Optimal. Leaf size=152 \[ \frac{2 \sqrt{d+e x} (2 b e g-5 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.445262, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{2 \sqrt{d+e x} (2 b e g-5 c d g+c e f)}{3 c^2 e^2 (2 c d-b e) \sqrt{d (c d-b e)-b e^2 x-c e^2 x^2}}+\frac{2 (d+e x)^{5/2} (-b e g+c d g+c e f)}{3 c e^2 (2 c d-b e) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}} \]
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)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 55.5957, size = 139, normalized size = 0.91 \[ \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (b e g - c d g - c e f\right )}{3 c e^{2} \left (b e - 2 c d\right ) \left (- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )\right )^{\frac{3}{2}}} - \frac{2 \sqrt{d + e x} \left (2 b e g - 5 c d g + c e f\right )}{3 c^{2} e^{2} \left (b e - 2 c d\right ) \sqrt{- b e^{2} x - c e^{2} x^{2} + d \left (- b e + c d\right )}} \]
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)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0993624, size = 76, normalized size = 0.5 \[ -\frac{2 \sqrt{d+e x} (2 b e g-2 c d g+c e (f+3 g x))}{3 c^2 e^2 (b e-c d+c e x) \sqrt{(d+e x) (c (d-e x)-b e)}} \]
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)^(5/2),x]
[Out]
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Maple [A] time = 0.13, size = 78, normalized size = 0.5 \[ -{\frac{ \left ( 2\,cex+2\,be-2\,cd \right ) \left ( 3\,cegx+2\,beg-2\,cdg+cef \right ) }{3\,{c}^{2}{e}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}} \left ( -c{e}^{2}{x}^{2}-b{e}^{2}x-bde+c{d}^{2} \right ) ^{-{\frac{5}{2}}}} \]
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)^(5/2),x)
[Out]
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Maxima [A] time = 0.743518, size = 139, normalized size = 0.91 \[ -\frac{2 \,{\left (3 \, c e x - 2 \, c d + 2 \, b e\right )} g}{3 \,{\left (c^{3} e^{3} x - c^{3} d e^{2} + b c^{2} e^{3}\right )} \sqrt{-c e x + c d - b e}} - \frac{2 \, f}{3 \,{\left (c^{2} e^{2} x - c^{2} d e + b c e^{2}\right )} \sqrt{-c e x + c d - b e}} \]
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)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.273393, size = 208, normalized size = 1.37 \[ \frac{2 \, \sqrt{-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}{\left (3 \, c e g x + c e f - 2 \,{\left (c d - b e\right )} g\right )} \sqrt{e x + d}}{3 \,{\left (c^{4} e^{5} x^{3} + c^{4} d^{3} e^{2} - 2 \, b c^{3} d^{2} e^{3} + b^{2} c^{2} d e^{4} -{\left (c^{4} d e^{4} - 2 \, b c^{3} e^{5}\right )} x^{2} -{\left (c^{4} d^{2} e^{3} - b^{2} c^{2} e^{5}\right )} x\right )}} \]
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)^(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)/(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.760789, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]
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)^(5/2),x, algorithm="giac")
[Out]