Optimal. Leaf size=51 \[ 2 \sqrt{x} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{3} x^{3/2} (A c+b B)+\frac{2}{5} B c x^{5/2} \]
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Rubi [A] time = 0.0669664, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ 2 \sqrt{x} (a B+A b)-\frac{2 a A}{\sqrt{x}}+\frac{2}{3} x^{3/2} (A c+b B)+\frac{2}{5} B c x^{5/2} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2))/x^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 8.40682, size = 54, normalized size = 1.06 \[ - \frac{2 A a}{\sqrt{x}} + \frac{2 B c x^{\frac{5}{2}}}{5} + x^{\frac{3}{2}} \left (\frac{2 A c}{3} + \frac{2 B b}{3}\right ) + \sqrt{x} \left (2 A b + 2 B a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)/x**(3/2),x)
[Out]
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Mathematica [A] time = 0.0314233, size = 44, normalized size = 0.86 \[ \frac{2 x (5 A (3 b+c x)+B x (5 b+3 c x))-30 a (A-B x)}{15 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2))/x^(3/2),x]
[Out]
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Maple [A] time = 0.006, size = 42, normalized size = 0.8 \[ -{\frac{-6\,Bc{x}^{3}-10\,Ac{x}^{2}-10\,Bb{x}^{2}-30\,Abx-30\,aBx+30\,aA}{15}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)/x^(3/2),x)
[Out]
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Maxima [A] time = 0.715718, size = 53, normalized size = 1.04 \[ \frac{2}{5} \, B c x^{\frac{5}{2}} + \frac{2}{3} \,{\left (B b + A c\right )} x^{\frac{3}{2}} - \frac{2 \, A a}{\sqrt{x}} + 2 \,{\left (B a + A b\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(B*x + A)/x^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.265604, size = 53, normalized size = 1.04 \[ \frac{2 \,{\left (3 \, B c x^{3} + 5 \,{\left (B b + A c\right )} x^{2} - 15 \, A a + 15 \,{\left (B a + A b\right )} x\right )}}{15 \, \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(B*x + A)/x^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.28334, size = 65, normalized size = 1.27 \[ - \frac{2 A a}{\sqrt{x}} + 2 A b \sqrt{x} + \frac{2 A c x^{\frac{3}{2}}}{3} + 2 B a \sqrt{x} + \frac{2 B b x^{\frac{3}{2}}}{3} + \frac{2 B c x^{\frac{5}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)/x**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.270552, size = 58, normalized size = 1.14 \[ \frac{2}{5} \, B c x^{\frac{5}{2}} + \frac{2}{3} \, B b x^{\frac{3}{2}} + \frac{2}{3} \, A c x^{\frac{3}{2}} + 2 \, B a \sqrt{x} + 2 \, A b \sqrt{x} - \frac{2 \, A a}{\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(B*x + A)/x^(3/2),x, algorithm="giac")
[Out]