Optimal. Leaf size=78 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^2 (b d-a e)}{3 b^2}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{4 b^2} \]
[Out]
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Rubi [A] time = 0.170191, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^2 (b d-a e)}{3 b^2}+\frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{4 b^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 18.0937, size = 60, normalized size = 0.77 \[ \frac{\left (d + e x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{4 b} - \frac{\left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{12 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)*((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0389653, size = 64, normalized size = 0.82 \[ \frac{x \sqrt{(a+b x)^2} \left (6 a^2 (2 d+e x)+4 a b x (3 d+2 e x)+b^2 x^2 (4 d+3 e x)\right )}{12 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(d + e*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.006, size = 66, normalized size = 0.9 \[{\frac{x \left ( 3\,{b}^{2}e{x}^{3}+8\,{x}^{2}bea+4\,{x}^{2}{b}^{2}d+6\,xe{a}^{2}+12\,abdx+12\,{a}^{2}d \right ) }{12\,bx+12\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)*((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)*(b*x + a)*(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.282393, size = 65, normalized size = 0.83 \[ \frac{1}{4} \, b^{2} e x^{4} + a^{2} d x + \frac{1}{3} \,{\left (b^{2} d + 2 \, a b e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b d + a^{2} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)*(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.209033, size = 49, normalized size = 0.63 \[ a^{2} d x + \frac{b^{2} e x^{4}}{4} + x^{3} \left (\frac{2 a b e}{3} + \frac{b^{2} d}{3}\right ) + x^{2} \left (\frac{a^{2} e}{2} + a b d\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)*((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.280409, size = 119, normalized size = 1.53 \[ \frac{1}{4} \, b^{2} x^{4} e{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, b^{2} d x^{3}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, a b x^{3} e{\rm sign}\left (b x + a\right ) + a b d x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a^{2} x^{2} e{\rm sign}\left (b x + a\right ) + a^{2} d x{\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)*(e*x + d),x, algorithm="giac")
[Out]