Optimal. Leaf size=53 \[ \frac{2}{3} x^{3/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{5} x^{5/2} (A c+b B)+\frac{2}{7} B c x^{7/2} \]
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Rubi [A] time = 0.0634002, antiderivative size = 53, 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 \[ \frac{2}{3} x^{3/2} (a B+A b)+2 a A \sqrt{x}+\frac{2}{5} x^{5/2} (A c+b B)+\frac{2}{7} B c x^{7/2} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + b*x + c*x^2))/Sqrt[x],x]
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Rubi in Sympy [A] time = 8.39045, size = 58, normalized size = 1.09 \[ 2 A a \sqrt{x} + \frac{2 B c x^{\frac{7}{2}}}{7} + x^{\frac{5}{2}} \left (\frac{2 A c}{5} + \frac{2 B b}{5}\right ) + x^{\frac{3}{2}} \left (\frac{2 A b}{3} + \frac{2 B a}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x+a)/x**(1/2),x)
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Mathematica [A] time = 0.0300634, size = 46, normalized size = 0.87 \[ \frac{2}{105} \sqrt{x} (35 a (3 A+B x)+x (7 A (5 b+3 c x)+3 B x (7 b+5 c x))) \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + b*x + c*x^2))/Sqrt[x],x]
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Maple [A] time = 0.006, size = 42, normalized size = 0.8 \[{\frac{30\,Bc{x}^{3}+42\,Ac{x}^{2}+42\,Bb{x}^{2}+70\,Abx+70\,aBx+210\,aA}{105}\sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x+a)/x^(1/2),x)
[Out]
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Maxima [A] time = 0.708306, size = 53, normalized size = 1. \[ \frac{2}{7} \, B c x^{\frac{7}{2}} + \frac{2}{5} \,{\left (B b + A c\right )} x^{\frac{5}{2}} + 2 \, A a \sqrt{x} + \frac{2}{3} \,{\left (B a + A b\right )} x^{\frac{3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(B*x + A)/sqrt(x),x, algorithm="maxima")
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Fricas [A] time = 0.289893, size = 53, normalized size = 1. \[ \frac{2}{105} \,{\left (15 \, B c x^{3} + 21 \,{\left (B b + A c\right )} x^{2} + 105 \, A a + 35 \,{\left (B a + A b\right )} x\right )} \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(B*x + A)/sqrt(x),x, algorithm="fricas")
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Sympy [A] time = 4.21101, size = 68, normalized size = 1.28 \[ 2 A a \sqrt{x} + \frac{2 A b x^{\frac{3}{2}}}{3} + \frac{2 A c x^{\frac{5}{2}}}{5} + \frac{2 B a x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{5}{2}}}{5} + \frac{2 B c x^{\frac{7}{2}}}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x+a)/x**(1/2),x)
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GIAC/XCAS [A] time = 0.268284, size = 58, normalized size = 1.09 \[ \frac{2}{7} \, B c x^{\frac{7}{2}} + \frac{2}{5} \, B b x^{\frac{5}{2}} + \frac{2}{5} \, A c x^{\frac{5}{2}} + \frac{2}{3} \, B a x^{\frac{3}{2}} + \frac{2}{3} \, A b x^{\frac{3}{2}} + 2 \, A a \sqrt{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)*(B*x + A)/sqrt(x),x, algorithm="giac")
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