Optimal. Leaf size=118 \[ \frac{2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{2 a A \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{2 b B x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]
[Out]
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Rubi [A] time = 0.150112, antiderivative size = 118, 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 \[ \frac{2 x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac{2 a A \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{2 b B x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/Sqrt[x],x]
[Out]
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Rubi in Sympy [A] time = 18.4377, size = 122, normalized size = 1.03 \[ \frac{B \sqrt{x} \left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5 b} + \frac{4 a \sqrt{x} \left (5 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 b \left (a + b x\right )} + \frac{2 \sqrt{x} \left (5 A b - B a\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{15 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**(1/2),x)
[Out]
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Mathematica [A] time = 0.0392552, size = 49, normalized size = 0.42 \[ \frac{2 \sqrt{x} \sqrt{(a+b x)^2} (5 a (3 A+B x)+b x (5 A+3 B x))}{15 (a+b x)} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/Sqrt[x],x]
[Out]
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Maple [A] time = 0.006, size = 44, normalized size = 0.4 \[{\frac{6\,Bb{x}^{2}+10\,Abx+10\,aBx+30\,aA}{15\,bx+15\,a}\sqrt{x}\sqrt{ \left ( bx+a \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*((b*x+a)^2)^(1/2)/x^(1/2),x)
[Out]
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Maxima [A] time = 0.70379, size = 46, normalized size = 0.39 \[ \frac{2}{15} \,{\left (3 \, b x^{2} + 5 \, a x\right )} B \sqrt{x} + \frac{2 \,{\left (b x^{2} + 3 \, a x\right )} A}{3 \, \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)/sqrt(x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.301607, size = 36, normalized size = 0.31 \[ \frac{2}{15} \,{\left (3 \, B b x^{2} + 15 \, A a + 5 \,{\left (B a + A b\right )} x\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)/sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \sqrt{\left (a + b x\right )^{2}}}{\sqrt{x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*((b*x+a)**2)**(1/2)/x**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.27196, size = 72, normalized size = 0.61 \[ \frac{2}{5} \, B b x^{\frac{5}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, B a x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + \frac{2}{3} \, A b x^{\frac{3}{2}}{\rm sign}\left (b x + a\right ) + 2 \, A a \sqrt{x}{\rm sign}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x + a)^2)*(B*x + A)/sqrt(x),x, algorithm="giac")
[Out]