Optimal. Leaf size=41 \[ \frac{2 (a+b x) (d+e x)^{5/2}}{5 e \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.11803, 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 \[ \frac{2 (a+b x) (d+e x)^{5/2}}{5 e \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(d + e*x)^(3/2))/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 15.9546, size = 37, normalized size = 0.9 \[ \frac{2 \left (a + b x\right ) \left (d + e x\right )^{\frac{5}{2}}}{5 e \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**(3/2)/((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0398465, size = 32, normalized size = 0.78 \[ \frac{2 (a+b x) (d+e x)^{5/2}}{5 e \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(d + e*x)^(3/2))/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.005, size = 27, normalized size = 0.7 \[{\frac{2\,bx+2\,a}{5\,e} \left ( ex+d \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^(3/2)/((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)*(e*x + d)^(3/2)/sqrt((b*x + a)^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.279649, size = 38, normalized size = 0.93 \[ \frac{2 \,{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{e x + d}}{5 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^(3/2)/sqrt((b*x + a)^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)**(3/2)/((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.291623, size = 69, normalized size = 1.68 \[ \frac{2}{15} \,{\left (5 \,{\left (x e + d\right )}^{\frac{3}{2}} d{\rm sign}\left (b x + a\right ) +{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )}{\rm sign}\left (b x + a\right )\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^(3/2)/sqrt((b*x + a)^2),x, algorithm="giac")
[Out]