Optimal. Leaf size=92 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{4 e^2 (a+b x)} \]
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Rubi [A] time = 0.116782, 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 \[ \frac{b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^5}{5 e^2 (a+b x)}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^4 (b d-a e)}{4 e^2 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Rubi in Sympy [A] time = 13.0178, size = 73, normalized size = 0.79 \[ \frac{\left (d + e x\right )^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 e} + \frac{\left (d + e x\right )^{4} \left (a e - b d\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{20 e^{2} \left (a + b x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3*((b*x+a)**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0546361, size = 89, normalized size = 0.97 \[ \frac{x \sqrt{(a+b x)^2} \left (5 a \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+b x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )}{20 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]
[Out]
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Maple [A] time = 0.005, size = 90, normalized size = 1. \[{\frac{x \left ( 4\,b{e}^{3}{x}^{4}+5\,{x}^{3}a{e}^{3}+15\,{x}^{3}bd{e}^{2}+20\,ad{e}^{2}{x}^{2}+20\,b{d}^{2}e{x}^{2}+30\,xa{d}^{2}e+10\,xb{d}^{3}+20\,a{d}^{3} \right ) }{20\,bx+20\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3*((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)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.201542, size = 93, normalized size = 1.01 \[ \frac{1}{5} \, b e^{3} x^{5} + a d^{3} x + \frac{1}{4} \,{\left (3 \, b d e^{2} + a e^{3}\right )} x^{4} +{\left (b d^{2} e + a d e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b d^{3} + 3 \, a d^{2} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.251069, size = 73, normalized size = 0.79 \[ a d^{3} x + \frac{b e^{3} x^{5}}{5} + x^{4} \left (\frac{a e^{3}}{4} + \frac{3 b d e^{2}}{4}\right ) + x^{3} \left (a d e^{2} + b d^{2} e\right ) + x^{2} \left (\frac{3 a d^{2} e}{2} + \frac{b d^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3*((b*x+a)**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.211994, size = 159, normalized size = 1.73 \[ \frac{1}{5} \, b x^{5} e^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, b d x^{4} e^{2}{\rm sign}\left (b x + a\right ) + b d^{2} x^{3} e{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, b d^{3} x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, a x^{4} e^{3}{\rm sign}\left (b x + a\right ) + a d x^{3} e^{2}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a d^{2} x^{2} e{\rm sign}\left (b x + a\right ) + a d^{3} x{\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(e*x + d)^3,x, algorithm="giac")
[Out]