Optimal. Leaf size=71 \[ -\frac{b d-a e}{4 b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac{e}{3 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.0740905, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{b d-a e}{4 b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac{e}{3 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 9.35045, size = 66, normalized size = 0.93 \[ - \frac{e}{3 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} + \frac{\left (2 a + 2 b x\right ) \left (a e - b d\right )}{8 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0324591, size = 39, normalized size = 0.55 \[ \frac{-a e-3 b d-4 b e x}{12 b^2 (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
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Maple [A] time = 0.008, size = 33, normalized size = 0.5 \[ -{\frac{ \left ( bx+a \right ) \left ( 4\,bex+ae+3\,bd \right ) }{12\,{b}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
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Maxima [A] time = 0.753135, size = 85, normalized size = 1.2 \[ -\frac{e}{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{2}} - \frac{d}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{a e}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206334, size = 82, normalized size = 1.15 \[ -\frac{4 \, b e x + 3 \, b d + a e}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{d + e x}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.585846, size = 4, normalized size = 0.06 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")
[Out]