Optimal. Leaf size=143 \[ \frac{3 a^2 b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
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Rubi [A] time = 0.10831, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{3 a^2 b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x,x]
[Out]
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Rubi in Sympy [A] time = 14.2079, size = 109, normalized size = 0.76 \[ \frac{a^{3} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} + a^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} + \frac{a \left (3 a + 3 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6} + \frac{\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{3} \]
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,x)
[Out]
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Mathematica [A] time = 0.0320345, size = 52, normalized size = 0.36 \[ \frac{\sqrt{(a+b x)^2} \left (6 a^3 \log (x)+b x \left (18 a^2+9 a b x+2 b^2 x^2\right )\right )}{6 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^(3/2)/x,x]
[Out]
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Maple [A] time = 0.008, size = 51, normalized size = 0.4 \[{\frac{2\,{b}^{3}{x}^{3}+9\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( x \right ) +18\,{a}^{2}bx}{6\, \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]
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,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,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232962, size = 42, normalized size = 0.29 \[ \frac{1}{3} \, b^{3} x^{3} + \frac{3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/x,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x}\, dx \]
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,x)
[Out]
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GIAC/XCAS [A] time = 0.210229, size = 76, normalized size = 0.53 \[ \frac{1}{3} \, b^{3} x^{3}{\rm sign}\left (b x + a\right ) + \frac{3}{2} \, a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 3 \, a^{2} b x{\rm sign}\left (b x + a\right ) + a^{3}{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)/x,x, algorithm="giac")
[Out]