Optimal. Leaf size=217 \[ \frac{4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{3 c^3 e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{3 c^2 e^2 (2 c d-b e)}+\frac{2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]
[Out]
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Rubi [A] time = 0.705782, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{4 \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{3 c^3 e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2} (-4 b e g+5 c d g+3 c e f)}{3 c^2 e^2 (2 c d-b e)}+\frac{2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{c e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[((d + e*x)^(5/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 80.8729, size = 206, normalized size = 0.95 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (b e g - c d g - c e f\right )}{c e^{2} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} + \frac{2 \sqrt{d + e x} \left (4 b e g - 5 c d g - 3 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{3 c^{2} e^{2} \left (b e - 2 c d\right )} - \frac{4 \left (4 b e g - 5 c d g - 3 c e f\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}}{3 c^{3} e^{2} \sqrt{d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(5/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.154233, size = 105, normalized size = 0.48 \[ -\frac{2 \sqrt{d+e x} \left (-8 b^2 e^2 g+2 b c e (11 d g+3 e f-2 e g x)+c^2 \left (-14 d^2 g+d e (7 g x-9 f)+e^2 x (3 f+g x)\right )\right )}{3 c^3 e^2 \sqrt{(d+e x) (c (d-e x)-b e)}} \]
Antiderivative was successfully verified.
[In] Integrate[((d + e*x)^(5/2)*(f + g*x))/(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.008, size = 139, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( -g{x}^{2}{c}^{2}{e}^{2}+4\,bc{e}^{2}gx-7\,{c}^{2}degx-3\,{c}^{2}{e}^{2}fx+8\,{b}^{2}{e}^{2}g-22\,bcdeg-6\,bc{e}^{2}f+14\,{c}^{2}{d}^{2}g+9\,{c}^{2}def \right ) }{3\,{c}^{3}{e}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(5/2)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2),x)
[Out]
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Maxima [A] time = 0.736334, size = 151, normalized size = 0.7 \[ -\frac{2 \,{\left (c e x - 3 \, c d + 2 \, b e\right )} f}{\sqrt{-c e x + c d - b e} c^{2} e} - \frac{2 \,{\left (c^{2} e^{2} x^{2} - 14 \, c^{2} d^{2} + 22 \, b c d e - 8 \, b^{2} e^{2} +{\left (7 \, c^{2} d e - 4 \, b c e^{2}\right )} x\right )} g}{3 \, \sqrt{-c e x + c d - b e} c^{3} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272152, size = 258, normalized size = 1.19 \[ -\frac{2 \,{\left (c^{2} e^{3} g x^{3} +{\left (3 \, c^{2} e^{3} f + 4 \,{\left (2 \, c^{2} d e^{2} - b c e^{3}\right )} g\right )} x^{2} - 3 \,{\left (3 \, c^{2} d^{2} e - 2 \, b c d e^{2}\right )} f - 2 \,{\left (7 \, c^{2} d^{3} - 11 \, b c d^{2} e + 4 \, b^{2} d e^{2}\right )} g -{\left (6 \,{\left (c^{2} d e^{2} - b c e^{3}\right )} f +{\left (7 \, c^{2} d^{2} e - 18 \, b c d e^{2} + 8 \, b^{2} e^{3}\right )} g\right )} x\right )}}{3 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt{e x + d} c^{3} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(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((e*x+d)**(5/2)*(g*x+f)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.643812, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(5/2)*(g*x + f)/(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2),x, algorithm="giac")
[Out]