Optimal. Leaf size=69 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (b d-a e)}{4 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2} \]
[Out]
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Rubi [A] time = 0.074788, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (b d-a e)}{4 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 9.25341, size = 66, normalized size = 0.96 \[ \frac{e \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{5 b^{2}} - \frac{\left (2 a + 2 b x\right ) \left (a e - b d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0606221, size = 83, normalized size = 1.2 \[ \frac{x \sqrt{(a+b x)^2} \left (10 a^3 (2 d+e x)+10 a^2 b x (3 d+2 e x)+5 a b^2 x^2 (4 d+3 e x)+b^3 x^3 (5 d+4 e x)\right )}{20 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.007, size = 90, normalized size = 1.3 \[{\frac{x \left ( 4\,e{b}^{3}{x}^{4}+15\,{x}^{3}ae{b}^{2}+5\,{x}^{3}d{b}^{3}+20\,{a}^{2}be{x}^{2}+20\,a{b}^{2}d{x}^{2}+10\,x{a}^{3}e+30\,x{a}^{2}bd+20\,{a}^{3}d \right ) }{20\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)*(b^2*x^2+2*a*b*x+a^2)^(3/2),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((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.204713, size = 93, normalized size = 1.35 \[ \frac{1}{5} \, b^{3} e x^{5} + a^{3} d x + \frac{1}{4} \,{\left (b^{3} d + 3 \, a b^{2} e\right )} x^{4} +{\left (a b^{2} d + a^{2} b e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d + a^{3} e\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.212023, size = 167, normalized size = 2.42 \[ \frac{1}{5} \, b^{3} x^{5} e{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, b^{3} d x^{4}{\rm sign}\left (b x + a\right ) + \frac{3}{4} \, a b^{2} x^{4} e{\rm sign}\left (b x + a\right ) + a b^{2} d x^{3}{\rm sign}\left (b x + a\right ) + a^{2} b x^{3} e{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a^{2} b d x^{2}{\rm sign}\left (b x + a\right ) + \frac{1}{2} \, a^{3} x^{2} e{\rm sign}\left (b x + a\right ) + a^{3} d x{\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d),x, algorithm="giac")
[Out]