Optimal. Leaf size=92 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6}{6 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)}{5 e^2 (a+b x)} \]
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Rubi [A] time = 0.130016, 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 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^6}{6 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5 (b d-a e)}{5 e^2 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 12.9963, size = 73, normalized size = 0.79 \[ \frac{\left (d + e x\right )^{5} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6 e} + \frac{\left (d + e x\right )^{5} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{30 e^{2} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**4*((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0849907, size = 111, normalized size = 1.21 \[ \frac{x \sqrt{(a+b x)^2} \left (6 a \left (5 d^4+10 d^3 e x+10 d^2 e^2 x^2+5 d e^3 x^3+e^4 x^4\right )+b x \left (15 d^4+40 d^3 e x+45 d^2 e^2 x^2+24 d e^3 x^3+5 e^4 x^4\right )\right )}{30 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.005, size = 114, normalized size = 1.2 \[{\frac{x \left ( 5\,b{e}^{4}{x}^{5}+6\,{x}^{4}a{e}^{4}+24\,{x}^{4}bd{e}^{3}+30\,{x}^{3}ad{e}^{3}+45\,{x}^{3}b{d}^{2}{e}^{2}+60\,{x}^{2}a{d}^{2}{e}^{2}+40\,{x}^{2}b{d}^{3}e+60\,xa{d}^{3}e+15\,xb{d}^{4}+30\,a{d}^{4} \right ) }{30\,bx+30\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^4*((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)*(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.202928, size = 130, normalized size = 1.41 \[ \frac{1}{6} \, b e^{4} x^{6} + a d^{4} x + \frac{1}{5} \,{\left (4 \, b d e^{3} + a e^{4}\right )} x^{5} + \frac{1}{2} \,{\left (3 \, b d^{2} e^{2} + 2 \, a d e^{3}\right )} x^{4} + \frac{2}{3} \,{\left (2 \, b d^{3} e + 3 \, a d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b d^{4} + 4 \, a d^{3} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.271463, size = 100, normalized size = 1.09 \[ a d^{4} x + \frac{b e^{4} x^{6}}{6} + x^{5} \left (\frac{a e^{4}}{5} + \frac{4 b d e^{3}}{5}\right ) + x^{4} \left (a d e^{3} + \frac{3 b d^{2} e^{2}}{2}\right ) + x^{3} \left (2 a d^{2} e^{2} + \frac{4 b d^{3} e}{3}\right ) + x^{2} \left (2 a d^{3} e + \frac{b d^{4}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**4*((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.212031, size = 207, normalized size = 2.25 \[ \frac{1}{6} \, b x^{6} e^{4}{\rm sign}\left (b x + a\right ) + \frac{4}{5} \, b d x^{5} e^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, b d^{2} x^{4} e^{2}{\rm sign}\left (b x + a\right ) + \frac{4}{3} \, b d^{3} x^{3} e{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, b d^{4} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{5} \, a x^{5} e^{4}{\rm sign}\left (b x + a\right ) + a d x^{4} e^{3}{\rm sign}\left (b x + a\right ) + 2 \, a d^{2} x^{3} e^{2}{\rm sign}\left (b x + a\right ) + 2 \, a d^{3} x^{2} e{\rm sign}\left (b x + a\right ) + a d^{4} x{\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(e*x + d)^4,x, algorithm="giac")
[Out]