Optimal. Leaf size=41 \[ -\frac{2 (a+b x)}{3 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}} \]
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Rubi [A] time = 0.118784, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086 \[ -\frac{2 (a+b x)}{3 e \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)/((d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 16.1793, size = 41, normalized size = 1. \[ - \frac{2 \left (a + b x\right )}{3 e \left (d + e x\right )^{\frac{3}{2}} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)/(e*x+d)**(5/2)/((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0286116, size = 32, normalized size = 0.78 \[ -\frac{2 (a+b x)}{3 e \sqrt{(a+b x)^2} (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)/((d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]),x]
[Out]
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Maple [A] time = 0.004, size = 27, normalized size = 0.7 \[ -{\frac{2\,bx+2\,a}{3\,e} \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)/(e*x+d)^(5/2)/((b*x+a)^2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(sqrt((b*x + a)^2)*(e*x + d)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.279231, size = 27, normalized size = 0.66 \[ -\frac{2}{3 \,{\left (e^{2} x + d e\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(sqrt((b*x + a)^2)*(e*x + d)^(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((b*x+a)/(e*x+d)**(5/2)/((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.285079, size = 24, normalized size = 0.59 \[ -\frac{2 \, e^{\left (-1\right )}{\rm sign}\left (b x + a\right )}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(sqrt((b*x + a)^2)*(e*x + d)^(5/2)),x, algorithm="giac")
[Out]