Optimal. Leaf size=125 \[ \frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (b d-a e)}{2 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^2 (b d-a e)^2}{3 b^3}+\frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{5 b^3} \]
[Out]
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Rubi [A] time = 0.254557, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{e \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3 (b d-a e)}{2 b^3}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^2 (b d-a e)^2}{3 b^3}+\frac{e^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{5 b^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 60.0956, size = 100, normalized size = 0.8 \[ \frac{\left (d + e x\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{5 b} - \frac{\left (d + e x\right ) \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{10 b^{2}} + \frac{\left (a e - b d\right )^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{30 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**2*((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0658509, size = 97, normalized size = 0.78 \[ \frac{x \sqrt{(a+b x)^2} \left (10 a^2 \left (3 d^2+3 d e x+e^2 x^2\right )+5 a b x \left (6 d^2+8 d e x+3 e^2 x^2\right )+b^2 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )}{30 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)*(d + e*x)^2*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.008, size = 107, normalized size = 0.9 \[{\frac{x \left ( 6\,{b}^{2}{e}^{2}{x}^{4}+15\,{x}^{3}ab{e}^{2}+15\,{x}^{3}{b}^{2}de+10\,{x}^{2}{a}^{2}{e}^{2}+40\,{x}^{2}abde+10\,{x}^{2}{b}^{2}{d}^{2}+30\,{a}^{2}dex+30\,ab{d}^{2}x+30\,{a}^{2}{d}^{2} \right ) }{30\,bx+30\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^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(sqrt((b*x + a)^2)*(b*x + a)*(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.269277, size = 109, normalized size = 0.87 \[ \frac{1}{5} \, b^{2} e^{2} x^{5} + a^{2} d^{2} x + \frac{1}{2} \,{\left (b^{2} d e + a b e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} d^{2} + 4 \, a b d e + a^{2} e^{2}\right )} x^{3} +{\left (a b d^{2} + a^{2} d e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)*(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.263533, size = 87, normalized size = 0.7 \[ a^{2} d^{2} x + \frac{b^{2} e^{2} x^{5}}{5} + x^{4} \left (\frac{a b e^{2}}{2} + \frac{b^{2} d e}{2}\right ) + x^{3} \left (\frac{a^{2} e^{2}}{3} + \frac{4 a b d e}{3} + \frac{b^{2} d^{2}}{3}\right ) + x^{2} \left (a^{2} d e + a b d^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)**2*((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.289241, size = 193, normalized size = 1.54 \[ \frac{1}{5} \, b^{2} x^{5} e^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, b^{2} d x^{4} e{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, b^{2} d^{2} x^{3}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a b x^{4} e^{2}{\rm sign}\left (b x + a\right ) + \frac{4}{3} \, a b d x^{3} e{\rm sign}\left (b x + a\right ) + a b d^{2} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{3} \, a^{2} x^{3} e^{2}{\rm sign}\left (b x + a\right ) + a^{2} d x^{2} e{\rm sign}\left (b x + a\right ) + a^{2} d^{2} x{\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(b*x + a)*(e*x + d)^2,x, algorithm="giac")
[Out]