Optimal. Leaf size=94 \[ \frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt{d+e x}} \]
[Out]
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Rubi [A] time = 0.115679, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{3 e^2 (a+b x) (d+e x)^{3/2}}-\frac{2 b \sqrt{a^2+2 a b x+b^2 x^2}}{e^2 (a+b x) \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 12.7989, size = 78, normalized size = 0.83 \[ - \frac{2 \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{e \left (d + e x\right )^{\frac{3}{2}}} + \frac{4 \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e^{2} \left (a + b x\right ) \left (d + e x\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((b*x+a)**2)**(1/2)/(e*x+d)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0419418, size = 47, normalized size = 0.5 \[ -\frac{2 \sqrt{(a+b x)^2} (a e+2 b d+3 b e x)}{3 e^2 (a+b x) (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^(5/2),x]
[Out]
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Maple [A] time = 0.004, size = 42, normalized size = 0.5 \[ -{\frac{6\,bex+2\,ae+4\,bd}{3\, \left ( bx+a \right ){e}^{2}}\sqrt{ \left ( bx+a \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((b*x+a)^2)^(1/2)/(e*x+d)^(5/2),x)
[Out]
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Maxima [A] time = 0.75572, size = 47, normalized size = 0.5 \[ -\frac{2 \,{\left (3 \, b e x + 2 \, b d + a e\right )}}{3 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)/(e*x + d)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.207087, size = 47, normalized size = 0.5 \[ -\frac{2 \,{\left (3 \, b e x + 2 \, b d + a e\right )}}{3 \,{\left (e^{3} x + d e^{2}\right )} \sqrt{e x + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(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)**2)**(1/2)/(e*x+d)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.213652, size = 65, normalized size = 0.69 \[ -\frac{2 \,{\left (3 \,{\left (x e + d\right )} b{\rm sign}\left (b x + a\right ) - b d{\rm sign}\left (b x + a\right ) + a e{\rm sign}\left (b x + a\right )\right )} e^{\left (-2\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)/(e*x + d)^(5/2),x, algorithm="giac")
[Out]