Optimal. Leaf size=32 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b} \]
[Out]
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Rubi [A] time = 0.0194233, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 1.94354, size = 31, normalized size = 0.97 \[ \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{8 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0213473, size = 23, normalized size = 0.72 \[ \frac{(a+b x) \left ((a+b x)^2\right )^{3/2}}{4 b} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.004, size = 49, normalized size = 1.5 \[{\frac{x \left ({b}^{3}{x}^{3}+4\,a{b}^{2}{x}^{2}+6\,{a}^{2}bx+4\,{a}^{3} \right ) }{4\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((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),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.204561, size = 42, normalized size = 1.31 \[ \frac{1}{4} \, b^{3} x^{4} + a b^{2} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210697, size = 93, normalized size = 2.91 \[ \frac{1}{4} \, b^{3} x^{4}{\rm sign}\left (b x + a\right ) + a b^{2} x^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a^{2} b x^{2}{\rm sign}\left (b x + a\right ) + a^{3} x{\rm sign}\left (b x + a\right ) + \frac{a^{4}{\rm sign}\left (b x + a\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2),x, algorithm="giac")
[Out]