Optimal. Leaf size=25 \[ \frac{(a+b x) \sqrt{c (a+b x)^2}}{2 b} \]
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Rubi [A] time = 0.0257906, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{(a+b x) \sqrt{c (a+b x)^2}}{2 b} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[c*(a + b*x)^2],x]
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Rubi in Sympy [A] time = 2.74969, size = 36, normalized size = 1.44 \[ \frac{\left (2 a + 2 b x\right ) \sqrt{a^{2} c + 2 a b c x + b^{2} c x^{2}}}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*(b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0223195, size = 31, normalized size = 1.24 \[ \frac{c x (a+b x) (2 a+b x)}{2 \sqrt{c (a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[c*(a + b*x)^2],x]
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Maple [A] time = 0.004, size = 29, normalized size = 1.2 \[{\frac{x \left ( bx+2\,a \right ) }{2\,bx+2\,a}\sqrt{c \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*(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(sqrt((b*x + a)^2*c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211785, size = 55, normalized size = 2.2 \[ \frac{\sqrt{b^{2} c x^{2} + 2 \, a b c x + a^{2} c}{\left (b x^{2} + 2 \, a x\right )}}{2 \,{\left (b x + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2*c),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c \left (a + b x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*(b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.217353, size = 49, normalized size = 1.96 \[ \frac{1}{2} \,{\left ({\left (b x^{2} + 2 \, a x\right )}{\rm sign}\left (b x + a\right ) + \frac{a^{2}{\rm sign}\left (b x + a\right )}{b}\right )} \sqrt{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2*c),x, algorithm="giac")
[Out]