Optimal. Leaf size=92 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^2 (a+b x) (d+e x)^4}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3} \]
[Out]
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Rubi [A] time = 0.125727, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)}{4 e^2 (a+b x) (d+e x)^4}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^2 (a+b x) (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 12.8726, size = 73, normalized size = 0.79 \[ - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 e \left (d + e x\right )^{4}} + \frac{\left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{12 e^{2} \left (a + b x\right ) \left (d + e x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((b*x+a)**2)**(1/2)/(e*x+d)**5,x)
[Out]
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Mathematica [A] time = 0.0348337, size = 45, normalized size = 0.49 \[ -\frac{\sqrt{(a+b x)^2} (3 a e+b (d+4 e x))}{12 e^2 (a+b x) (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/(d + e*x)^5,x]
[Out]
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Maple [A] time = 0.005, size = 42, normalized size = 0.5 \[ -{\frac{4\,bex+3\,ae+bd}{12\,{e}^{2} \left ( ex+d \right ) ^{4} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((b*x+a)^2)^(1/2)/(e*x+d)^5,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(sqrt((b*x + a)^2)/(e*x + d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205153, size = 82, normalized size = 0.89 \[ -\frac{4 \, b e x + b d + 3 \, a e}{12 \,{\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)/(e*x + d)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.38849, size = 65, normalized size = 0.71 \[ - \frac{3 a e + b d + 4 b e x}{12 d^{4} e^{2} + 48 d^{3} e^{3} x + 72 d^{2} e^{4} x^{2} + 48 d e^{5} x^{3} + 12 e^{6} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x+a)**2)**(1/2)/(e*x+d)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.211104, size = 61, normalized size = 0.66 \[ -\frac{{\left (4 \, b x e{\rm sign}\left (b x + a\right ) + b d{\rm sign}\left (b x + a\right ) + 3 \, a e{\rm sign}\left (b x + a\right )\right )} e^{\left (-2\right )}}{12 \,{\left (x e + d\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)/(e*x + d)^5,x, algorithm="giac")
[Out]