Optimal. Leaf size=116 \[ \frac{2 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)}+\frac{2 b B x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]
[Out]
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Rubi [A] time = 0.152162, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{2 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}-\frac{2 a A \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)}+\frac{2 b B x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 18.4261, size = 117, normalized size = 1.01 \[ - \frac{A \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a \sqrt{x}} + \frac{\sqrt{x} \left (4 A b + \frac{4 B a}{3}\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{a + b x} + \frac{2 \sqrt{x} \left (3 A b + B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**(3/2),x)
[Out]
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Mathematica [A] time = 0.0350177, size = 47, normalized size = 0.41 \[ \frac{2 \sqrt{(a+b x)^2} (b x (3 A+B x)-3 a (A-B x))}{3 \sqrt{x} (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/x^(3/2),x]
[Out]
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Maple [A] time = 0.006, size = 44, normalized size = 0.4 \[ -{\frac{-2\,Bb{x}^{2}-6\,Abx-6\,aBx+6\,aA}{3\,bx+3\,a}\sqrt{ \left ( bx+a \right ) ^{2}}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*((b*x+a)^2)^(1/2)/x^(3/2),x)
[Out]
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Maxima [A] time = 0.70617, size = 45, normalized size = 0.39 \[ \frac{2 \,{\left (b x^{2} + 3 \, a x\right )} B}{3 \, \sqrt{x}} + \frac{2 \,{\left (b x^{2} - a x\right )} A}{x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)/x^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.302811, size = 35, normalized size = 0.3 \[ \frac{2 \,{\left (B b x^{2} - 3 \, A a + 3 \,{\left (B a + A b\right )} x\right )}}{3 \, \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)/x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \sqrt{\left (a + b x\right )^{2}}}{x^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.270054, size = 72, normalized size = 0.62 \[ \frac{2}{3} \, B b x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 2 \, B a \sqrt{x}{\rm sign}\left (b x + a\right ) + 2 \, A b \sqrt{x}{\rm sign}\left (b x + a\right ) - \frac{2 \, A a{\rm sign}\left (b x + a\right )}{\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)/x^(3/2),x, algorithm="giac")
[Out]