Optimal. Leaf size=52 \[ -\frac{2 d-3 e}{24 (2 x+3) \left (4 x^2+12 x+9\right )^{5/2}}-\frac{e}{20 \left (4 x^2+12 x+9\right )^{5/2}} \]
[Out]
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Rubi [A] time = 0.0478515, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{2 d-3 e}{24 (2 x+3) \left (4 x^2+12 x+9\right )^{5/2}}-\frac{e}{20 \left (4 x^2+12 x+9\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(9 + 12*x + 4*x^2)^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 5.41348, size = 42, normalized size = 0.81 \[ - \frac{e}{20 \left (4 x^{2} + 12 x + 9\right )^{\frac{5}{2}}} - \frac{\left (\frac{d}{48} - \frac{e}{32}\right ) \left (8 x + 12\right )}{\left (4 x^{2} + 12 x + 9\right )^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(4*x**2+12*x+9)**(7/2),x)
[Out]
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Mathematica [A] time = 0.031262, size = 34, normalized size = 0.65 \[ \frac{-10 d-3 (4 e x+e)}{120 (2 x+3)^5 \sqrt{(2 x+3)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(9 + 12*x + 4*x^2)^(7/2),x]
[Out]
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Maple [A] time = 0.007, size = 28, normalized size = 0.5 \[ -{\frac{ \left ( 2\,x+3 \right ) \left ( 12\,ex+10\,d+3\,e \right ) }{120} \left ( \left ( 2\,x+3 \right ) ^{2} \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(4*x^2+12*x+9)^(7/2),x)
[Out]
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Maxima [A] time = 0.828535, size = 49, normalized size = 0.94 \[ -\frac{e}{20 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}}} - \frac{d}{12 \,{\left (2 \, x + 3\right )}^{6}} + \frac{e}{8 \,{\left (2 \, x + 3\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(4*x^2 + 12*x + 9)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.200197, size = 61, normalized size = 1.17 \[ -\frac{12 \, e x + 10 \, d + 3 \, e}{120 \,{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(4*x^2 + 12*x + 9)^(7/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d + e x}{\left (\left (2 x + 3\right )^{2}\right )^{\frac{7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(4*x**2+12*x+9)**(7/2),x)
[Out]
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GIAC/XCAS [A] time = 0.600044, size = 4, normalized size = 0.08 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(4*x^2 + 12*x + 9)^(7/2),x, algorithm="giac")
[Out]