671, size = 382, normalized size = 3.06 \[ \frac{2 \,{\left (5 \, c^{4} d^{4} e g x^{5} + 7 \, a^{3} c d^{2} e^{3} f - 2 \, a^{4} d e^{4} g +{\left (7 \, c^{4} d^{4} e f +{\left (5 \, c^{4} d^{5} + 13 \, a c^{3} d^{3} e^{2}\right )} g\right )} x^{4} +{\left (7 \,{\left (c^{4} d^{5} + 3 \, a c^{3} d^{3} e^{2}\right )} f +{\left (13 \, a c^{3} d^{4} e + 9 \, a^{2} c^{2} d^{2} e^{3}\right )} g\right )} x^{3} +{\left (21 \,{\left (a c^{3} d^{4} e + a^{2} c^{2} d^{2} e^{3}\right )} f +{\left (9 \, a^{2} c^{2} d^{3} e^{2} - a^{3} c d e^{4}\right )} g\right )} x^{2} +{\left (7 \,{\left (3 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} f -{\left (a^{3} c d^{2} e^{3} + 2 \, a^{4} e^{5}\right )} g\right )} x\right )}}{35 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} c^{2} d^{2}} \]
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)*(g*x + f)/(e*x + d)^(3/2),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((g*x+f)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**(3/2),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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)*(g*x + f)/(e*x + d)^(3/2),x, algorithm="giac")
[Out]